Latent Lie-Poisson Neural Networks (LLPNNs): Discovering the motion of Lie-Poisson systems through observable data and latent dynamics

# Latent Lie-Poisson Neural Networks (LLPNNs): Discovering the motion of Lie-Poisson systems through observable data and latent dynamics
Source: [https://arxiv.org/html/2607.28939](https://arxiv.org/html/2607.28939)
###### Abstract

Structure\-preserving neural networks are essential for the long\-term prediction of Hamiltonian systems from data\. Many important Hamiltonian systems in mechanics and control admit symmetry reduction to Lie–Poisson systems, including rigid bodies, underwater vehicles, fluids, plasmas, and optimal control problems\. A fundamental challenge in learning such systems is that their dynamics evolve in momentum variables that are typically unobservable, while available data consist only of observable quantities such as configurations and velocities\. In optimal control applications, the situation is further complicated because the latent variables contain unobservable co\-states \(Lagrange multipliers\) and the reduced Hamiltonian may be degenerate, preventing the existence of a corresponding Lagrangian and rendering encoder\-decoder approaches inapplicable\.

We introduce*Latent Lie–Poisson Neural Networks*\(LLPNNs\), a structure\-preserving framework for learning Lie–Poisson dynamics directly from observable data\. The proposed approach exploits three geometric ingredients: \(i\) learning either a Hamiltonian decoder or a pseudo\-Lagrangian encoder on the active variables, \(ii\) constructing latent trajectories through a universal Noether invariant arising from Lie–Poisson symmetry reduction, and \(iii\) reconstructing observable and latent dynamics through Lie–Poisson flows combined with Magnus\-based Lie\-group updates\. The resulting method preserves the underlying geometric structure and remains applicable to both regular and degenerate Hamiltonian systems\.

We demonstrate the method on three representative examples: a generalized rigid body onSO​\(3\)\\mathrm\{SO\}\(3\), Kirchhoff’s underwater vehicle onSE​\(3\)\\mathrm\{SE\}\(3\), and an optimal\-control problem for interacting vehicles onSE​\(2\)N\\mathrm\{SE\}\(2\)^\{N\}\. Numerical experiments show excellent long\-term predictive accuracy, strong robustness to noise, and competitive performance using only modest datasets and lightweight neural\-network architectures\.

###### keywords:

Structure\-preserving learning , Symmetry reduction , Lie–Poisson dynamics , Geometric deep learning , Latent dynamics , Degenerate Hamiltonians

††journal:Neural Networks\\affiliation

\[ua\]organization=Department of Mathematics, The University of Alabama, addressline=Gordon Palmer Hall, city=Tuscaloosa, state=AL, postcode=35401, country=USA

## 1Introduction and Problem Statement

### 1\.1Physics\-Informed Learning of Hamiltonian Systems

##### Previous work on learning general dynamical systems

The goal of data\-driven methods of modeling is to approximate the governing dynamics directly from observations\. Physics\-Informed Neural Networks \(PINNs\) incorporate differential equations, initial conditions, and boundary conditions into the training loss and use automatic differentiation to evaluate the corresponding residuals\[[52](https://arxiv.org/html/2607.28939#bib.bib149),[38](https://arxiv.org/html/2607.28939#bib.bib154),[14](https://arxiv.org/html/2607.28939#bib.bib144)\]\. This approach has been successfully applied to a broad range of scientific computing problems\.

Neural Ordinary Differential Equations \(Neural ODEs\)\[[9](https://arxiv.org/html/2607.28939#bib.bib130)\]represent the vector field𝐮˙=𝐟​\(𝐮,t\)\\dot\{\\mathbf\{u\}\}=\\mathbf\{f\}\(\\mathbf\{u\},t\)by a neural network and recover trajectories through numerical integration\. By training the vector field directly, Neural ODEs provide a flexible framework for learning continuous\-time latent dynamics and have inspired numerous extensions in scientific machine learning\.

A complementary paradigm is operator learning\. Deep Operator Networks \(DeepONets\)\[[45](https://arxiv.org/html/2607.28939#bib.bib132)\]approximate nonlinear operators mapping input functions to solution fields and can generalize across large families of initial conditions, forcing terms, or model parameters\. Their ability to perform rapid inference without retraining has made them attractive for many computational physics applications\.

Although these methods often achieve excellent predictive performance, they do not generally preserve the geometric structures underlying Hamiltonian and Poisson systems\. Consequently, long\-term predictions may violate conserved quantities and drift away from physically admissible trajectories\. This observation has motivated the development of geometric machine\-learning architectures that explicitly incorporate Hamiltonian, symplectic, or Poisson structure\.

A more fundamental limitation arises in many control and symmetry\-reduced mechanical systems\. In these problems the governing equations are written for latent momentum variables, whereas the available data typically consist only of observable configuration and velocity variables\. Consequently, the observable variables need not form a closed dynamical system\. In particular, for degenerate Hamiltonians arising in optimal control, there may not exist an autonomous evolution equation for the observable variables alone, as we demonstrate on explicit examples in this paper\. This hidden\-state dynamics that is governing the evolution of the observable data motivates the framework developed in this paper\.

##### Previous work in Structure\-Preserving Data\-Driven Modeling: Continuous Approach

One dominant strategy in this field aims to directly infer either the Hamiltonian function or the underlying Poisson bracket from experimental data\. For canonical systems, this paradigm was established by*Hamiltonian Neural Networks \(HNNs\)*\[[23](https://arxiv.org/html/2607.28939#bib.bib97)\], which approximateH​\(𝐪,𝐩\)H\(\\mathbf\{q\},\\mathbf\{p\}\)by training a network to satisfy the relations in \([1](https://arxiv.org/html/2607.28939#S1.E1)\)\. By embedding this geometric structure directly into the loss function, HNNs achieve significantly greater robustness and trajectory accuracy than standard black\-box neural networks\. This formulation has since been expanded to accommodate chaotic regimes and adaptive parameters\[[25](https://arxiv.org/html/2607.28939#bib.bib103)\], with formal proofs of the existence of such learned Hamiltonians detailed in\[[15](https://arxiv.org/html/2607.28939#bib.bib101)\]\.

An alternative formulation is offered by*Lagrangian Neural Networks \(LNNs\)*\[[13](https://arxiv.org/html/2607.28939#bib.bib102)\], which model the Euler\-Lagrange equations directly in coordinate\-velocity space\(𝐪,𝐪˙\)\(\\mathbf\{q\},\\dot\{\\mathbf\{q\}\}\), thereby eliminating the need for a Legendre transform to momentum space\. For non\-canonical structures, direct learning of the vector fields was investigated in\[[55](https://arxiv.org/html/2607.28939#bib.bib98)\], though rigorously enforcing the Jacobi identity remains a major open challenge\. More recently, kernel ridge regression was proposed to learn Hamiltonians of Poisson systems from noisy observations\[[31](https://arxiv.org/html/2607.28939#bib.bib148)\]\. However, a key limitation of these continuous methods is that the true Hamiltonian is only identifiable up to the addition of an arbitrary function of the Casimirs, which complicates its unique isolation using standard machine learning optimization\.

Furthermore, continuous\-time methods assume that once the vector field is learned, trajectories can be obtained via standard numerical integration\. However, simulating Hamiltonian systems over extended time horizons with generic numerical solvers typically violates fundamental conservation laws\. Preserving these geometric properties requires the use of variational integrators\[[48](https://arxiv.org/html/2607.28939#bib.bib114),[42](https://arxiv.org/html/2607.28939#bib.bib100),[24](https://arxiv.org/html/2607.28939#bib.bib99)\]\. While these specialized integrators preserve momentum\-like maps to machine precision, they are often computationally expensive compared to standard explicit solvers\.

##### Learning Hamiltonian systems through structure\-preserving discrete maps

A major research direction in geometric machine learning focuses on designing data\-driven architectures that respect the underlying mathematical structures of physical systems\. The majority of existing works address*canonical*Hamiltonian systems, where the state vector is defined on a2​n2n\-dimensional phase space as𝐮=\(𝐪,𝐩\)\\mathbf\{u\}=\(\\mathbf\{q\},\\mathbf\{p\}\), with𝐪\\mathbf\{q\}and𝐩\\mathbf\{p\}representing the generalized coordinates and momenta, respectively\. Under this formulation, the state’s time evolution is governed by a Hamiltonian functionH​\(𝐪,𝐩\)H\(\\mathbf\{q\},\\mathbf\{p\}\)through the vector field𝐮˙=𝐟​\(𝐮\)\\dot\{\\mathbf\{u\}\}=\\mathbf\{f\}\(\\mathbf\{u\}\), where the operator𝐟\\mathbf\{f\}is expressed as:

𝐟=𝕁​∇𝐮H,𝕁=\(0𝕀n−𝕀n0\),⇔𝐪˙=∂H∂𝐩,𝐩˙=−∂H∂𝐪,\\mathbf\{f\}=\\mathbb\{J\}\\nabla\_\{\\mathbf\{u\}\}H\\,,\\quad\\mathbb\{J\}=\\left\(\\begin\{array\}\[\]\{cc\}0&\\mathbb\{I\}\_\{n\}\\\\ \-\\mathbb\{I\}\_\{n\}&0\\end\{array\}\\right\)\\,,\\quad\\Leftrightarrow\\quad\\dot\{\\mathbf\{q\}\}=\\frac\{\\partial H\}\{\\partial\\mathbf\{p\}\}\\,,\\quad\\dot\{\\mathbf\{p\}\}=\-\\frac\{\\partial H\}\{\\partial\\mathbf\{q\}\}\\,,\(1\)and𝕀n\\mathbb\{I\}\_\{n\}denotes then×nn\\times nidentity matrix\. For any arbitrary smooth functionF​\(𝐪,𝐩\)F\(\\mathbf\{q\},\\mathbf\{p\}\), its evolution along a trajectory of \([1](https://arxiv.org/html/2607.28939#S1.E1)\) is dictated by the canonical Poisson bracket:

d​Fd​t=\{F,H\}=∂F∂𝐪⋅∂H∂𝐩−∂H∂𝐪⋅∂F∂𝐩\.\\frac\{dF\}\{dt\}=\\\{F,H\\\}=\\frac\{\\partial F\}\{\\partial\\mathbf\{q\}\}\\cdot\\frac\{\\partial H\}\{\\partial\\mathbf\{p\}\}\-\\frac\{\\partial H\}\{\\partial\\mathbf\{q\}\}\\cdot\\frac\{\\partial F\}\{\\partial\\mathbf\{p\}\}\.\(2\)In addition to the canonical brackets \([2](https://arxiv.org/html/2607.28939#S1.E2)\), there are also non\-canonical Poisson brackets, see Section[2\.1](https://arxiv.org/html/2607.28939#S2.SS1)for precise definition\. The dynamical systems governed by these non\-canonical brackets are known as Poisson systems\. A key feature of non\-canonical Poisson systems is the presence of a non\-trivial null space, which yields conserved quantities known as*Casimir invariants*\(or*Casimirs*\)\. These invariants are intrinsic to the structure of the Poisson bracket itself and remain conserved regardless of the specific Hamiltonian governing the system\. This work focuses on developing data\-driven computational methods for a highly important class of these non\-canonical Poisson systems\.

To overcome the challenges of continuous integration, we adopt a different perspective: learning discrete, structure\-preserving phase space transformations directly\. For canonical dynamics, Poincaré’s theorem states that the flowϕt​\(𝐮\)\\boldsymbol\{\\phi\}\_\{t\}\(\\mathbf\{u\}\)mapping initial states to their configurations at timettis a symplectic transformation\[[1](https://arxiv.org/html/2607.28939#bib.bib115),[46](https://arxiv.org/html/2607.28939#bib.bib116),[51](https://arxiv.org/html/2607.28939#bib.bib172)\]\. Mathematically, for𝐮=\(𝐪,𝐩\)\\mathbf\{u\}=\(\\mathbf\{q\},\\mathbf\{p\}\), the flow mappingϕt​\(𝐮\)\\boldsymbol\{\\phi\}\_\{t\}\(\\mathbf\{u\}\)must satisfy:

\(∂ϕ∂𝐮\)T​𝕁​\(∂ϕ∂𝐮\)=𝕁\.\\left\(\\frac\{\\partial\\boldsymbol\{\\phi\}\}\{\\partial\\mathbf\{u\}\}\\right\)^\{T\}\\mathbb\{J\}\\left\(\\frac\{\\partial\\boldsymbol\{\\phi\}\}\{\\partial\\mathbf\{u\}\}\\right\)=\\mathbb\{J\}\\,\.\(3\)Instead of learning the vector fields and subsequently integrating them, several works propose learning these symplectic maps directly from data\.

Early frameworks in this domain include*Symplectic Recurrent Neural Networks \(SRNNs\)*\[[11](https://arxiv.org/html/2607.28939#bib.bib94)\], which enforce symplecticity by embedding discretized symplectic numerical schemes into the network architecture\. This approach showed that structure preservation significantly mitigates sensitivity to observational noise\. Similarly,*Non\-Separable Symplectic Neural Networks \(NSSNNs\)*\[[62](https://arxiv.org/html/2607.28939#bib.bib104)\]were developed to handle non\-separable Hamiltonians by utilizing the explicit integration algorithms from\[[56](https://arxiv.org/html/2607.28939#bib.bib96)\]\. HNNs based on symplectic integrators were recently developed to compute the effective Hamiltonians from noisy data\[[12](https://arxiv.org/html/2607.28939#bib.bib111)\]\. From a Lagrangian perspective, discrete mapping frameworks in the velocity\-phase space\(𝐪,𝐪˙\)\(\\mathbf\{q\},\\dot\{\\mathbf\{q\}\}\)were explored in\[[54](https://arxiv.org/html/2607.28939#bib.bib192)\]and recently extended to include non\-conservative dissipative and thermodynamic systems\[[18](https://arxiv.org/html/2607.28939#bib.bib139)\]\.

Further progress in learning symplectic maps has been driven by three main architectures\[[35](https://arxiv.org/html/2607.28939#bib.bib121),[8](https://arxiv.org/html/2607.28939#bib.bib78)\]\. First,*SympNets*\[[35](https://arxiv.org/html/2607.28939#bib.bib121)\]construct symplectic maps by composing specialized, structure\-preserving neural network layers\. Second,*Generating Function Neural Networks \(GFNNs\)*\[[8](https://arxiv.org/html/2607.28939#bib.bib78)\]parameterize the discrete flow using generating functions, which provides rigorous bounds on long\-term tracking errors\. Third,*HénonNets*\[[7](https://arxiv.org/html/2607.28939#bib.bib151)\]build symplectic maps by composing Hénon\-like transformations to learn Poincaré maps\. These architectures have demonstrated exceptional performance in simulating both integrable systems \(such as pendulums and orbital mechanics\) and chaotic dynamics \(such as the three\-body problem\)\.

##### Previous work on data\-based learning Poisson systems

If the symmetry\-reduced momentaμ\\muare known \(which is not the case for the problem we consider here\), several approaches have been suggested to construct a structure\-preserving learning procedure\.

Recent work\[[19](https://arxiv.org/html/2607.28939#bib.bib140),[20](https://arxiv.org/html/2607.28939#bib.bib127)\]exploits the property that the phase flow of any Hamiltonian function constitutes a Poisson map, meaning that it preserves the Poisson bracket\[[46](https://arxiv.org/html/2607.28939#bib.bib116)\]\. These architectures consider flows generated by “test” Hamiltonians of special forms, for which the equations of motion simplify and can be solved analytically, providing the mathematical foundation for Lie–Poisson neural network layers\.

By composing multiple Poisson maps of this type, the neural network in\[[19](https://arxiv.org/html/2607.28939#bib.bib140)\]optimizes the parameters of these test Hamiltonians to match observed phase\-space trajectories\. This sequence enables the network to reconstruct the flow across the entire phase space with high accuracy, using small datasets and lightweight parameterizations\. Crucially, this construction ensures that Casimir invariants are conserved to machine precision, independent of the training data or optimization quality\. This work was recently extended to optimal\-control problems for interacting autonomous vehicles\[[32](https://arxiv.org/html/2607.28939#bib.bib129)\], where Lie–Poisson transformations were generalized to incorporate individual neural networks depending on the symmetry\-reduced momenta\. This structure enabled a completeness result for networks built from Lie–Poisson transformations\.

These previous works highlight the value of exploiting the exact Lie–Poisson structure rather than treating the dynamics as an unconstrained vector\-field learning problem\. In the present work, we adopt the symmetry group and its Lie–Poisson bracket as known geometric data, while learning the physical or control Hamiltonian from observations\. This separation is natural in many physical and control systems, where the symmetry structure is fixed by the configuration space, whereas the Hamiltonian may be unknown or only approximately modeled\. For a related discussion in the context of physical systems, see\[[57](https://arxiv.org/html/2607.28939#bib.bib170)\]\.

For non\-canonical Poisson systems, the SympNet framework was extended under the name*Poisson Neural Networks \(PNNs\)*\[[34](https://arxiv.org/html/2607.28939#bib.bib137)\]\. By leveraging the Lie\-Darboux theorem, PNNs locally map non\-canonical coordinates to canonical ones under the assumption of a constant\-rank Poisson structure\. However, this strategy faces two major limitations: the Lie\-Darboux transformation is typically local rather than global, and PNNs do not guarantee exact preservation of Casimir invariants\. Because preserving Casimirs is vital for the statistical validity of long\-term ensemble simulations\[[16](https://arxiv.org/html/2607.28939#bib.bib167)\], there is a strong demand for methods that enforce Casimir conservation to machine precision\.

An alternative formulation presented in\[[59](https://arxiv.org/html/2607.28939#bib.bib185),[58](https://arxiv.org/html/2607.28939#bib.bib147)\]constructs maps that exactly conserve momentum maps and Casimirs by solving approximate Hamilton–Jacobi equations for symmetry\-reduced systems\. However, solving Hamilton–Jacobi equations for high\-dimensional, complex systems is often computationally and analytically intractable\. More recently,*LocSympNets*and*SymLocSympNets*\[[2](https://arxiv.org/html/2607.28939#bib.bib95)\]were introduced, modeling flows by composing volume\-preserving mappings\. While these models have proven highly effective for simulating rigid\-body dynamics, linear advection, and charged particles in magnetic fields, their theoretical completeness remains an open question\.

In a related study,\[[49](https://arxiv.org/html/2607.28939#bib.bib122)\]considered the use of purely visual streams to model 3D rotational dynamics by projecting images onto anS​O​\(3\)SO\(3\)orientation manifold while simultaneously learning the moment\-of\-inertia tensor\. While their technical framework is different from ours, their work highlights the feasibility of learning underlying latent dynamics purely from observable pixel data without relying on full state measurements including the latent variables\.

Another recent work\[[31](https://arxiv.org/html/2607.28939#bib.bib148)\]addressed learning of a general Poisson system using a structure\-preserving kernel ridge regression\. The method allowed to find Hamilto\- nian functions on general Poisson manifolds out of datasets made of noisy observations of Hamiltonian vector fields\.

It is important to note that all these works assumed that the information about the flow on the momentum \(latent\) manifold is known\. In contrast, we do not assume that knowledge as in practical realizations, especially in control systems, the latent variables are hidden, and only the information about the observable variables is available\. This observation motivates our development of the*Latent Lie–Poisson Neural Network \(LLPNN\)*, a framework that learns structure\-preserving Lie–Poisson dynamics directly from observable variables while reconstructing the hidden momentum dynamics\.

##### Optimal Control via Neural Network Architectures and Symmetry Reduction to Lie\-Poisson Systems

Let us also briefly describe the efforts of using neural networks for optimal control problems\. In addition to the paper\[[32](https://arxiv.org/html/2607.28939#bib.bib129)\]we have mentioned earlier, several works considered the optimal control problem using neural networks\. These models are mainly focused on finding the solutions to the optimal control problems, using neural networks as an alternative way to finding solutions to the equations of optimal control\. Neural networks have also been applied extensively to optimal control, although the focus of these works \(planning\) is different from the goal of this paper \(learning from observations\)\. In particular,\[[17](https://arxiv.org/html/2607.28939#bib.bib248)\]incorporated the state and co\-state equations from Pontryagin’s Minimum \(or Maximum\) Principle \(PMP\)\[[22](https://arxiv.org/html/2607.28939#bib.bib8)\]as soft constraints within a loss function, utilizing a formulation that conceptually anticipated Physics\-Informed Neural Networks \(PINNs\)\. To ensure control tractability,\[[10](https://arxiv.org/html/2607.28939#bib.bib251)\]deployed input\-convex neural networks\. Neural network controllers have also been applied to practical engineering challenges, including robotic manipulation\[[39](https://arxiv.org/html/2607.28939#bib.bib250)\]and autonomous aeronautical landing maneuvers\[[53](https://arxiv.org/html/2607.28939#bib.bib249)\]\. For a comprehensive overview of machine learning paradigms in this domain, we refer the reader to the review by\[[60](https://arxiv.org/html/2607.28939#bib.bib252)\]\. Despite these notable advancements, existing methodologies generally overlook structure\-preserving network designs, thereby failing to exploit the intrinsic geometric structure of Pontryagin’s equations\. Enforcing this underlying mathematical structure within the network architecture is essential for achieving accurate, long\-term dynamical predictions\[[35](https://arxiv.org/html/2607.28939#bib.bib121),[19](https://arxiv.org/html/2607.28939#bib.bib140)\], which motivates the framework introduced in this work\. However, the most important feature of the optimal control is that the observable variables are different from the dynamics latent variables carrying the dynamics, which makes direct machine learning challenging\. Finding the latent dynamics from observable data is the main point of this paper\.

### 1\.2Novelty of this paper

Our paper is different from previous work on the subject as it presents the following novel results:

1. 1\.We design a structure\-preserving method \(LLPNN\) for learning the system based on the observable variables only,
2. 2\.Our method guarantees conservation of Casimirs with machine precision in the latent space, even though it only uses the data from the observable space\.
3. 3\.LLPNNs are applicable to both regular and degenerate Lie–Poisson Hamiltonian systems, including optimal\-control problems for which a corresponding Lagrangian formulation in terms of observables may not exist\.

## 2Mathematical Foundations of the Theory

### 2\.1A brief introduction to Lie–Poisson systems

To develop our framework, we draw upon general properties of Lie–Poisson systems\. Here, we present a brief summary; for a complete description, see[B](https://arxiv.org/html/2607.28939#A2), as well as\[[26](https://arxiv.org/html/2607.28939#bib.bib119),[46](https://arxiv.org/html/2607.28939#bib.bib116)\]for detailed expositions\.

Lie–Poisson systems are a special class of general Poisson systems\. They arise naturally through symmetry reduction with respect to the action of a Lie group, where the resulting Poisson bracket inherits its structure from the underlying Lie algebra\.

A general Poisson manifold consists of a smooth manifoldℳ\\mathcal\{M\}equipped with a bracket\{⋅,⋅\}\\\{\\cdot,\\cdot\\\}that maps a pair of smooth functionsF,G∈C∞​\(ℳ\)F,G\\in C^\{\\infty\}\(\\mathcal\{M\}\)to another smooth function\{F,G\}∈C∞​\(ℳ\)\\\{F,G\\\}\\in C^\{\\infty\}\(\\mathcal\{M\}\)\. To constitute a valid Poisson bracket, it must satisfy the following properties for allF,G,H∈C∞​\(ℳ\)F,G,H\\in C^\{\\infty\}\(\\mathcal\{M\}\):

1. 1\.*Antisymmetry:*\{F,G\}=−\{G,F\}\\\{F,G\\\}=\-\\\{G,F\\\},
2. 2\.*Bilinearity:*\{a​F\+b​G,H\}=a​\{F,H\}\+b​\{G,H\}\\\{aF\+bG,H\\\}=a\\\{F,H\\\}\+b\\\{G,H\\\}for all real constantsa,b∈ℝa,b\\in\\mathbb\{R\},
3. 3\.*Leibniz rule*:\{F​G,H\}=G​\{F,H\}\+F​\{G,H\}\\\{FG,H\\\}=G\\\{F,H\\\}\+F\\\{G,H\\\},
4. 4\.*Jacobi identity*:\{\{F,G\},H\}\+\{\{H,F\},G\}\+\{\{G,H\},F\}=0\\\{\\\{F,G\\\},H\\\}\+\\\{\\\{H,F\\\},G\\\}\+\\\{\\\{G,H\\\},F\\\}=0\.

For finite\-dimensional systems, Poisson brackets can be expressed in terms of a Poisson tensor matrix𝔹​\(μ\)\\mathbb\{B\}\(\\mu\)\[[46](https://arxiv.org/html/2607.28939#bib.bib116)\]:

\{F,G\}=\(∂F∂μ\)T​𝔹​\(μ\)​∂G∂μ,\\\{F,G\\\}=\\left\(\\frac\{\\partial F\}\{\\partial\\mu\}\\right\)^\{T\}\\mathbb\{B\}\(\\mu\)\\frac\{\\partial G\}\{\\partial\\mu\}\\,,\(4\)where𝔹​\(μ\)\\mathbb\{B\}\(\\mu\)is an antisymmetric matrix satisfying a system of non\-linear first\-order partial differential equations required by the Jacobi identity\. Because these PDE constraints are generally non\-trivial to solve analytically in closed form, one typically verifies whether a given matrix defines a valid Poisson structure rather than deriving a general parameterization\.

Lie–Poisson systems emerge when a dynamical system exhibits symmetry under a Lie group action\. Suppose the configuration manifold of the system is a Lie groupGG, and the Lagrangian or Hamiltonian is invariant with respect to the action ofGG\. For instance, rigid body motion is modeled on the special orthogonal groupG=SO​\(3\)G=\\mathrm\{SO\}\(3\), where rotational invariance holds with respect to the same group\. Just as unreduced canonical systems admit both Lagrangian and Hamiltonian formulations, symmetry\-reduced systems preserve this duality\. The standard \(unreduced\) Lagrangian description depends on configuration coordinates and velocities\(g,g˙\)∈T​G\(g,\\dot\{g\}\)\\in TG, whereas the canonical Hamiltonian description uses coordinates and momenta\(g,p\)∈T∗​G\(g,p\)\\in T^\{\*\}G\.

The symmetry\-reduced Lagrangian formulation is expressed in terms of Lie algebra\-valued velocitiesξ∈𝔤\\xi\\in\\mathfrak\{g\}, obtained by trivializing the tangent bundle\. For matrix Lie groups considered here, these velocities are defined through one of the two standard trivializations,

g˙=g​ξorg˙=ξ​g\.\\dot\{g\}=g\\xi\\qquad\\text\{or\}\\qquad\\dot\{g\}=\\xi g\.\(5\)The first convention corresponds to the body\-frame velocity \(left trivialization\),ξ=g−1​g˙\\xi=g^\{\-1\}\\dot\{g\}, while the second corresponds to the spatial velocity \(right trivialization\),ξ=g˙​g−1\\xi=\\dot\{g\}g^\{\-1\}\. The sign in the reduced equations depends on whether the Lagrangian is left\- or right\-invariant\. For a reduced Lagrangianℓ​\(ξ\)\\ell\(\\xi\), the symmetry\-reduced equations of motion are given by the*Euler–Poincaré equations*\[[26](https://arxiv.org/html/2607.28939#bib.bib119),[46](https://arxiv.org/html/2607.28939#bib.bib116)\]:

dd​t​∂ℓ∂ξ∓adξ∗⁡∂ℓ∂ξ=0,\\frac\{d\}\{dt\}\\frac\{\\partial\\ell\}\{\\partial\\xi\}\\mp\\operatorname\{ad\}^\{\*\}\_\{\\xi\}\\frac\{\\partial\\ell\}\{\\partial\\xi\}=0,\(6\)wheread∗\\operatorname\{ad\}^\{\*\}denotes the coadjoint operator for the Lie algebra𝔤\\mathfrak\{g\}, and the minus \(*resp\.*, plus\) sign corresponds to left \(*resp\.*, right\) invariance\.

In Lie–Poisson dynamics, we define the symmetry\-reduced momentumμ=g−1​p\\mu=g^\{\-1\}p\(for left\-invariant systems\) orμ=p​g−1\\mu=pg^\{\-1\}\(for right\-invariant systems\), whereμ∈𝔤∗\\mu\\in\\mathfrak\{g\}^\{\*\}belongs to the dual Lie algebra\. For example, satellite attitude dynamics are formulated onSO​\(3\)\\mathrm\{SO\}\(3\), whereμ∈𝔰​𝔬​\(3\)∗\\mu\\in\\mathfrak\{so\}\(3\)^\{\*\}is represented as a three\-dimensional vector\. The intrinsic equations of motion and Poisson bracket forμ\\muare given by:

dd​t​μ∓ad∂h∂μ∗⁡μ=0,\\frac\{d\}\{dt\}\\mu\\mp\\operatorname\{ad\}^\{\*\}\_\{\\frac\{\\partial h\}\{\\partial\\mu\}\}\\mu=0\\,,\(7\)where we focus primarily on left\-invariant systems \(right\-invariant systems are treated analogously\)\. Equation \([7](https://arxiv.org/html/2607.28939#S2.E7)\) can be expressed using the Lie–Poisson bracket:

μ˙=\{μ,h\}⇔μ˙=Λ​\(μ\)​∇h​\(μ\),\{F,G\}=∓⟨μ,\[∂F∂μ,∂G∂μ\]⟩,\\dot\{\\mu\}=\\\{\\mu,h\\\}\\quad\\Leftrightarrow\\quad\\dot\{\\mu\}=\\Lambda\(\\mu\)\\nabla h\(\\mu\)\\,,\\quad\\\{F,G\\\}=\\mp\\left\\langle\\mu,\\left\[\\frac\{\\partial F\}\{\\partial\\mu\},\\frac\{\\partial G\}\{\\partial\\mu\}\\right\]\\right\\rangle,\(8\)where\[⋅,⋅\]\[\\cdot,\\cdot\]denotes the Lie algebra commutator\. Again, the−\-sign in the Poisson bracket \([8](https://arxiv.org/html/2607.28939#S2.E8)\) corresponds to left\-invariant systems, while the\+\+sign corresponds to right\-invariant systems\. Here, the Poisson tensorΛ​\(μ\)\\Lambda\(\\mu\)depends linearly onμ\\mu, with its structural form determined entirely by the structure constants of the Lie algebra\. These brackets often admit Casimir invariants,*i\.e\.*, functionsC​\(μ\)C\(\\mu\)satisfying\{C,h\}=0\\\{C,h\\\}=0for any Hamiltonianhh, which play a central role in our structure\-preserving approach\.

Optimal control problems with symmetry naturally inherit a Lie–Poisson structure\[[40](https://arxiv.org/html/2607.28939#bib.bib20),[4](https://arxiv.org/html/2607.28939#bib.bib117)\], including multi\-agent formation control systems\[[36](https://arxiv.org/html/2607.28939#bib.bib15),[37](https://arxiv.org/html/2607.28939#bib.bib17)\]\. Notably, symmetry reduction of controlled Hamiltonian systems yields a complete Lie–Poisson reduction onto the direct product of individual Lie algebras serving as reduced phase spaces\. While symmetry reduction in multi\-body systems is frequently incomplete, requiring relative Lie group configurations alongside algebra\-level dynamics\[[21](https://arxiv.org/html/2607.28939#bib.bib55)\], this complexity is avoided in Lie–Poisson optimal control due to the specific algebraic structure of control Hamiltonians\[[36](https://arxiv.org/html/2607.28939#bib.bib15),[37](https://arxiv.org/html/2607.28939#bib.bib17)\]\.

For classical mechanical systems, Lagrangian and Hamiltonian formulations are*mathematically*equivalent via the Legendre transform\. However, they differ fundamentally regarding data accessibility, as illustrated in Figure[1](https://arxiv.org/html/2607.28939#S2.F1)\. Positions, orientations, and group configurationsg​\(t\)g\(t\)are directly observable from measurements\. Ifg​\(t\)g\(t\)is observed, its rate of changeg˙​\(t\)\\dot\{g\}\(t\)and the reduced velocityξ​\(t\)\\xi\(t\)defined by \([5](https://arxiv.org/html/2607.28939#S2.E5)\) are likewise observable\.

In contrast, the latent momentumμ\\mucannot be measured directly without prior knowledge of the system’s underlying functional formℓ​\(ξ\)\\ell\(\\xi\)orh​\(μ\)h\(\\mu\)\. Thus, empirical simulations of observable states naturally operate in velocity spaceξ\\xi, whereas structure\-preserving dynamics are intrinsically formulated in Hamiltonian spaceμ\\mu, where the Poisson/symplectic geometric structure resides\[[26](https://arxiv.org/html/2607.28939#bib.bib119),[46](https://arxiv.org/html/2607.28939#bib.bib116),[1](https://arxiv.org/html/2607.28939#bib.bib115)\]\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x1.png)Figure 1:Illustration of the structural relationship between observable configuration/velocity variables and unobservable latent momentum variables\.In principle, when the system is non\-degenerate, one can use the reduced Lagrangian and Hamiltonian to define encoder and decoder mappings between observable and latent spaces:

μ=μ​\(ξ\)=∂ℓ∂ξ\(encoder\),ξ=ξ​\(μ\)=∂h∂μ\(decoder\)\.\\mu=\\mu\(\\xi\)=\\frac\{\\partial\\ell\}\{\\partial\\xi\}\\quad\\text\{\(encoder\)\},\\qquad\\xi=\\xi\(\\mu\)=\\frac\{\\partial h\}\{\\partial\\mu\}\\quad\\text\{\(decoder\)\}\\,\.\(9\)This allows standard latent space architectures to learn bi\-directional mappings between observables and latent variables before applying structure\-preserving integrators\. While this encoder\-decoder paradigm \([9](https://arxiv.org/html/2607.28939#S2.E9)\) works well for regular mechanical systems, it frequently fails for optimal control systems\.

Although symmetry\-reduced optimal control problems reduce to Lie–Poisson equations of the form \([8](https://arxiv.org/html/2607.28939#S2.E8)\), they differ in a crucial aspect: optimal control problems frequently yield reduced Hamiltoniansh​\(μ\)h\(\\mu\)that are*degenerate*—specifically, linear in components ofμ\\muassociated with drift controls\. For example, in the ground\-vehicle model studied in Section[4\.4](https://arxiv.org/html/2607.28939#S4.SS4), vehicles drive forward at constant speed, cannot slide laterally, and control their turning rate\. Consequently,h​\(μ\)h\(\\mu\)is linear in the forward\-drive momentum, independent of the lateral momentum, and non\-linear only in the angular momentum component\. For the corresponding velocity components,ξi=const\\xi\_\{i\}=\\mathrm\{const\}\. The Hessian∇2h​\(μ\)\\nabla^\{2\}h\(\\mu\)is therefore singular, rendering the Legendre transform non\-invertible and preventing the existence of a corresponding reduced Lagrangianℓ​\(ξ\)\\ell\(\\xi\)\. Under such degeneracy, only the decoderξ=ξ​\(μ\)\\xi=\\xi\(\\mu\)exists; the encoderμ=μ​\(ξ\)\\mu=\\mu\(\\xi\)does not\. That is, no functionℓ​\(ξ\)\\ell\(\\xi\)exists such that the Euler–Poincaré equations \([6](https://arxiv.org/html/2607.28939#S2.E6)\) close autonomously inξ\\xi\-space\. In contrast, the Lie–Poisson equations \([7](https://arxiv.org/html/2607.28939#S2.E7)\) remain well\-defined regardless of Hamiltonian degeneracy\.

One might consider regularizing a degenerate Hamiltonian by adding arbitrary functions of Casimir invariants, which do not alter momentum\-space dynamics\. For instance, in rigid body mechanics where𝝁∈ℝ3\\boldsymbol\{\\mu\}\\in\\mathbb\{R\}^\{3\}, Casimirs are functions of‖𝝁‖\\\|\\boldsymbol\{\\mu\}\\\|\. Adding a quadratic Casimir termκ​‖𝝁‖2\\kappa\\\|\\boldsymbol\{\\mu\}\\\|^\{2\}\(κ\>0\\kappa\>0\) can render the Hessian non\-singular, locally restoring an invertible Lagrangianℓ​\(ξ\)\\ell\(\\xi\)\. However, this regularization alters the reconstructed velocity fieldξ\\xifor a given momentumμ\\mu\. Because data\-driven models must fit observed velocity trajectories, modifying the Hamiltonian with Casimir terms to force invertibility is non\-viable\.

This lack of an invertible mapping between observables and latent states represents a key challenge for structure\-preserving machine learning\. Moreover, Section[2\.2](https://arxiv.org/html/2607.28939#S2.SS2)demonstrates that for some degenerate Hamiltonians that are of practical importance for control theory, the evolution in observable space cannot even be expressed as an autonomous differential equation\.

To overcome this, our framework demonstrates that latent Lie–Poisson dynamics can be reconstructed and predicted solely from observable configuration and velocity trajectories\. By integrating geometric mechanics with structure\-preserving deep learning, we propose*Latent Lie\-Poisson Neural Networks \(LLPNNs\)*\.

A foundational property of Lie–Poisson systems leveraged in our architecture is the existence of a conserved spatial momentum\. By Noether’s theorem, left\-invariance guarantees that the spatial momentump0∈𝔤∗p\_\{0\}\\in\\mathfrak\{g\}^\{\*\}is conserved for all time\. This yields the coadjoint reconstruction relationship:

dd​t​\(Adg​\(t\)−1∗⁡μ​\(t\)\)=0⟹μ​\(t\)=Adg​\(t\)∗⁡p0,\\frac\{d\}\{dt\}\\left\(\\operatorname\{Ad\}^\{\*\}\_\{g\(t\)^\{\-1\}\}\\mu\(t\)\\right\)=0\\implies\\mu\(t\)=\\operatorname\{Ad\}^\{\*\}\_\{g\(t\)\}p\_\{0\}\\,,\(10\)whereAdg∗\\operatorname\{Ad\}^\{\*\}\_\{g\}denotes the coadjoint action ofGGon𝔤∗\\mathfrak\{g\}^\{\*\}\(see Definition \([73](https://arxiv.org/html/2607.28939#A1.E73)\)\)\. Equation \([10](https://arxiv.org/html/2607.28939#S2.E10)\) provides an exact, structure\-preserving bridge: it maps the unobservable, time\-varying body momentumμ​\(t\)\\mu\(t\)to a constant spatial momentum vectorp0p\_\{0\}using only the observed configuration trajectoryg​\(t\)g\(t\)\.

### 2\.2The fundamental difference between evolution in latent and observable variables

For classical, physically based mechanical systems, one assumes that the Hessian of the Hamiltonian functionh​\(μ\)h\(\\mu\)is non\-degenerate\. Under this condition, the \(inverse\) Legendre transform

ℓ​\(ξ\)=⟨ξ,μ⟩−h​\(μ\),ξ=∂h∂μ↔μ=μ​\(ξ\)\\ell\(\\xi\)=\\langle\\xi,\\mu\\rangle\-h\(\\mu\),\\quad\\xi=\\frac\{\\partial h\}\{\\partial\\mu\}\\leftrightarrow\\mu=\\mu\(\\xi\)\(12\)maps the Hamiltonian representationh​\(μ\)h\(\\mu\)to the Lagrangian representationℓ​\(ξ\)\\ell\(\\xi\)\. The Euler–Poincaré equations \([6](https://arxiv.org/html/2607.28939#S2.E6)\) then constitute a well\-defined dynamical system in the observable space\.

##### Degenerate Hamiltonians and the absence of closed observable dynamics

The situation is substantially more difficult whenh​\(μ\)h\(\\mu\)is degenerate\. The failure of a Lagrangian formulation for degenerate Hamiltonians is only part of the challenge; in some cases, the observable variables do not even admit a closed autonomous dynamical system\.

As an example, consider the Lie–Poisson equations on the dual Lie algebra𝔰​𝔢​\(2\)∗\\mathfrak\{se\}\(2\)^\{\*\}with Hamiltonian\[[36](https://arxiv.org/html/2607.28939#bib.bib15),[37](https://arxiv.org/html/2607.28939#bib.bib17)\]

h​\(μ\)=μ1\+12​μ32\.h\(\\mu\)=\\mu\_\{1\}\+\\frac\{1\}\{2\}\\mu\_\{3\}^\{2\}\.\(13\)This control Hamiltonian describes the motion of an autonomous vehicle on a plane: motion occurs at constant speed along the main body axis \(corresponding toμ1\\mu\_\{1\}\), no motion is allowed along the transverse axis \(corresponding to the absence ofμ2\\mu\_\{2\}\), and control is exerted via the rotation angle \(expressed byμ3\\mu\_\{3\}\)\. We will study this multi\-vehicle system in substantially more detail in Section[4\.4](https://arxiv.org/html/2607.28939#S4.SS4)\.

The Lie–Poisson equations \([8](https://arxiv.org/html/2607.28939#S2.E8)\) on𝔰​𝔢​\(2\)∗\\mathfrak\{se\}\(2\)^\{\*\}for the Hamiltonian \([13](https://arxiv.org/html/2607.28939#S2.E13)\) take the form

μ˙1=−μ2​μ3,μ˙2=μ1​μ3,μ˙3=μ2\.\\dot\{\\mu\}\_\{1\}=\-\\mu\_\{2\}\\mu\_\{3\},\\qquad\\dot\{\\mu\}\_\{2\}=\\mu\_\{1\}\\mu\_\{3\},\\qquad\\dot\{\\mu\}\_\{3\}=\\mu\_\{2\}\.\(14\)These equations are well defined; in fact, \([8](https://arxiv.org/html/2607.28939#S2.E8)\) definesμ˙=𝐟​\(μ\)\\dot\{\\mu\}=\\mathbf\{f\}\(\\mu\)for any time\-independent Hamiltonianh​\(μ\)h\(\\mu\)\. Moreover, the solutions forμ\\muare readily mapped into the observable domain via

ξ=∂h∂μ=\(10μ3\),\\xi=\\frac\{\\partial h\}\{\\partial\\mu\}=\\begin\{pmatrix\}1\\\\ 0\\\\ \\mu\_\{3\}\\end\{pmatrix\},\(15\)reflecting constant velocity along the forward axis, zero motion along the transverse axis, and control over the rotation angle\. This mapping exists for every Hamiltonian, whether degenerate or not\. However, we can readily prove that for the degenerate Hamiltonian \([13](https://arxiv.org/html/2607.28939#S2.E13)\), no autonomous differential equation exists forξ\\xi, as illustrated by the following lemma\.

###### Lemma 2\.2

For the Hamiltonian \([13](https://arxiv.org/html/2607.28939#S2.E13)\), there is no well\-defined autonomous first\-order system

ξ˙=𝐠​\(ξ\)\\dot\{\\xi\}=\\mathbf\{g\}\(\\xi\)describing the evolution in the purely observable velocity space\.

Proof\.Sinceξ3=μ3\\xi\_\{3\}=\\mu\_\{3\}, we obtain

ξ˙3=μ˙3=μ2\.\\dot\{\\xi\}\_\{3\}=\\dot\{\\mu\}\_\{3\}=\\mu\_\{2\}\.\(16\)However, the value ofμ2\\mu\_\{2\}cannot be recovered from the observable variablesξ=\(1,0,μ3\)\\xi=\(1,0,\\mu\_\{3\}\)\. Consequently, two different momentum states may produce identical observable velocities while possessing different time derivatives\. For example,

\(μ1,μ2,μ3\)=\(1,0,2\)and\(μ1,μ2,μ3\)=\(1,1,2\)\(\\mu\_\{1\},\\mu\_\{2\},\\mu\_\{3\}\)=\(1,0,2\)\\quad\\text\{and\}\\quad\(\\mu\_\{1\},\\mu\_\{2\},\\mu\_\{3\}\)=\(1,1,2\)\(17\)both yield

ξ=\(1,0,2\),\\xi=\(1,0,2\),\(18\)but lead to

ξ˙3=0andξ˙3=1,\\dot\{\\xi\}\_\{3\}=0\\quad\\text\{and\}\\quad\\dot\{\\xi\}\_\{3\}=1,\(19\)respectively\. Therefore, there does not exist a first\-order autonomous systemξ˙=𝐠​\(ξ\)\\dot\{\\xi\}=\\mathbf\{g\}\(\\xi\)describing the evolution of the observable variables alone\.■\\blacksquare

##### Consequences of non\-existence of a unique vector field in observable space

While the result of Lemma[2\.2](https://arxiv.org/html/2607.28939#S2.Thmtheorem2)may seem surprising, it stems from the fundamental fact that observable velocity is not a state variable, but merely an output of the latent momentum dynamics\. This distinction is critical for control problems involving degenerate Hamiltonians\. Standard data\-driven methods, such as Neural ODEs or DeepONets, attempt to learn the vector field directly in terms of observable variables\. However, to close the dynamics inξ\\xiand resolve state ambiguity, one must incorporate additional structural knowledge—such as restricting motion to a specific symplectic leaf defined by a fixed Casimir value\. Consequently, while Neural ODEs and DeepONets perform reasonably well when the Hamiltonian is non\-degenerate, as illustrated by our benchmark examples of generalized rigid body dynamics in Section[4\.2](https://arxiv.org/html/2607.28939#S4.SS2)and the generalized Kirchhoff underwater vehicle problem in Section[4\.3](https://arxiv.org/html/2607.28939#S4.SS3), they fail rapidly when the Hamiltonian is degenerate, as demonstrated for the autonomous vehicles studied in Section[4\.4](https://arxiv.org/html/2607.28939#S4.SS4)\. In contrast, the LLPNN framework developed in this work reconstructs the latent Lie–Poisson state, where dynamics obeys a well\-defined differential equation regardless of Hamiltonian degeneracy, relying solely on observable data and intrinsic geometric properties of Lie–Poisson systems\.

### 2\.3Observable Data, System Input and Problem Formulation

To design a neural network for learning latent dynamics, we must account for the fact that, in general, no invertible mapping exists between observable states \(positions and velocities\) and latent momenta\. Consequently, we must infer the functional form of the Hamiltonianh​\(μ\)h\(\\mu\)directly from observable measurements of group elements and velocities\. More precisely, we assume access to a dataset comprisingMMshort trajectories\. For each trajectoryj=1,…,Mj=1,\\dots,M, we observe configuration group elements\(g1j,…,gNj\)\(g\_\{1\}^\{j\},\\dots,g\_\{N\}^\{j\}\), alongside their corresponding Lie algebra\-valued velocities\(ξ1j,…,ξNj\)\(\\xi\_\{1\}^\{j\},\\dots,\\xi\_\{N\}^\{j\}\)acrossNNdiscrete time instances\.

For each trajectory, we treat this dataset as two separate collections: one containing configuration statesgi∈Gg\_\{i\}\\in G, and another containing Lie algebra\-valued velocitiesξi∈𝔤\\xi\_\{i\}\\in\\mathfrak\{g\}\. In practice, velocities can often be reconstructed directly from group elements—for instance, the Cayley transform provides a map from the special orthogonal groupSO​\(3\)\\mathrm\{SO\}\(3\)to its Lie algebra𝔰​𝔬​\(3\)\\mathfrak\{so\}\(3\)\[[33](https://arxiv.org/html/2607.28939#bib.bib118)\], while matrix logarithms can be employed for general Lie groups\. In such cases, velocity errors stem from both deterministic discretization errors and stochastic measurement noise in the configuration states\. Alternatively, velocity dataξ\\ximay be acquired independently of orientation data, such as via onboard telemetry from gyroscopes\. Thus, in what follows, we assume that both group element measurementsgj​\(ti\)≃gijg^\{j\}\(t\_\{i\}\)\\simeq g\_\{i\}^\{j\}and velocity measurementsξj​\(ti\)≃ξij\\xi^\{j\}\(t\_\{i\}\)\\simeq\\xi\_\{i\}^\{j\}are available fori=1,…,Ni=1,\\dots,Nandj=1,…,Mj=1,\\dots,M\. When analyzing noise effects, we assume the error distributions in these two datasets to be independent\.

Given these available measurements, the core problem addressed in this work is formulated as follows\.

##### Problem Formulation

Consider a dataset\(g1j,…,gNj\)\(g\_\{1\}^\{j\},\\dots,g\_\{N\}^\{j\}\)of configuration group elementsgij∈Gg\_\{i\}^\{j\}\\in Gand corresponding symmetry\-reduced velocitiesξ1j,…,ξNj∈𝔤\\xi\_\{1\}^\{j\},\\dots,\\xi\_\{N\}^\{j\}\\in\\mathfrak\{g\}sampled acrossNNdiscrete time points forMMtrajectories\. Assume that the dynamics in the latent spaceμ∈𝔤∗\\mu\\in\\mathfrak\{g\}^\{\*\}is governed by the Lie–Poisson equations \([8](https://arxiv.org/html/2607.28939#S2.E8)\) with a known Lie–Poisson matrixΛ​\(μ\)\\Lambda\(\\mu\)but an unknown Hamiltonian functionH​\(μ\)H\(\\mu\)\. Our goal is to learn this underlying dynamics and reconstruct both the observable trajectories\(ξ,g\)\(\\xi,g\)and the latent momentum trajectoriesμ\\mu\.

## 3Latent Lie\-Poisson Neural Networks: two approaches

### 3\.1LLPNNs: The decoder \(Hamiltonian\) approach

#### 3\.1\.1Mathematical background

In this first approach, we formulate the learning method for the Hamiltonian functionh​\(μ\)h\(\\mu\)based on Noether’s theorem\. The primary advantage of this procedure is that it operates directly in the dynamical space of the momentaμ\\mu\. Because we only learn the decoderh​\(μ\)h\(\\mu\), we never need to deduce a corresponding encoderℓ​\(ξ\)\\ell\(\\xi\), thus bypassing the need to consider whether a Lagrangian formulation exists\.

We begin by identifying degenerate directions, which yield constant velocities\. These degenerate directions correspond to components ofξ=∂h∂μ\\xi=\\frac\{\\partial h\}\{\\partial\\mu\}that are either identically zero or constant\. Identifying these directions directly from data is a natural task in systems with time\-independent Hamiltonians\. We formalize this distinction as follows:

###### Definition 3\.1

Suppose coordinate componentkkexhibits a constant velocity, i\.e\.,ξk=∂h∂μk=Ck\\xi\_\{k\}=\\frac\{\\partial h\}\{\\partial\\mu\_\{k\}\}=C\_\{k\}\(which may be zero\) for allμ\\mu\. We define such coordinates as*passive*, while all remaining coordinates are defined as*active*\.

A standard method for discovering constant linear combinations of functions from data is Singular Value Decomposition \(SVD\)\[[6](https://arxiv.org/html/2607.28939#bib.bib133)\]\. If a linear combination of coordinates remains constant during evolution, we can perform a change of variables to align with these new coordinates, placing the passive coordinates at the end of the coordinate vector\. In practice, degenerate coordinates can often be identified directly; for instance, in the ground vehicle model discussed in Section[4\.4](https://arxiv.org/html/2607.28939#S4.SS4), they are readily observed by computing the body\-frame angular velocity from orientation measurements\.

Because the Hamiltonian depends linearly on passive variables, the neural network parameterizing the Hamiltonian needs to depend only on the active variables\. Suppose the passive components are defined byξkpass=ak∈ℝ\\xi\_\{k\}^\{\\text\{pass\}\}=a\_\{k\}\\in\\mathbb\{R\}, which are directly observed from data\. We posit the following structure for the Hamiltonian:

hN​N​\(μ\)=∑kak​μkpass\+FN​N​\(μact\)\.h\_\{NN\}\(\\mu\)=\\sum\_\{k\}a\_\{k\}\\mu\_\{k\}^\{\\text\{pass\}\}\+F\_\{NN\}\(\\mu^\{\\text\{act\}\}\)\\,\.\(20\)Learning the Hamiltonian in \([20](https://arxiv.org/html/2607.28939#S3.E20)\) proceeds in three steps, while dynamic trajectory reconstruction requires an additional step utilizing a Magnus expansion combined with Noether’s theorem\.

##### Step 1\.

Given observable configuration data\(g1j,…,gNj\)\(g\_\{1\}^\{j\},\\ldots,g\_\{N\}^\{j\}\)for trajectoryj∈\{1,…,M\}j\\in\\\{1,\\ldots,M\\\}, conservation law \([10](https://arxiv.org/html/2607.28939#S2.E10)\) guarantees the existence of a trajectory\-specific constant vectorp0jp\_\{0\}^\{j\}such that the latent momentum trajectory satisfies:

μij=Adgij∗⁡p0j\.\\mu\_\{i\}^\{j\}=\\operatorname\{Ad\}^\{\*\}\_\{g\_\{i\}^\{j\}\}p\_\{0\}^\{j\}\\,\.\(21\)We therefore assign a constant vectorp0jp\_\{0\}^\{j\}to each trajectoryj=1,…,Mj=1,\\ldots,M\. These vectors, optimized during network training, allow exact reconstruction of latent trajectories\. Note that in \([21](https://arxiv.org/html/2607.28939#S3.E21)\),μ\\muincludes all components, both active and passive\.

##### Step 2

Given observable velocity data\(ξ1j,…,ξNj\)\(\\xi\_\{1\}^\{j\},\\ldots,\\xi\_\{N\}^\{j\}\)for the same trajectories, we parameterize the Hamiltonian using the neural networkh=hN​N​\(μ\)h=h\_\{NN\}\(\\mu\)and compute predicted velocities using a projection operatorℙact\\mathbb\{P\}^\{\\text\{act\}\}onto the active components:

ℙact​ξ~ij=ℙact​∂hN​N∂μ​\(Adgij∗⁡p0j\)\.\\mathbb\{P\}^\{\\text\{act\}\}\\widetilde\{\\xi\}\_\{i\}^\{j\}=\\mathbb\{P\}^\{\\text\{act\}\}\\frac\{\\partial h\_\{NN\}\}\{\\partial\\mu\}\\left\(\\operatorname\{Ad\}^\{\*\}\_\{g\_\{i\}^\{j\}\}p\_\{0\}^\{j\}\\right\)\\,\.\(22\)Projecting onto active components viaℙact\\mathbb\{P\}^\{\\text\{act\}\}is sufficient, as passive velocity components are satisfied automatically by the design of \([20](https://arxiv.org/html/2607.28939#S3.E20)\)\.

##### Step 3

We define the loss function with respect to the network weights𝐖\\mathbf\{W\}and the set of trajectory constantsp¯0=\(p01,…,p0M\)\\bar\{p\}\_\{0\}=\(p\_\{0\}^\{1\},\\ldots,p\_\{0\}^\{M\}\):

L​\(𝐖,p¯0\)=∑j=1M∑i=1N‖ℙact​\(ξij−ξ~ij\)‖2,with​ξ~ij​defined by \([22](https://arxiv.org/html/2607.28939#S3.E22)\)\.L\(\\mathbf\{W\},\\bar\{p\}\_\{0\}\)=\\sum\_\{j=1\}^\{M\}\\sum\_\{i=1\}^\{N\}\\left\\\|\\mathbb\{P\}^\{\\text\{act\}\}\\left\(\\xi\_\{i\}^\{j\}\-\\widetilde\{\\xi\}\_\{i\}^\{j\}\\right\)\\right\\\|^\{2\}\\,,\\quad\\text\{with \}\\widetilde\{\\xi\}\_\{i\}^\{j\}\\text\{ defined by \\eqref\{eq:xi\_i\_approx\}\}\.\(23\)Optimizing loss function \([23](https://arxiv.org/html/2607.28939#S3.E23)\) yields an approximation for the Hamiltonianh​\(μ\)h\(\\mu\)\. While optimization also determines the constant vectorsp0jp\_\{0\}^\{j\}for training trajectories, these values are dataset\-specific and not used directly for predicting new trajectories\.

#### 3\.1\.2Reconstruction Algorithm: Hamiltonian Approach

##### Step 1: Finding Noether’s constant

OncehN​Nh\_\{NN\}has been learned, we use it to reconstruct system trajectories\. In momentum space, trajectory reconstruction is straightforward, requiring forward integration of \([7](https://arxiv.org/html/2607.28939#S2.E7)\) using structure\-preserving integrators\. However, our primary goal is to reconstruct dynamics in observable space\. Whenh​\(μ\)h\(\\mu\)is regular with no passive directions, one can use the Legendre transform to map Hamiltonian dynamics into Lagrangian dynamics\. For degenerate Hamiltonians, however, constructing a Lagrangian that directly governs velocity evolution is generally impossible\.

Rather than attempting to find a Lagrangian that may not exist, we operate entirely within the Hamiltonian framework\. Given initial observationsξ0\\xi\_\{0\}andg0g\_\{0\}for a trajectory to be reconstructed, we determine the conserved spatial momentump0p\_\{0\}satisfying:

ℙact​ξ0=ℙact​∂hN​N∂μ​\(Adg0∗⁡p0\)\.\\mathbb\{P\}^\{\\text\{act\}\}\\xi\_\{0\}=\\mathbb\{P\}^\{\\text\{act\}\}\\frac\{\\partial h\_\{NN\}\}\{\\partial\\mu\}\\left\(\\operatorname\{Ad\}^\{\*\}\_\{g\_\{0\}\}p\_\{0\}\\right\)\.\(24\)Although solving \([24](https://arxiv.org/html/2607.28939#S3.E24)\) from a single snapshot is theoretically possible for regular Hamiltonians, fittingp0p\_\{0\}over a short initial window ofNfitN\_\{\\text\{fit\}\}states significantly improves robustness against measurement noise\. Because accurate estimation ofp0p\_\{0\}is crucial for long\-term trajectory prediction, relying on a single data point often leads to numerical degradation\. We thus determinep0p\_\{0\}from a short history window rather than a single point to improve the stability and accuracy of the method\.

##### Step 2: Finding the update for the group element

Oncep0p\_\{0\}is determined from Step 1, letΔk=gk−1​gk\+1\\Delta\_\{k\}=g\_\{k\}^\{\-1\}g\_\{k\+1\}denote the relative group evolution element, yielding the configuration updategk\+1=gk​Δkg\_\{k\+1\}=g\_\{k\}\\Delta\_\{k\}\. Using Noether’s conservation law and the coadjoint anti\-homomorphism property \(Adg​h∗=Adh∗⁡Adg∗\\operatorname\{Ad\}^\{\*\}\_\{gh\}=\\operatorname\{Ad\}^\{\*\}\_\{h\}\\operatorname\{Ad\}^\{\*\}\_\{g\}\), we obtain:

μ~k\+1​\(Δk\)=Adgk\+1∗⁡p0=Adgk​Δk∗⁡p0=AdΔk∗⁡Adgk∗⁡p0=AdΔk∗⁡μk\.\\widetilde\{\\mu\}\_\{k\+1\}\(\\Delta\_\{k\}\)=\\operatorname\{Ad\}^\{\*\}\_\{g\_\{k\+1\}\}p\_\{0\}=\\operatorname\{Ad\}^\{\*\}\_\{g\_\{k\}\\Delta\_\{k\}\}p\_\{0\}=\\operatorname\{Ad\}^\{\*\}\_\{\\Delta\_\{k\}\}\\operatorname\{Ad\}^\{\*\}\_\{g\_\{k\}\}p\_\{0\}=\\operatorname\{Ad\}^\{\*\}\_\{\\Delta\_\{k\}\}\\mu\_\{k\}\\,\.\(25\)Simultaneously, the Lie algebra velocity at stepk\+1k\+1must satisfyξ~k\+1=∂hN​N∂μ​\(μ~k\+1\)\\widetilde\{\\xi\}\_\{k\+1\}=\\frac\{\\partial h\_\{NN\}\}\{\\partial\\mu\}\(\\widetilde\{\\mu\}\_\{k\+1\}\)\. For the time stepΔ​t\\Delta t, the relative stepΔk\\Delta\_\{k\}is computed via a second\-order Magnus expansion:

Θ\(2\)​\(Δ​t,Δk\)=Δ​t2​\(ξk\+ξ~k\+1​\(Δk\)\)\+\(Δ​t\)212​\[ξk,ξ~k\+1​\(Δk\)\],\\Theta^\{\(2\)\}\(\\Delta t,\\Delta\_\{k\}\)=\\frac\{\\Delta t\}\{2\}\\left\(\\xi\_\{k\}\+\\widetilde\{\\xi\}\_\{k\+1\}\(\\Delta\_\{k\}\)\\right\)\+\\frac\{\(\\Delta t\)^\{2\}\}\{12\}\\left\[\\xi\_\{k\},\\widetilde\{\\xi\}\_\{k\+1\}\(\\Delta\_\{k\}\)\\right\],\(26\)which yields an implicit algebraic relation forΔk\\Delta\_\{k\}\. Note that the Lie algebra commutator in \([26](https://arxiv.org/html/2607.28939#S3.E26)\) involves all components ofξk\\xi\_\{k\}andξ~k\+1\\widetilde\{\\xi\}\_\{k\+1\}, not just active ones\. Setting the residual to zero creates the equation:

res​\(Δk\)=Δk−exp⁡\(Θ\(2\)​\(h,Δk\)\)=0\.\\text\{res\}\(\\Delta\_\{k\}\)=\\Delta\_\{k\}\-\\exp\\left\(\\Theta^\{\(2\)\}\(h,\\Delta\_\{k\}\)\\right\)=0\.\(27\)BecauseΔk\\Delta\_\{k\}is close to the identity, we parameterizeΔk\\Delta\_\{k\}via its Lie algebra counterpartUkU\_\{k\}asΔk=exp⁡\(h​Uk\)\\Delta\_\{k\}=\\exp\(hU\_\{k\}\)\. Equation \([27](https://arxiv.org/html/2607.28939#S3.E27)\) can be solved using standard root\-finding algorithms or non\-linear least squares; we employ the Newton–Raphson method for computational efficiency\. Solving forUkU\_\{k\}at each time step yields the updated relative group elementΔk\\Delta\_\{k\}\.

##### Step 3: Solving for observable and latent variables

ObtainingΔk\\Delta\_\{k\}allows us to update the dynamic state variables at stepk\+1k\+1:

gk\+1=gk​Δk,μk\+1=AdΔk∗⁡μk,ξk\+1=∂hN​N∂μ​\(μk\+1\)\.g\_\{k\+1\}=g\_\{k\}\\Delta\_\{k\},\\quad\\mu\_\{k\+1\}=\\operatorname\{Ad\}^\{\*\}\_\{\\Delta\_\{k\}\}\\mu\_\{k\},\\quad\\xi\_\{k\+1\}=\\frac\{\\partial h\_\{NN\}\}\{\\partial\\mu\}\(\\mu\_\{k\+1\}\)\\,\.\(28\)

##### Step 4: Full reconstruction

Withp0p\_\{0\}fixed from Step 1 for the entire trajectory, repeat Steps 2 and 3 sequentially across all time steps to complete trajectory reconstruction\.

### 3\.2LLPNNs: The encoder \(Lagrangian\) approach

#### 3\.2\.1Mathematical background

In this second approach, we formulate the learning procedure using a Lagrangian\-like functionℓ​\(ξ\)\\ell\(\\xi\)\. Care must be taken here: unlike in the Hamiltonian formulation, we cannot assume that a physically meaningful Lagrangian always exists\. We can, however, assume that, after excluding the passive coordinates, the mapping between active momentum and velocity variables is locally one\-to\-one\. Specifically, the key assumption of the Lagrangian LLPNN approach is that the restriction of the mapΦ​\(μ\)=ξ​\(μ\)=∂h/∂μ\\Phi\(\\mu\)=\\xi\(\\mu\)=\\partial h/\\partial\\muto the active coordinates has a non\-singular Hessian:

det∂2h∂μact​∂μact≠0⟹Ψ​\(ξact\)=∂ℓ∂ξact=μact,Ψ=Φ−1\.\\det\\frac\{\\partial^\{2\}h\}\{\\partial\\mu^\{\\rm act\}\\partial\\mu^\{\\rm act\}\}\\neq 0\\quad\\Longrightarrow\\quad\\Psi\(\\xi^\{\\rm act\}\)=\\frac\{\\partial\\ell\}\{\\partial\\xi^\{\\rm act\}\}=\\mu^\{\\rm act\},\\qquad\\Psi=\\Phi^\{\-1\}\.\(29\)Indeed, by the inverse function theorem, the non\-singularity condition in \([29](https://arxiv.org/html/2607.28939#S3.E29)\) guarantees the existence of a local inverseΨ=Φ−1\\Psi=\\Phi^\{\-1\}\. Furthermore, the Jacobian ofΨ\\Psiis the inverse Hessian ofhhin active components, which is symmetric\. Consequently,Ψ\\Psiis locally a gradient mapping and therefore admits a scalar potentialℓ​\(ξ\)\\ell\(\\xi\)satisfyingμact=∂ℓ/∂ξact\\mu^\{\\rm act\}=\\partial\\ell/\\partial\\xi^\{\\rm act\}\.

We emphasize thatℓ​\(ξ\)\\ell\(\\xi\)is generally not a true physical Lagrangian unless all coordinates are active and the Hamiltonian is non\-degenerate\. Condition \([29](https://arxiv.org/html/2607.28939#S3.E29)\) merely defines an invertible change of coordinates between active momentum and velocity variables and does not imply the existence of an Euler–Poincaré formulation\. To reflect its purely formal role as a partial encoder, we refer toℓ​\(ξ\)\\ell\(\\xi\)as a*pseudo\-Lagrangian*\.

The main advantage of this formulation is that it operates directly on observable velocitiesξ\\xi\. Here, we only learn the encoder in active variablesℓ​\(ξact\)\\ell\(\\xi^\{\\text\{act\}\}\)and do not require a corresponding decoderh​\(μ\)h\(\\mu\)\. Learning the phase space dynamics proceeds in two steps\.

##### Step 1

Using conservation law \([10](https://arxiv.org/html/2607.28939#S2.E10)\) for observable configurations\(g1j,…,gNj\)\(g\_\{1\}^\{j\},\\ldots,g\_\{N\}^\{j\}\)along trajectoryj∈\{1,…,M\}j\\in\\\{1,\\ldots,M\\\}, there exists a constant spatial momentump0jp\_\{0\}^\{j\}such that the active momentum components satisfy:

ℙact​μij​\(ξij\)=∂ℓ∂ξact​\(ξij,act\)=ℙact​Adgij∗⁡p0j\.\\mathbb\{P\}^\{\\text\{act\}\}\\mu\_\{i\}^\{j\}\(\\xi\_\{i\}^\{j\}\)=\\frac\{\\partial\\ell\}\{\\partial\\xi^\{\\text\{act\}\}\}\\left\(\\xi\_\{i\}^\{j,\\text\{act\}\}\\right\)=\\mathbb\{P\}^\{\\text\{act\}\}\\operatorname\{Ad\}^\{\*\}\_\{g\_\{i\}^\{j\}\}p\_\{0\}^\{j\}\\,\.\(30\)Note that \([30](https://arxiv.org/html/2607.28939#S3.E30)\) is formulated in terms of velocity coordinatesξ\\xi, unlike the momentum formulation in \([21](https://arxiv.org/html/2607.28939#S3.E21)\)\.

##### Step 2

We seek to approximate the pseudo\-Lagrangianℓ​\(ξ\)≈ℓN​N​\(ξ\)\\ell\(\\xi\)\\approx\\ell\_\{NN\}\(\\xi\)using a neural network while simultaneously determining the constant vectorsp0jp\_\{0\}^\{j\}for each trajectory\. A key technical difference arises when determiningp0jp\_\{0\}^\{j\}in the Lagrangian setting versus the Hamiltonian setting\. Unlike in \([21](https://arxiv.org/html/2607.28939#S3.E21)\), the constantsp0jp\_\{0\}^\{j\}appear*outside*the neural network evaluation in \([30](https://arxiv.org/html/2607.28939#S3.E30)\)\. Consequently, for a given networkℓN​N​\(ξ\)\\ell\_\{NN\}\(\\xi\), \([30](https://arxiv.org/html/2607.28939#S3.E30)\) forms a*linear*system forp0jp\_\{0\}^\{j\}, which can be solved efficiently via linear least squares\. Thus, for network weights𝐖\\mathbf\{W\}, the optimal constantsp0j​\(𝐖\)p\_\{0\}^\{j\}\(\\mathbf\{W\}\)are determined directly from data\. The predicted active momentum components are then given by:

μ~ij,act=ℙact​∂ℓN​N∂ξ​\(ξij\)\.\\widetilde\{\\mu\}\_\{i\}^\{j,\\text\{act\}\}=\\mathbb\{P\}^\{\\text\{act\}\}\\frac\{\\partial\\ell\_\{NN\}\}\{\\partial\\xi\}\\left\(\\xi\_\{i\}^\{j\}\\right\)\\,\.\(31\)

##### Step 3

We construct a loss function depending solely on the network weights𝐖\\mathbf\{W\}:

L​\(𝐖\)=∑j=1M∑i=1N‖ℙact​Adgij∗⁡p0j​\(𝐖\)−μ~ij,act‖2,with​μ~ij,act​defined by \([31](https://arxiv.org/html/2607.28939#S3.E31)\)\.L\(\\mathbf\{W\}\)=\\sum\_\{j=1\}^\{M\}\\sum\_\{i=1\}^\{N\}\\left\\\|\\mathbb\{P\}^\{\\text\{act\}\}\\operatorname\{Ad\}^\{\*\}\_\{g\_\{i\}^\{j\}\}p\_\{0\}^\{j\}\(\\mathbf\{W\}\)\-\\widetilde\{\\mu\}\_\{i\}^\{j,\\text\{act\}\}\\right\\\|^\{2\},\\quad\\text\{with \}\\widetilde\{\\mu\}\_\{i\}^\{j,\\text\{act\}\}\\text\{ defined by \\eqref\{eq:mu\_i\_approx\_Lagr\}\}\.\(32\)Optimizing loss function \([32](https://arxiv.org/html/2607.28939#S3.E32)\) yields the learned pseudo\-LagrangianℓN​N​\(ξact\)\\ell\_\{NN\}\(\\xi^\{\\text\{act\}\}\), providing the forward encoder mapping to active momentum components\. This optimization also determines the Noether constantsp0j​\(𝐖\)p\_\{0\}^\{j\}\(\\mathbf\{W\}\)for each training trajectory\.

#### 3\.2\.2Reconstruction Algorithm: Pseudo\-Lagrangian Approach

##### Step 1: Finding Noether’s constant

Similar to the Hamiltonian approach, suppose we observe a short window of initial conditions\(gi,ξi\)\(g\_\{i\},\\xi\_\{i\}\)for0≤i≤Nfit0\\leq i\\leq N\_\{\\mathrm\{fit\}\}along a trajectory to be reconstructed\. We seek the conserved spatial momentump0p\_\{0\}satisfying

∂ℓN​N∂ξact​\(ξiact\)=ℙact​Adgi∗⁡p0,0≤i≤Nfit\.\\frac\{\\partial\\ell\_\{NN\}\}\{\\partial\\xi^\{\\text\{act\}\}\}\\left\(\\xi\_\{i\}^\{\\text\{act\}\}\\right\)=\\mathbb\{P\}^\{\\mathrm\{act\}\}\\operatorname\{Ad\}^\{\*\}\_\{g\_\{i\}\}p\_\{0\},\\qquad 0\\leq i\\leq N\_\{\\mathrm\{fit\}\}\.\(33\)
SinceℓN​N\\ell\_\{NN\}has already been learned, equation \([33](https://arxiv.org/html/2607.28939#S3.E33)\) is linear in the unknown vectorp0p\_\{0\}\. This should be contrasted with its Hamiltonian counterpart \([24](https://arxiv.org/html/2607.28939#S3.E24)\), wherep0p\_\{0\}appears nonlinearly inside the learned HamiltonianhN​Nh\_\{NN\}\.

BecauseℓN​N\\ell\_\{NN\}acts only on the active components, whereasp0p\_\{0\}contains all momentum components in the dual Lie algebra, a single observation is generally insufficient to uniquely determinep0p\_\{0\}\. Instead, a short fitting window ofNfitN\_\{\\mathrm\{fit\}\}observations produces an overdetermined linear system that can be solved efficiently in the least\-squares sense\.

##### Step 2: Determining velocity at the next step using Magnus residual

Oncep0p\_\{0\}is determined, we also know full initial momentumμ0=A​dg0∗​p0\\mu\_\{0\}=Ad^\{\*\}\_\{g\_\{0\}\}p\_\{0\}since we know the initial Lie group elementg0g\_\{0\}\. Suppose we knowμk\\mu\_\{k\},gkg\_\{k\}andξk\\xi\_\{k\}on the stepkkand would like to find out these values at the stepk\+1k\+1\.

LetΔk=gk−1​gk\+1\\Delta\_\{k\}=g\_\{k\}^\{\-1\}g\_\{k\+1\}denote the relative configuration updategk\+1=gk​Δkg\_\{k\+1\}=g\_\{k\}\\Delta\_\{k\}\. Given current velocityξk\\xi\_\{k\}and a candidate velocityξ~k\+1=\(ξpass,ξ~k\+1act\)\\widetilde\{\\xi\}\_\{k\+1\}=\\left\(\\xi^\{\\text\{pass\}\},\\widetilde\{\\xi\}\_\{k\+1\}^\{\\text\{act\}\}\\right\)at stepk\+1k\+1, we obtain:

μ~k\+1act​\(Δk\)=∂ℓN​N∂ξact​\(ξ~k\+1act\)=ℙact​AdΔk∗⁡μk\.\\widetilde\{\\mu\}\_\{k\+1\}^\{\\text\{act\}\}\(\\Delta\_\{k\}\)=\\frac\{\\partial\\ell\_\{NN\}\}\{\\partial\\xi^\{\\text\{act\}\}\}\(\\widetilde\{\\xi\}\_\{k\+1\}^\{\\text\{act\}\}\)=\\mathbb\{P\}^\{\\text\{act\}\}\\operatorname\{Ad\}^\{\*\}\_\{\\Delta\_\{k\}\}\\mu\_\{k\}\\,\.\(34\)As before, we relateΔk\\Delta\_\{k\}to velocitiesξk\\xi\_\{k\}andξ~k\+1\\widetilde\{\\xi\}\_\{k\+1\}using the second\-order Magnus expansion \([26](https://arxiv.org/html/2607.28939#S3.E26)\)\. Equating the two representations of the momentum at time stepk\+1k\+1defines the residual equation:

res​\(ξ~k\+1\)=∂ℓN​N∂ξact​\(ξ~k\+1act\)−ℙact​AdΔk​\(ξk,ξ~k\+1\)∗⁡μk=0\.\\text\{res\}\(\\widetilde\{\\xi\}\_\{k\+1\}\)=\\frac\{\\partial\\ell\_\{NN\}\}\{\\partial\\xi^\{\\text\{act\}\}\}\(\\widetilde\{\\xi\}\_\{k\+1\}^\{\\text\{act\}\}\)\-\\mathbb\{P\}^\{\\text\{act\}\}\\operatorname\{Ad\}^\{\*\}\_\{\\Delta\_\{k\}\(\\xi\_\{k\},\\widetilde\{\\xi\}\_\{k\+1\}\)\}\\mu\_\{k\}=0\.\(35\)SinceΔk\\Delta\_\{k\}lies near the identity, we parameterize it through its Lie\-algebra element\. Equation \([35](https://arxiv.org/html/2607.28939#S3.E35)\) is then solved for

ξk\+1act=ξ~k\+1act\\xi\_\{k\+1\}^\{\\mathrm\{act\}\}=\\widetilde\{\\xi\}\_\{k\+1\}^\{\\mathrm\{act\}\}using a root\-finding or nonlinear least\-squares procedure\. The full velocityξk\+1\\xi\_\{k\+1\}is then obtained by combining the solved active components with the fixed passive components\.

##### Step 3: Prediction of observable and latent variables at the next time step

Once the full velocityξk\+1\\xi\_\{k\+1\}is determined from \([35](https://arxiv.org/html/2607.28939#S3.E35)\), we compute the relative configuration stepΔk\\Delta\_\{k\}and update the latent momentum at stepk\+1k\+1as

μk\+1=AdΔk∗⁡μk\.\\mu\_\{k\+1\}=\\operatorname\{Ad\}^\{\*\}\_\{\\Delta\_\{k\}\}\\mu\_\{k\}\.\(36\)Note that \([36](https://arxiv.org/html/2607.28939#S3.E36)\), like its Hamiltonian counterpart \([28](https://arxiv.org/html/2607.28939#S3.E28)\), iterates the full latent momentum forward using coadjoint group actions, thereby preserving Casimir invariants to machine precision\.

##### Step 4: Full reconstruction

Withp0p\_\{0\}determined from Step 1 for the entire trajectory, repeat Steps 2 and 3 sequentially across all time steps to complete trajectory reconstruction\.

A schematic overview comparing the Lagrangian and Hamiltonian approaches is shown in Figure[2](https://arxiv.org/html/2607.28939#S3.F2)\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x2.png)Figure 2:Schematic overview of the proposed LLPNN framework in both Lagrangian and Hamiltonian settings\.

### 3\.3Warm Start for the Learning Process

To improve robustness and convergence during optimization, we initialize our algorithm with a physically and computationally viable warm start\. This is particularly crucial for the Hamiltonian approach, where we must simultaneously train the Hamiltonian neural network and estimate the Noether constantsp0p\_\{0\}from data\. When Noether constants enter non\-linearly into network evaluations, optimization initialized from random weights can encounter complex loss landscapes and converge to spurious local minima\. An inaccurate initial state for the residual network can similarly cause convergence to false minima\. Although random initialization often yields correct results, implementing a warm\-start procedure significantly improves optimization reliability for practical applications\. We design this procedure to provide accurate initial approximations for common control Hamiltonians, specifically those exhibiting quadratic behavior near the origin in active variables\.

To construct our warm\-start procedure, we assume that the Hamiltonian and pseudo\-Lagrangian in active coordinates are dominated by quadratic terms, with neural networks modeling non\-quadratic residuals:

ℓ​\(ξact\)\\displaystyle\\ell\\left\(\\xi^\{\\text\{act\}\}\\right\)=12​ξact⋅𝔹​ξact\+ℓN​Nres​\(ξact\),\\displaystyle=\\frac\{1\}\{2\}\\xi^\{\\text\{act\}\}\\cdot\\mathbb\{B\}\\xi^\{\\text\{act\}\}\+\\ell^\{\\text\{res\}\}\_\{NN\}\\left\(\\xi^\{\\text\{act\}\}\\right\)\\,,\(37\)h​\(μ\)\\displaystyle h\(\\mu\)=a⋅μpassive\+12​μact⋅𝕂​μact\+hN​Nres​\(μact\),\\displaystyle=a\\cdot\\mu^\{\\text\{passive\}\}\+\\frac\{1\}\{2\}\\mu^\{\\text\{act\}\}\\cdot\\mathbb\{K\}\\mu^\{\\text\{act\}\}\+h^\{\\text\{res\}\}\_\{NN\}\\left\(\\mu^\{\\text\{act\}\}\\right\)\\,,\(38\)where𝔹\\mathbb\{B\}and𝕂\\mathbb\{K\}are trainable symmetric matrices of sizeK×KK\\times K, withKKdenoting the number of active variables\. During the warm\-start phase, we freeze the residual neural networks in \([37](https://arxiv.org/html/2607.28939#S3.E37)\) and \([38](https://arxiv.org/html/2607.28939#S3.E38)\) to zero and solve for purely quadratic representations\. Under this assumption, parameter initialization for the Hamiltonian, pseudo\-Lagrangian, and Noether constants simplifies dramatically\.

In the Hamiltonian setting, \([22](https://arxiv.org/html/2607.28939#S3.E22)\) simplifies to a biquadratic equation inp0p\_\{0\}and matrix𝕂\\mathbb\{K\}, which can be solved efficiently using Newton iterations\. In the Lagrangian setting, \([31](https://arxiv.org/html/2607.28939#S3.E31)\) reduces to a homogeneous linear system in𝔹\\mathbb\{B\}andp0p\_\{0\}, solvable as an overdetermined least\-squares problem\. A unique solution is obtained by requiring, for example,tr⁡𝔹=1\\operatorname\{tr\}\\mathbb\{B\}=1, which yields a Karush–Kuhn–Tucker \(KKT\) linear system\[[50](https://arxiv.org/html/2607.28939#bib.bib123)\]\. This formulation establishes a unique scale for matrix𝔹\\mathbb\{B\}in \([37](https://arxiv.org/html/2607.28939#S3.E37)\) and all Noether constantsp0p\_\{0\}\.

Once initial parameter estimates are established, we unfreeze the neural network weights and proceed with full training\. For the Hamiltonian formulation, this step also refines the estimated Noether constantsp0p\_\{0\}\. This warm\-start strategy provides a fast, robust initialization for the subsequent optimization phase\.

### 3\.4Energy Definition and Regularity

For autonomous Hamiltonian systems, the Hamiltonian function evaluated on momenta is a conserved quantity\. Consequently, when using the Hamiltonian LLPNN formulation, energy conservation along reconstructed trajectories can be verified directly, as both the Hamiltonian function and state trajectories in observable and latent spaces are explicitly reconstructed\.

The situation differs in the Lagrangian setting\. For a regular Lagrangian system, the conserved energy\-like functionEEis obtained through the Legendre transform:

E​\(ξ\)=⟨ξ,∂ℓ∂ξ⟩−ℓ​\(ξ\),E\(\\xi\)=\\left\\langle\\xi,\\frac\{\\partial\\ell\}\{\\partial\\xi\}\\right\\rangle\-\\ell\(\\xi\),\(39\)where⟨⋅,⋅⟩\\langle\\cdot,\\cdot\\rangledenotes the natural pairing between the Lie algebra𝔤\\mathfrak\{g\}and its dual𝔤∗\\mathfrak\{g\}^\{\*\}\. Note that energy function \([39](https://arxiv.org/html/2607.28939#S3.E39)\) is formulated in terms of velocityξ\\xirather than momentumμ\\mu; it represents the quantity that becomes Hamiltonianh​\(μ\)h\(\\mu\)upon substitutingξ=ξ​\(μ\)\\xi=\\xi\(\\mu\)\.

If bothℓ​\(ξ\)\\ell\(\\xi\)andh​\(μ\)h\(\\mu\)are regular \(i\.e\., the Hessians∇2ℓ\\nabla^\{2\}\\elland∇2h\\nabla^\{2\}hare non\-singular in the region of interest\), the Legendre transformation is locally invertible\. Consequently, the energy functionalE​\(ξ\)E\(\\xi\)in \([39](https://arxiv.org/html/2607.28939#S3.E39)\) exists at least locally, a property that fails for degenerate or constrained systems\. If the Hamiltonian is singular, the observable dynamics need not admit an Euler–Poincaré representation \([6](https://arxiv.org/html/2607.28939#S2.E6)\), as a valid Lagrangian satisfying the observed velocity data may not exist, and there may not even exist a closed autonomous dynamical system in the observable variablesξ\\xi, as we discussed above in Section[2\.2](https://arxiv.org/html/2607.28939#S2.SS2)\.

Accordingly, we compute energy in the Lagrangian formulation only when the underlying Hamiltonian is regular\. In this work, regularity holds for the Kirchhoff underwater vehicle problem onS​E​\(3\)SE\(3\)detailed in Section[4\.3](https://arxiv.org/html/2607.28939#S4.SS3)\. In contrast, planar drone control onS​E​\(2\)NSE\(2\)^\{N\}\(Section[4\.4](https://arxiv.org/html/2607.28939#S4.SS4)\) involves a control Hamiltonian with two singular directions and one regular direction per agent\. Consequently, the energy coming from the Legendre\-transform \([39](https://arxiv.org/html/2607.28939#S3.E39)\) is not defined and no corresponding energy\-conservation test is available in the pseudo\-Lagrangian formulation\.

### 3\.5Overall Comparison Between Hamiltonian and Lagrangian LLPNNs

Both Lagrangian and Hamiltonian formulations offer distinct trade\-offs, making each suitable for specific problem classes\. A comparative summary is provided in Table[1](https://arxiv.org/html/2607.28939#S3.T1)\.

QuantityHamiltonian LLPNNLagrangian LLPNNPhase spaceLatentObservableApplicabilityAny HamiltonianLegendre invertiblein active variablesNoether Integralp0p\_\{0\}computationImplicit root\-findinginside NN evaluationOverdeterminedlinear systemGauge\-invariantlatent spaceYesYesEnergy conservationDirectly availablefor any HamiltonianRegular HamiltoniansTable 1:Summary of differences between Hamiltonian and Lagrangian LLPNN approaches\.The Hamiltonian formulation operates directly in the latent state space where the dynamics is closed, while the pseudo\-Lagrangian formulation operates in the observable variables through an invertible active\-sector encoding\. The Hamiltonian LLPNN therefore applies to arbitrary Lie–Poisson systems, whereas the Lagrangian LLPNN benefits from substantially simpler reconstruction whenever the active velocity–momentum map is invertible\. Consequently, both formulations are relevant in practical applications: the Hamiltonian approach offers the greatest generality and remains applicable to a wide class of degenerate systems, while the Lagrangian approach provides a computationally simpler learning and reconstruction procedure whenever a suitable active\-variable inverse mapping exists\.

Having established the theoretical framework, we now turn to practical applications: rigid\-body motion, Kirchhoff’s underwater vehicle problem, and interacting planar drone dynamics\. An interesting feature of the proposed warm\-start procedure is that, when the underlying Hamiltonian and Lagrangian are quadratic, the initialization already recovers the exact solution, and further neural\-network training provides little or no improvement\. Consequently, purely quadratic systems, whether degenerate or non\-degenerate, are of limited interest as neural\-network benchmarks\. While classical formulations of these problems typically employ quadratic Hamiltonians, we consider generalized non\-quadratic models in order to demonstrate the full expressive power of the LLPNN framework\.

### 3\.6Comparison with other methods

To the best of our knowledge, no existing structure\-preserving framework is capable of learning arbitrary latent Hamiltonian dynamics directly from observable variables\(ξ,g\)\(\\xi,g\)like our approach\. We therefore evaluate alternative baseline methods from the literature that can be adapted to provide a meaningful benchmark comparison\.

If no additional information about the system is provided, one could use non\-structure\-preserving approaches such as Neural ODEs\[[9](https://arxiv.org/html/2607.28939#bib.bib130)\]and DeepONets\[[45](https://arxiv.org/html/2607.28939#bib.bib132)\]\. However, these methods operate purely on the observable variables and therefore implicitly assume that the observables constitute a closed dynamical system\. As discussed in Section[2\.2](https://arxiv.org/html/2607.28939#S2.SS2), this assumption may fail for degenerate Lie–Poisson Hamiltonians\. In such cases, the observable variables do not uniquely determine the latent momentum state and may not satisfy any autonomous evolution equation\. Consequently, Neural ODEs and DeepONets are forced to approximate an ill\-defined dynamics without the knowledge of the latent states, whereas LLPNNs reconstruct the latent state space in which the dynamics is actually closed\.

Since we work with velocity data, an alternative would be to consider purely Lagrangian methods\[[13](https://arxiv.org/html/2607.28939#bib.bib102)\]adapted to the symmetry\-reduced Euler\-Poincaré equations \([6](https://arxiv.org/html/2607.28939#S2.E6)\)\. However, as previously discussed, one cannot in general expect the motion of a degenerate symmetry\-reduced Hamiltonian to reduce to a corresponding Euler\-Poincaré system for a Lagrangian, as a Lagrangian representation may not even exist\.

Instead, we adopt a compromise: rather than searching for a structure\-preserving algorithm that predicts latent\-variable Hamiltonian dynamics using only observable variables, we generate latent\-space data sequences to train an alternative structure\-preserving baseline\. Specifically, we choose an algorithm based on Lie\-Poisson Neural Networks \(LPNets\)\[[19](https://arxiv.org/html/2607.28939#bib.bib140)\], extended to interacting objects\[[20](https://arxiv.org/html/2607.28939#bib.bib127),[32](https://arxiv.org/html/2607.28939#bib.bib129)\]\. This comparison should be interpreted with care, since LPNet training requires latent momentum trajectories that are not directly available from the observations\. Thus, the LPNet baseline relies heavily on our method’s learned Hamiltonian, which performs the preliminary work required to generate those latent trajectories\.

In summary, we compare LLPNNs against three representative alternatives\. Neural ODEs and DeepONets require only observable data but do not exploit the latent Lie–Poisson structure\. LPNets preserve the Lie–Poisson geometry exactly, but require access to latent momentum trajectories that are not available in practical applications\. Together, these baselines provide a useful comparison between observable\-space learning, latent\-space structure\-preserving learning, and the proposed latent reconstruction framework\.

The details of the learning procedure, which is common for all cases, is summarized on Table[2](https://arxiv.org/html/2607.28939#S3.T2)\.

PhaseNetworkEpochsInitial LRDecay RateDecay StepsL\-NetLnetL\_\{\\text\{net\}\}20,00020,0000\.0050\.0050\.900\.90500500H\-NetHnet,p0H\_\{\\text\{net\}\},p\_\{0\}20,00020,0000\.0100\.0100\.900\.90500500LPNetsLPNet10,00010,0000\.0050\.005Fixed LRN/ATable 2:Training Schedules and Optimization Hyperparameters

## 4Experimental results

### 4\.1Ground Truth Data Generation

To generate physically faithful long\-horizon reference trajectories without introducing numerical dissipation or artificial manifold drift, we employ a structure\-preserving geometric integrator formulated for a general Lie groupGGwith dual Lie algebra𝔤∗\\mathfrak\{g\}^\{\*\}\(e\.g\.,S​E​\(3\)SE\(3\),S​O​\(3\)SO\(3\), or product groups such asS​E​\(2\)NSE\(2\)^\{N\}\)\. The ground\-truth dynamics are integrated in two coupled stages: a symplectic implicit midpoint update for the body momentumμ∈𝔤∗\\mu\\in\\mathfrak\{g\}^\{\*\}, followed by a second\-order Magnus expansion to reconstruct group configurationsg∈Gg\\in G\.

##### 1\. Body Momentum Step \(Implicit Midpoint\)

The reduced Lie–Poisson dynamics on𝔤∗\\mathfrak\{g\}^\{\*\}obeyμ˙=Λ​\(μ\)​∇h​\(μ\)\\dot\{\\mu\}=\\Lambda\(\\mu\)\\nabla h\(\\mu\), whereΛi​j​\(μ\)=−∑kμk​γi​jk\\Lambda\_\{ij\}\(\\mu\)=\-\\sum\_\{k\}\\mu\_\{k\}\\gamma^\{k\}\_\{ij\}represents the Poisson structure tensor induced by the Lie algebra structure constantsγi​jk\\gamma^\{k\}\_\{ij\}\. To advance the body momentum fromμk\\mu\_\{k\}toμk\+1\\mu\_\{k\+1\}across an internal step sizeΔ​tint\\Delta t\_\{\\text\{int\}\}, we solve the implicit midpoint scheme:

μk\+1−μk−Δ​tint​Λ​\(μk\+μk\+12\)​∇h​\(μk\+μk\+12\)=𝟎\.\\mu\_\{k\+1\}\-\\mu\_\{k\}\-\\Delta t\_\{\\text\{int\}\}\\,\\Lambda\\left\(\\frac\{\\mu\_\{k\}\+\\mu\_\{k\+1\}\}\{2\}\\right\)\\nabla h\\left\(\\frac\{\\mu\_\{k\}\+\\mu\_\{k\+1\}\}\{2\}\\right\)=\\mathbf\{0\}\\,\.\(40\)Equation \([40](https://arxiv.org/html/2607.28939#S4.E40)\) is solved numerically at each step via a high\-precision multi\-dimensional root finder with a residual tolerance of10−1210^\{\-12\}\. The body velocity vectorξ=∇h​\(μ\)∈𝔤\\xi=\\nabla h\(\\mu\)\\in\\mathfrak\{g\}is obtained via exact gradient evaluations of the Hamiltonianh​\(μ\)h\(\\mu\), which may be non\-quadratic\. This method preserves all quadratic Casimirs \(which comprise all Casimirs in the systems considered here\) to machine precision\.

##### 2\. Configuration Reconstruction \(Magnus Expansion\)

Given momentum statesμk\\mu\_\{k\}andμk\+1\\mu\_\{k\+1\}, their associated body velocitiesξk=∇h​\(μk\)\\xi\_\{k\}=\\nabla h\(\\mu\_\{k\}\)andξk\+1=∇h​\(μk\+1\)∈𝔤\\xi\_\{k\+1\}=\\nabla h\(\\mu\_\{k\+1\}\)\\in\\mathfrak\{g\}are used to advance the group configurationgk∈Gg\_\{k\}\\in G\. The algebra update elementΩk∈𝔤\\Omega\_\{k\}\\in\\mathfrak\{g\}is constructed via a second\-order Magnus expansion:

Ωk=Δ​tint2​\(ξk\+ξk\+1\)\+\(Δ​tint\)212​\[ξk,ξk\+1\]𝔤,\\Omega\_\{k\}=\\frac\{\\Delta t\_\{\\text\{int\}\}\}\{2\}\(\\xi\_\{k\}\+\\xi\_\{k\+1\}\)\+\\frac\{\(\\Delta t\_\{\\text\{int\}\}\)^\{2\}\}\{12\}\[\\xi\_\{k\},\\xi\_\{k\+1\}\]\_\{\\mathfrak\{g\}\}\\,,\(41\)where\[⋅,⋅\]𝔤\[\\cdot,\\cdot\]\_\{\\mathfrak\{g\}\}denotes the Lie bracket on𝔤\\mathfrak\{g\}\. The group state is then updated via the Lie exponential map:

gk\+1=gk​exp⁡\(Ω^k\),g\_\{k\+1\}=g\_\{k\}\\exp\\left\(\\widehat\{\\Omega\}\_\{k\}\\right\)\\,,\(42\)whereΩ^k\\widehat\{\\Omega\}\_\{k\}denotes the matrix Lie algebra representation ofΩk\\Omega\_\{k\}\.

##### 3\. Spatial Momentum Reconstruction

By Noether’s theorem, spatial momentump∈𝔤∗p\\in\\mathfrak\{g\}^\{\*\}is conserved underGG\-invariance\. At each recorded step, spatial momentumpkp\_\{k\}is computed from body momentumμk\\mu\_\{k\}via the coadjoint action:

pk=Adgk−1∗⁡μk\.p\_\{k\}=\\operatorname\{Ad\}^\{\*\}\_\{g\_\{k\}^\{\-1\}\}\\mu\_\{k\}\\,\.\(43\)The spatial momentapkp\_\{k\}, conserved by Noether’s theorem, are saved throughout integration to monitor and verify numerical accuracy\.

In our data generation pipeline, integrations are executed using a fine sub\-stepping time stepΔ​tint=0\.01​s\\Delta t\_\{\\text\{int\}\}=0\.01\\,\\text\{s\}and downsampled to an observational resolution ofΔ​t=0\.1​s\\Delta t=0\.1\\,\\text\{s\}\. This dual\-stage scheme guarantees exact geometric preservation of coadjoint orbits, high\-accuracy conservation of spatial momentumpp, preservation of system Casimirs, and long\-term stability of the total energy across general Lie group phase spaces\.

The color scheme and line styles used across all experimental figures in subsequent sections adhere to the conventions summarized in Table[3](https://arxiv.org/html/2607.28939#S4.T3)\.

MethodColorLine StyleGround truthBlueSolid \(\-\)Standard LPNetRedDashed \(\-\-\)Hamiltonian LLPNNGreenDash\-Dotted \(\-\.\)Lagrangian LLPNNMagentaDotted \(:\)DeepONetOrangeDashed \(\-\-\)Neural ODECyanDash\-Dotted \(\-\.\)Table 3:Summary of universal color and line style codes used in figure plots reporting experimental results\.

### 4\.2A Generalized Rigid Body:S​O​\(3\)SO\(3\)dynamics

#### 4\.2\.1Problem formulation

As a first numerical application, we consider rigid body dynamics modeled via Lie–Poisson equations onS​O​\(3\)SO\(3\)\. Although classical governing relations for rigid body motion simplify under standard symmetries, they continue to exhibit rich dynamical behavior\. In this benchmark, we evaluate a non\-quadratic modification of the Hamiltonian to demonstrate the learning capability and geometric fidelity of the LLPNN framework\.

The phase\-space geometry of the rigid body is dictated by the Lie–Poisson bracket associated with the special orthogonal groupS​O​\(3\)SO\(3\), encapsulating three\-dimensional rotations\. ForS​O​\(3\)SO\(3\), Lie algebra velocities and dual Lie algebra momenta are identified with vectors inℝ3\\mathbb\{R\}^\{3\}, denoted by bold symbols𝝁\\boldsymbol\{\\mu\}and𝝃\\boldsymbol\{\\xi\}, respectively\. For an inertia matrix𝕀=diag⁡\(1,2,3\)\\mathbb\{I\}=\\operatorname\{diag\}\(1,2,3\), the classical quadratic kinetic energy is:

Hquad​\(𝝁\)=12​𝝁⋅𝕀−1​𝝁=12​\(μ12\+12​μ22\+13​μ32\)\.H\_\{\\text\{quad\}\}\(\\boldsymbol\{\\mu\}\)=\\frac\{1\}\{2\}\\boldsymbol\{\\mu\}\\cdot\\mathbb\{I\}^\{\-1\}\\boldsymbol\{\\mu\}=\\frac\{1\}\{2\}\\left\(\\mu\_\{1\}^\{2\}\+\\frac\{1\}\{2\}\\mu\_\{2\}^\{2\}\+\\frac\{1\}\{3\}\\mu\_\{3\}^\{2\}\\right\)\\,\.\(44\)For smooth functionsFFandHHon𝔰​𝔬​\(3\)∗\\mathfrak\{so\}\(3\)^\{\*\}, the Lie–Poisson bracket is defined through a standard cross\-product:

\{F,H\}=−𝝁⋅\(∂F∂𝝁×∂H∂𝝁\)\.\\\{F,H\\\}=\-\\boldsymbol\{\\mu\}\\cdot\\left\(\\frac\{\\partial F\}\{\\partial\\boldsymbol\{\\mu\}\}\\times\\frac\{\\partial H\}\{\\partial\\boldsymbol\{\\mu\}\}\\right\)\\,\.\(45\)
Applying Lie–Poisson bracket \([45](https://arxiv.org/html/2607.28939#S4.E45)\) to state variables yields the equations of motion:

𝝁˙=𝝁×∂H∂𝝁\.\\dot\{\\boldsymbol\{\\mu\}\}=\\boldsymbol\{\\mu\}\\times\\frac\{\\partial H\}\{\\partial\\boldsymbol\{\\mu\}\}\\,\.\(46\)Beyond total energy conservation, system \([46](https://arxiv.org/html/2607.28939#S4.E46)\) inherently preserves a Casimir invariant linked to𝔰​𝔬​\(3\)∗\\mathfrak\{so\}\(3\)^\{\*\}, given by the squared momentum magnitude:

C=‖𝝁‖2=μ12\+μ22\+μ32\.C=\\\|\\boldsymbol\{\\mu\}\\\|^\{2\}=\\mu\_\{1\}^\{2\}\+\\mu\_\{2\}^\{2\}\+\\mu\_\{3\}^\{2\}\\,\.\(47\)
As noted previously, purely quadratic Hamiltonians can be learned with minimal neural network complexity\. However, in general optimal control applications, quadratic Hamiltonians cannot be assumed\. We therefore consider the non\-quadratic Hamiltonian:

H​\(𝝁\)=Hquad​\(𝝁\)​\(1\+B​\(μ12\+μ22\+2​μ32\)\),H\(\\boldsymbol\{\\mu\}\)=H\_\{\\text\{quad\}\}\(\\boldsymbol\{\\mu\}\)\\left\(1\+B\(\\mu\_\{1\}^\{2\}\+\\mu\_\{2\}^\{2\}\+2\\mu\_\{3\}^\{2\}\)\\right\)\\,,\(48\)whereB=0\.1B=0\.1in all simulations\. The coefficient22multiplyingμ32\\mu\_\{3\}^\{2\}ensures that the non\-quadratic scaling factor is not simply a function of the Casimir invariant \([47](https://arxiv.org/html/2607.28939#S4.E47)\)\.

Even with non\-quadratic Hamiltonian \([48](https://arxiv.org/html/2607.28939#S4.E48)\), the two conserved quantities \(the Casimir and the Hamiltonian\) define one\-dimensional invariant curves at their intersection surfaces in three\-dimensional momentum space𝝁\\boldsymbol\{\\mu\}\(and correspondingly in velocity space𝝃\\boldsymbol\{\\xi\}\)\. Consequently, this generalized rigid body system remains integrable for any smoothH​\(𝝁\)H\(\\boldsymbol\{\\mu\}\), although the solutions, in general, cannot be expressed in terms of classical elliptic functions as is the case with the standard rigid body\[[61](https://arxiv.org/html/2607.28939#bib.bib124),[41](https://arxiv.org/html/2607.28939#bib.bib125)\]\.

#### 4\.2\.2Experimental Results:S​O​\(3\)SO\(3\)Non\-Quadratic Rigid Body

##### Ground truth data generation for training and testing

For training, we generate 200 trajectories, each containing 101 time points\. The time step is fixed atΔ​t=0\.1\\Delta t=0\.1throughout\. The initial conditions are sampled from the standard normal distribution in the three\-dimensional momentum space, namely

μ0train∼𝒩​\(0,I3\),\\mu^\{\\mathrm\{train\}\}\_\{0\}\\sim\\mathcal\{N\}\(0,I\_\{3\}\),whereI3I\_\{3\}is the3×33\\times 3identity matrix\. For the test trajectories, the initial momenta are sampled from a distribution with half the standard deviation,

μ0test∼𝒩​\(0,0\.25​I3\)\.\\mu^\{\\mathrm\{test\}\}\_\{0\}\\sim\\mathcal\{N\}\(0,0\.25I\_\{3\}\)\.The initial orientationsg0∈S​O​\(3\)g\_\{0\}\\in SO\(3\)are sampled independently from the uniform distribution onS​O​\(3\)SO\(3\)\. In total, we generate 50 test trajectories, each containing 2001 time points\.

##### Neural Network Architectures and Residual Formulations

Both Hamiltonian and Lagrangian neural networks are defined over active coordinates and initialized using quadratic warm\-start baselines:

1. 1\.Hamiltonian Model \(HNNH\_\{\\text\{NN\}\}\):Residual networkHnetH\_\{\\text\{net\}\}consists of33dense hidden layers with3232units per layer, utilizingtanh\\tanhactivation functions\. The output layer is a single linear unit without bias\. All layers are zero\-initialized\. To eliminate gauge ambiguity at𝝁=𝟎\\boldsymbol\{\\mu\}=\\mathbf\{0\}, the network subtracts its value and gradient evaluated at the origin\.
2. 2\.Lagrangian Model \(LNNL\_\{\\text\{NN\}\}\):Similarly, the direct Lagrangian model operates on active body velocities\. Residual networkLnetL\_\{\\text\{net\}\}mirrorsHnetH\_\{\\text\{net\}\}with33hidden layers of3232units,tanh\\tanhactivations, and identical zero\-point ambiguity removal at𝝃=𝟎\\boldsymbol\{\\xi\}=\\mathbf\{0\}\.

Both Lagrangian and Hamiltonian networks are trained for20,00020\{,\}000epochs\. During training, the loss functions \([23](https://arxiv.org/html/2607.28939#S3.E23)\) and \([32](https://arxiv.org/html/2607.28939#S3.E32)\) decrease by approximately four orders of magnitude, dropping from about10−310^\{\-3\}to∼10−6−10−7\\sim 10^\{\-6\}\-10^\{\-7\}, or lower depending on the run\.

Figure[3](https://arxiv.org/html/2607.28939#S4.F3)compares trajectory tracking for observable velocity components𝝃\\boldsymbol\{\\xi\}across evaluated methods for a representative trajectory\. The Hamiltonian LLPNN achieves near\-perfect tracking, remaining visually indistinguishable from ground truth across the full time horizon\. The Lagrangian LLPNN and standard LPNet also maintain close agreement, whereas non\-structure\-preserving baselines \(DeepONet and Neural ODE\) diverge significantly over long integration times\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x3.png)Figure 3:Predicted versus ground\-truth velocity components𝝃​\(t\)\\boldsymbol\{\\xi\}\(t\)for non\-quadraticS​O​\(3\)SO\(3\)rigid body dynamics\. Hamiltonian LLPNN \(green dash\-dotted line\) remains visually indistinguishable from ground truth \(blue solid line\), whereas non\-structure\-preserving baselines diverge over time\.Figure[4](https://arxiv.org/html/2607.28939#S4.F4)displays corresponding latent momentum trajectories𝝁\\boldsymbol\{\\mu\}reconstructed across momentum\-capable formulations\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x4.png)Figure 4:Reconstructed scaled latent momentum trajectories𝝁​\(t\)=\(μ1,μ2,μ3\)\\boldsymbol\{\\mu\}\(t\)=\(\\mu\_\{1\},\\mu\_\{2\},\\mu\_\{3\}\)for the non\-quadraticS​O​\(3\)SO\(3\)rigid body, shown for all models capable of predicting momentum evolution\.To evaluate preservation of integrability, Figure[5](https://arxiv.org/html/2607.28939#S4.F5)displays 3D phase\-space trajectories corresponding to Figures[3](https://arxiv.org/html/2607.28939#S4.F3)and[4](https://arxiv.org/html/2607.28939#S4.F4)\. Because the 3D phase space possesses two independent constants of motion \(energy and Casimir\), non\-singular trajectories form closed orbits\. The proposed Hamiltonian and Lagrangian LLPNNs reconstruct closed orbits matching ground truth data with high precision\. Conversely, standard LPNets exhibit gradual orbit drift, while Neural ODEs and DeepONets rapidly depart from invariant manifolds in velocity space\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x5.png)Figure 5:3D phase\-space trajectories in latent momentum space𝝁\\boldsymbol\{\\mu\}\(left\) and observable velocity space𝝃\\boldsymbol\{\\xi\}\(right\) for non\-quadraticS​O​\(3\)SO\(3\)rigid body dynamics\. The proposed LLPNNs maintain closed invariant orbits matching the ground truth, while the standard LPNet \(dashed red line\) and Neural ODE \(dashed blue line\) solutions drift slowly away from the true trajectory; the DeepONet baseline \(dashed orange line\) diverges rapidly\.Figure[6](https://arxiv.org/html/2607.28939#S4.F6)generalizes single\-trajectory observations by plotting Mean Absolute Error \(MAE\) evaluated across5050test trajectories with random initial conditions\. The Hamiltonian LLPNN consistently achieves the lowest error in both latent \(𝝁\\boldsymbol\{\\mu\}\) and observable \(𝝃\\boldsymbol\{\\xi\}\) spaces, followed closely by the Lagrangian LLPNN, confirming superior accuracy in long\-term reconstruction\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x6.png)Figure 6:Time\-evolution of Mean Absolute Error \(MAE\) for momentum𝝁\\boldsymbol\{\\mu\}\(left\) and velocity𝝃\\boldsymbol\{\\xi\}\(right\), averaged over 50 test trajectories with random initial conditions\. Hamiltonian LLPNN achieves superior long\-term accuracy\.Figure[7](https://arxiv.org/html/2607.28939#S4.F7)evaluates the preservation of conserved quantities across all evaluated methods\. As shown in the left panel, all Lie–Poisson formulations \(Hamiltonian LLPNN, Lagrangian LLPNN, and LPNet\) preserve Casimir invariants to machine precision\. In contrast, non\-structure\-preserving baselines that lack momentum representations \(Neural ODE and DeepONet\) cannot compute Casimir or energy values\. The right panel demonstrates energy conservation error over time\. Here, the Hamiltonian LLPNN achieves near\-zero energy drift, whereas the Lagrangian LLPNN exhibits a small constant offset due to numerical differentiation in the inverse Legendre transformation, though it still maintains excellent long\-term energy stability\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x7.png)

![Refer to caption](https://arxiv.org/html/2607.28939v1/x8.png)

Figure 7:Conservation properties for non\-quadraticS​O​\(3\)SO\(3\)rigid body dynamics averaged over 50 test trajectories\. Left: Casimir conservation error over time, showing machine\-precision preservation for all Lie–Poisson methods\. Right: Relative energy drift over time, highlighting highly accurate energy conservation for both the Hamiltonian and Lagrangian LLPNNs\.

### 4\.3Kirchhoff’s Underwater Vehicle:S​E​\(3\)SE\(3\)Dynamics

#### 4\.3\.1Problem formulation

As a second application of our framework, we consider the motion of a submerged, neutrally buoyant rigid body governed by Kirchhoff’s equations\. A detailed exposition on their Lie–Poisson structure and geometric stability analysis can be found in\[[44](https://arxiv.org/html/2607.28939#bib.bib49),[43](https://arxiv.org/html/2607.28939#bib.bib76)\]\. When the vehicle’s center of gravity coincides with its center of buoyancy, the governing relations simplify while exhibiting rich dynamical behavior, spanning both integrable regimes and chaotic dynamics\[[29](https://arxiv.org/html/2607.28939#bib.bib70)\]\. This system represents a significant step up in difficulty compared toS​O​\(3\)SO\(3\)rigid body dynamics, providing a demanding test of our learning methodology due to the complex coupling between rotational and translational modes\.

The motion of Kirchhoff’s underwater vehicle is described by Lie–Poisson equations onS​E​\(3\)SE\(3\)with dual Lie algebra momentum stateμ=\(𝐉,𝐏\)∈𝔰​𝔢​\(3\)∗\\mu=\(\\mathbf\{J\},\\mathbf\{P\}\)\\in\\mathfrak\{se\}\(3\)^\{\*\}, where𝐉\\mathbf\{J\}represents angular momentum and𝐏\\mathbf\{P\}denotes linear momentum\. The classical kinetic energy HamiltonianH0​\(𝐉,𝐏\)H\_\{0\}\(\\mathbf\{J\},\\mathbf\{P\}\)splits into rotational energy with inertia tensor𝕀\\mathbb\{I\}and translational energy with mass tensor𝕄\\mathbb\{M\}:

H0​\(μ\)=H0​\(𝐉,𝐏\)=12​𝐉⋅𝕀−1​𝐉\+12​𝐏⋅𝕄−1​𝐏\.H\_\{0\}\(\\mu\)=H\_\{0\}\(\\mathbf\{J\},\\mathbf\{P\}\)=\\frac\{1\}\{2\}\\mathbf\{J\}\\cdot\\mathbb\{I\}^\{\-1\}\\mathbf\{J\}\+\\frac\{1\}\{2\}\\mathbf\{P\}\\cdot\\mathbb\{M\}^\{\-1\}\\mathbf\{P\}\\,\.\(49\)In standard formulations, the inertia matrix𝕀\\mathbb\{I\}and generalized mass matrix𝕄\\mathbb\{M\}are taken to be diagonal\. The standard physical Hamiltonian for this problem \([49](https://arxiv.org/html/2607.28939#S4.E49)\) is quadratic in momentum\.

The phase\-space geometry of the vehicle is governed by the Lie–Poisson bracket associated with the special Euclidean groupS​E​\(3\)SE\(3\), encapsulating three\-dimensional rotations and translations\. For arbitrary smooth functionsFFandHHon𝔰​𝔢​\(3\)∗\\mathfrak\{se\}\(3\)^\{\*\}, this bracket reflects the underlying semidirect product structure\[[27](https://arxiv.org/html/2607.28939#bib.bib75),[44](https://arxiv.org/html/2607.28939#bib.bib49),[28](https://arxiv.org/html/2607.28939#bib.bib145)\]:

\{F,H\}=−𝐉⋅\(∂F∂𝐉×∂H∂𝐉\)−𝐏⋅\(∂F∂𝐉×∂H∂𝐏−∂H∂𝐉×∂F∂𝐏\)\.\\\{F,H\\\}=\-\\mathbf\{J\}\\cdot\\left\(\\frac\{\\partial F\}\{\\partial\\mathbf\{J\}\}\\times\\frac\{\\partial H\}\{\\partial\\mathbf\{J\}\}\\right\)\-\\mathbf\{P\}\\cdot\\left\(\\frac\{\\partial F\}\{\\partial\\mathbf\{J\}\}\\times\\frac\{\\partial H\}\{\\partial\\mathbf\{P\}\}\-\\frac\{\\partial H\}\{\\partial\\mathbf\{J\}\}\\times\\frac\{\\partial F\}\{\\partial\\mathbf\{P\}\}\\right\)\\,\.\(50\)
Applying Lie–Poisson bracket \([50](https://arxiv.org/html/2607.28939#S4.E50)\) to state variables yields Kirchhoff’s equations of motion:

𝐉˙\\displaystyle\\dot\{\\mathbf\{J\}\}=−∂H∂𝐉×𝐉−∂H∂𝐏×𝐏,\\displaystyle=\-\\frac\{\\partial H\}\{\\partial\\mathbf\{J\}\}\\times\\mathbf\{J\}\-\\frac\{\\partial H\}\{\\partial\\mathbf\{P\}\}\\times\\mathbf\{P\}\\,,\(51\)𝐏˙\\displaystyle\\dot\{\\mathbf\{P\}\}=−∂H∂𝐉×𝐏\.\\displaystyle=\-\\frac\{\\partial H\}\{\\partial\\mathbf\{J\}\}\\times\\mathbf\{P\}\\,\.In addition to total energy conservation, system \([51](https://arxiv.org/html/2607.28939#S4.E51)\) preserves two Casimir functions associated with the dual algebra𝔰​𝔢​\(3\)∗\\mathfrak\{se\}\(3\)^\{\*\}: the squared linear momentum magnitudeC1=‖𝐏‖2C\_\{1\}=\\\|\\mathbf\{P\}\\\|^\{2\}and the momentum pairingC2=𝐉⋅𝐏C\_\{2\}=\\mathbf\{J\}\\cdot\\mathbf\{P\}\.

To test the learning capabilities of our framework on non\-quadratic Hamiltonians, which naturally arise in controlled or fluid\-coupled body dynamics, we modify the kinetic Hamiltonian to include non\-quadratic state dependencies:

H​\(μ\)=H​\(𝐉,𝐏\)=\(12​𝐉⋅𝕀−1​𝐉\+12​𝐏⋅𝕄−1​𝐏\)​\(1\+B​\(‖𝐉‖2\+‖𝐏‖2\)\),H\(\\mu\)=H\(\\mathbf\{J\},\\mathbf\{P\}\)=\\left\(\\frac\{1\}\{2\}\\mathbf\{J\}\\cdot\\mathbb\{I\}^\{\-1\}\\mathbf\{J\}\+\\frac\{1\}\{2\}\\mathbf\{P\}\\cdot\\mathbb\{M\}^\{\-1\}\\mathbf\{P\}\\right\)\\left\(1\+B\\left\(\\\|\\mathbf\{J\}\\\|^\{2\}\+\\\|\\mathbf\{P\}\\\|^\{2\}\\right\)\\right\)\\,,\(52\)whereB=0\.1B=0\.1in all numerical simulations\. Unlike lower\-dimensional examples, all six degrees of freedom in active coordinates are coupled non\-linearly\.

#### 4\.3\.2Experimental Results:S​E​\(3\)SE\(3\)Underwater Vehicle

##### Ground truth data generation for training and testing

For training, we generate 200 trajectories, each containing 101 time points\. The initial momenta are sampled from the standard normal distribution in the six\-dimensional momentum space, namely

μ0train∼𝒩​\(0,I6\),\\mu^\{\\mathrm\{train\}\}\_\{0\}\\sim\\mathcal\{N\}\(0,I\_\{6\}\),whereI6I\_\{6\}is the6×66\\times 6identity matrix\. For testing, we generate 50 trajectories, each containing 2001 time points, from the same initial momentum distribution,

μ0test∼𝒩​\(0,I6\)\.\\mu^\{\\mathrm\{test\}\}\_\{0\}\\sim\\mathcal\{N\}\(0,I\_\{6\}\)\.The time step is fixed atΔ​t=0\.1\\Delta t=0\.1throughout\. The initial orientationsR0∈S​O​\(3\)R\_\{0\}\\in SO\(3\)are sampled independently from the uniform distribution onS​O​\(3\)SO\(3\), while the initial positions are sampled from the standard normal distribution inℝ3\\mathbb\{R\}^\{3\}\.

##### Neural Network Architectures and Residual Formulations

Both Hamiltonian and Lagrangian formulations operate on all six active coordinates \(K=6K=6\) and are initialized using warm\-start baselines:

1. 1\.Hamiltonian Model \(hNNh\_\{\\text\{NN\}\}\):The Hamiltonianh​\(μ\)h\(\\mu\)is decomposed into a quadratic baseline core and a neural network residual: h​\(μ\)=12​μ⊤​𝕂​μ\+hnet​\(μ\),h\(\\mu\)=\\frac\{1\}\{2\}\\mu^\{\\top\}\\mathbb\{K\}\\mu\+h\_\{\\text\{net\}\}\(\\mu\),\(53\)where𝕂=F​F⊤∈ℝ6×6\\mathbb\{K\}=FF^\{\\top\}\\in\\mathbb\{R\}^\{6\\times 6\}is a positive\-definite matrix initialized via Cholesky factorization of the warm\-start estimate for the inverse inertia matrix\. The residual networkhnet:ℝ6→ℝh\_\{\\text\{net\}\}:\\mathbb\{R\}^\{6\}\\to\\mathbb\{R\}comprises33hidden layers with3232units per layer andtanh\\tanhactivation functions\. The output layer is a single linear unit without bias, initialized with zero weights\. To enforce strict consistency at the origin \(hnet​\(𝟎\)=0h\_\{\\text\{net\}\}\(\\mathbf\{0\}\)=0and∇hnet​\(𝟎\)=𝟎\\nabla h\_\{\\text\{net\}\}\(\\mathbf\{0\}\)=\\mathbf\{0\}\), a linear correction is applied to raw network outputrhr\_\{h\}: hnet​\(μ\)=rh​\(μ\)−rh​\(𝟎\)−∇rh​\(𝟎\)⊤​μ\.h\_\{\\text\{net\}\}\(\\mu\)=r\_\{h\}\(\\mu\)\-r\_\{h\}\(\\mathbf\{0\}\)\-\\nabla r\_\{h\}\(\\mathbf\{0\}\)^\{\\top\}\\mu\.\(54\)The predicted velocity field is thusξpred=∇μh​\(μ\)=A0​μ\+∇hnet​\(μ\)\\xi\_\{\\text\{pred\}\}=\\nabla\_\{\\mu\}h\(\\mu\)=A\_\{0\}\\mu\+\\nabla h\_\{\\text\{net\}\}\(\\mu\)\.
2. 2\.Lagrangian Model \(ℓNN\\ell\_\{\\text\{NN\}\}\):Similarly, the direct Lagrangian modelℓNN:ℝ6→ℝ\\ell\_\{\\text\{NN\}\}:\\mathbb\{R\}^\{6\}\\to\\mathbb\{R\}acts on body velocitiesξ=\(𝛀,𝐯\)∈𝔰​𝔢​\(3\)\\xi=\(\\boldsymbol\{\\Omega\},\\mathbf\{v\}\)\\in\\mathfrak\{se\}\(3\): ℓNN​\(ξ\)=12​ξ⊤​𝔹​ξ\+ℓnet​\(ξ\),\\ell\_\{\\text\{NN\}\}\(\\xi\)=\\frac\{1\}\{2\}\\xi^\{\\top\}\\mathbb\{B\}\\xi\+\\ell\_\{\\text\{net\}\}\(\\xi\),\(55\)where𝔹=12​\(𝔹core\+𝔹core⊤\)∈ℝ6×6\\mathbb\{B\}=\\frac\{1\}\{2\}\(\\mathbb\{B\}\_\{\\text\{core\}\}\+\\mathbb\{B\}\_\{\\text\{core\}\}^\{\\top\}\)\\in\\mathbb\{R\}^\{6\\times 6\}is a trainable symmetric mass matrix initialized toMwarmM\_\{\\text\{warm\}\}\. Residual networkℓnet\\ell\_\{\\text\{net\}\}mirrorshneth\_\{\\text\{net\}\}with identical hidden layer sizing, zero\-initialized output weights, and origin constraints: ℓnet​\(ξ\)=rℓ​\(ξ\)−rℓ​\(𝟎\)−∇rℓ​\(𝟎\)⊤​ξ\.\\ell\_\{\\text\{net\}\}\(\\xi\)=r\_\{\\ell\}\(\\xi\)\-r\_\{\\ell\}\(\\mathbf\{0\}\)\-\\nabla r\_\{\\ell\}\(\\mathbf\{0\}\)^\{\\top\}\\xi\.\(56\)

##### Optimization and Training Performance

All models are implemented in 64\-bit precision \(tf\.float64\) and trained for20,00020\{,\}000epochs using the Adam optimizer, with training parameters detailed in Table[2](https://arxiv.org/html/2607.28939#S3.T2)\. Each trajectory reconstruction usesNfit=50N\_\{\\text\{fit\}\}=50fit steps to determine the Noether constantp0p\_\{0\}for that trajectory\. Throughout optimization, the loss functions \([23](https://arxiv.org/html/2607.28939#S3.E23)\) and \([32](https://arxiv.org/html/2607.28939#S3.E32)\) typically drop by approximately four orders of magnitude, decaying from initial values around10−310^\{\-3\}to the range10−610^\{\-6\}–10−710^\{\-7\}, with some runs converging to even lower loss values\.

##### Energy and Legendre Regularity

For the learned Lagrangian modelℓ​\(ξ\)\\ell\(\\xi\), the total energyE​\(ξ\)E\(\\xi\)is evaluated via the Legendre transform:

E​\(ξ\)=⟨ξ,∂ℓ∂ξ⟩−ℓ​\(ξ\),E\(\\xi\)=\\left\\langle\\xi,\\frac\{\\partial\\ell\}\{\\partial\\xi\}\\right\\rangle\-\\ell\(\\xi\),\(57\)where

⟨\(𝛀,𝐯\),\(𝐉,𝐏\)⟩=𝛀⋅𝐉\+𝐯⋅𝐏\\left\\langle\(\\boldsymbol\{\\Omega\},\\mathbf\{v\}\),\(\\mathbf\{J\},\\mathbf\{P\}\)\\right\\rangle=\\boldsymbol\{\\Omega\}\\cdot\\mathbf\{J\}\+\\mathbf\{v\}\\cdot\\mathbf\{P\}is the natural pairing between the Lie algebra𝔰​𝔢​\(3\)\\mathfrak\{se\}\(3\)and its dual𝔰​𝔢​\(3\)∗\\mathfrak\{se\}\(3\)^\{\*\}\. In this formulation, bothℓ​\(ξ\)\\ell\(\\xi\)andh​\(μ\)h\(\\mu\)are regular \(their Hessians∇ξ2ℓ\\nabla\_\{\\xi\}^\{2\}\\elland∇μ2h\\nabla\_\{\\mu\}^\{2\}hare positive\-definite\)\. Consequently, the Legendre mappingξ↦μ=∂ℓ∂ξ\\xi\\mapsto\\mu=\\frac\{\\partial\\ell\}\{\\partial\\xi\}is locally invertible, ensuring that energyE​\(ξ\)E\(\\xi\)is well\-defined across the domain\.

##### Trajectory and Conservation Analysis

Figure[8](https://arxiv.org/html/2607.28939#S4.F8)demonstrates the time\-series for the six observable velocity componentsξ=\(𝛀,𝐯\)=\(Ωx,Ωy,Ωz,vx,vy,vz\)\\xi=\(\\boldsymbol\{\\Omega\},\\mathbf\{v\}\)=\(\\Omega\_\{x\},\\Omega\_\{y\},\\Omega\_\{z\},v\_\{x\},v\_\{y\},v\_\{z\}\)on a representative test trajectory\. The Hamiltonian LLPNN captures both high\-frequency rotational oscillations \(𝛀\\boldsymbol\{\\Omega\}\) and coupled translational velocity dynamics \(𝐯\\mathbf\{v\}\), maintaining near\-perfect alignment with ground\-truth curves\. Lagrangian LLPNN and standard LPNet models follow closely, whereas DeepONet and Neural ODE predictions degrade rapidly as integration time increases\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x9.png)Figure 8:Predicted versus ground\-truth body velocity componentsξ=\(𝛀,𝐯\)=\(Ωx,Ωy,Ωz,vx,vy,vz\)\\xi=\(\\boldsymbol\{\\Omega\},\\mathbf\{v\}\)=\(\\Omega\_\{x\},\\Omega\_\{y\},\\Omega\_\{z\},v\_\{x\},v\_\{y\},v\_\{z\}\)for non\-quadraticS​E​\(3\)SE\(3\)Kirchhoff vehicle dynamics over time\. Proposed Hamiltonian LLPNN \(green dash\-dotted line\) stays closely aligned with ground truth \(blue solid line\) across all six components\.Figure[9](https://arxiv.org/html/2607.28939#S4.F9)presents the reconstructed latent body momentum statesμ=\(𝐉,𝐏\)=\(Jx,Jy,Jz,Px,Py,Pz\)\\mu=\(\\mathbf\{J\},\\mathbf\{P\}\)=\(J\_\{x\},J\_\{y\},J\_\{z\},P\_\{x\},P\_\{y\},P\_\{z\}\)corresponding to the velocity trajectories shown in Figure[8](https://arxiv.org/html/2607.28939#S4.F8)\. The Hamiltonian and Lagrangian LLPNN models maintain phase accuracy across all momentum coordinates\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x10.png)Figure 9:Reconstructed latent body momentum trajectoriesμ=\(𝐉,𝐏\)=\(Jx,Jy,Jz,Px,Py,Pz\)\\mu=\(\\mathbf\{J\},\\mathbf\{P\}\)=\(J\_\{x\},J\_\{y\},J\_\{z\},P\_\{x\},P\_\{y\},P\_\{z\}\)for the non\-quadraticS​E​\(3\)SE\(3\)Kirchhoff vehicle across momentum\-capable formulations\.Figure[10](https://arxiv.org/html/2607.28939#S4.F10)summarizes generalization performance across5050test trajectories with randomized initial conditions\. Mean Absolute Error \(MAE\) is shown separately for latent momentum \(μ\\mu\) and observable velocity \(ξ\\xi\) coordinates\. The Hamiltonian LLPNN achieves the lowest error growth over long horizons, followed by the Lagrangian LLPNN\. On observable velocities, Neural ODE performs comparably to standard LPNet but accumulates error significantly faster than the proposed LLPNN models, while DeepONet diverges rapidly\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x11.png)Figure 10:Time evolution of Mean Absolute Error \(MAE\) for body momentumμ\\mu\(left\) and body velocityξ\\xi\(right\), averaged over 50 test trajectories with random initial conditions forS​E​\(3\)SE\(3\)Kirchhoff dynamics\.Figure[11](https://arxiv.org/html/2607.28939#S4.F11)evaluates conservation properties across5050test trajectories\. The left and middle panels demonstrate that all Lie–Poisson structure\-preserving formulations \(Hamiltonian LLPNN, Lagrangian LLPNN, and LPNet\) preserve both quadratic Casimir invariantsC1=‖𝐏‖2C\_\{1\}=\\\|\\mathbf\{P\}\\\|^\{2\}andC2=𝐉⋅𝐏C\_\{2\}=\\mathbf\{J\}\\cdot\\mathbf\{P\}to machine precision\. The right panel evaluates total energy conservation error over time\. The Hamiltonian LLPNN exhibits minimal energy drift, while the Lagrangian LLPNN maintains long\-term energy stability despite a minor initial offset caused by numerical gradients in the inverse Legendre map\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x12.png)Figure 11:Conservation error metrics for non\-quadraticS​E​\(3\)SE\(3\)Kirchhoff vehicle dynamics averaged over 50 test trajectories\. Left and middle panels: Casimir invariant errors \(C1=‖𝐏‖2C\_\{1\}=\\\|\\mathbf\{P\}\\\|^\{2\}andC2=𝐉⋅𝐏C\_\{2\}=\\mathbf\{J\}\\cdot\\mathbf\{P\}\), demonstrating machine\-precision preservation for all Lie–Poisson methods\. Right panel: Mean Absolute Error in energy conservation over time\.

### 4\.4Multi\-Vehicle Optimal Control:SE​\(2\)N\\mathrm\{SE\}\(2\)^\{N\}Dynamics

#### 4\.4\.1Background: Optimal Control and Symmetry Reduction

As a third and crucial benchmark demonstrating the capability of our framework, we consider interacting autonomous agents evolving on a Lie group\. Specifically, we study a system of planar autonomous vehicles evolving onS​E​\(2\)SE\(2\), the special Euclidean group of rotations and translations in the plane\. This example demonstrates the performance of our method for degenerate Hamiltonians, a structure typical of optimal\-control formulations but rarely encountered in purely mechanical systems\. We briefly summarize the underlying geometric framework to make the problem setting clear; detailed derivations can be found in\[[36](https://arxiv.org/html/2607.28939#bib.bib15),[37](https://arxiv.org/html/2607.28939#bib.bib17)\]\.

##### Pontryagin’s Maximum Principle

For a controlled dynamical systemq˙=a​\(q,u,t\)\\dot\{q\}=a\(q,u,t\), Pontryagin’s Maximum Principle \(PMP\) converts an optimal control problem into a Hamiltonian system by introducing an auxiliary co\-state \(or adjoint variable\)pp\. The resulting Hamiltonian generates coupled canonical evolution equations for stateqqand co\-statepp, accompanied by an optimality condition for control inputuuthrough∇uH=𝟎\\nabla\_\{u\}H=\\mathbf\{0\}\. In this sense, PMP plays the same fundamental role in optimal control as Hamilton’s equations play in analytical mechanics\.

Crucially, co\-states are mathematical artifacts generated by the optimization formulation\. In practical applications, they are generally unobservable unless full knowledge of the optimal control model is available; they are completely absent in our observational setting, where only configurationggor velocityξ\\xion trajectories are recorded\. Consequently, neither the explicit control law nor co\-state dynamics are assumed known during learning\.

##### Symmetry Reduction

When agents evolve on a Lie groupGG, the configuration manifold of anNN\-agent ensemble isGNG^\{N\}\. Direct application of PMP yields dynamics on the full state/co\-state space, whose dimension scales rapidly withNN\. However, when the underlying optimal control problem exhibits Lie\-group invariance, PMP dynamics can be reduced via geometric symmetry reduction theory\[[40](https://arxiv.org/html/2607.28939#bib.bib20),[26](https://arxiv.org/html/2607.28939#bib.bib119),[47](https://arxiv.org/html/2607.28939#bib.bib214)\]\.

For general interacting systems, symmetry reduction yields momentum variables coupled to relative spatial configurations\. A key result of\[[36](https://arxiv.org/html/2607.28939#bib.bib15),[37](https://arxiv.org/html/2607.28939#bib.bib17)\]establishes that for the class of optimal control problems considered here, the reduced dynamics close entirely on symmetry\-reduced momenta\. Consequently, the reduced evolution is governed by a Lie–Poisson system on\(𝔤∗\)N\(\\mathfrak\{g\}^\{\*\}\)^\{N\}, eliminating the need to explicitly track relative agent orientations\.

This reduction offers substantial practical savings: rather than learning dynamics on a high\-dimensional state/co\-state space, the evolution simplifies to a lower\-dimensional Lie–Poisson system structured by group symmetries\. Throughout this study, we assume the symmetry class of the configuration group is known, while control laws, cost functionals, and co\-state trajectories remain entirely unknown\.

##### Group Configuration and Inactive Momentum Variables

For each vehiclek∈\{1,…,N\}k\\in\\\{1,\\dots,N\\\}, the planar configuration matrixgk∈SE​\(2\)g\_\{k\}\\in\\mathrm\{SE\}\(2\)and its body\-fixed momentum vectorμk∈𝔰​𝔢​\(2\)∗≅ℝ3\\mu\_\{k\}\\in\\mathfrak\{se\}\(2\)^\{\*\}\\cong\\mathbb\{R\}^\{3\}are given by:

gk=\(cos⁡θk−sin⁡θkxksin⁡θkcos⁡θkyk001\),μk=\(μk​1μk​2μk​3\),g\_\{k\}=\\begin\{pmatrix\}\\cos\\theta\_\{k\}&\-\\sin\\theta\_\{k\}&x\_\{k\}\\\\ \\sin\\theta\_\{k\}&\\cos\\theta\_\{k\}&y\_\{k\}\\\\ 0&0&1\\end\{pmatrix\},\\quad\\mu\_\{k\}=\\begin\{pmatrix\}\\mu\_\{k1\}\\\\ \\mu\_\{k2\}\\\\ \\mu\_\{k3\}\\end\{pmatrix\},\(58\)whereμk​1\\mu\_\{k1\}andμk​2\\mu\_\{k2\}denote longitudinal and lateral linear body momenta, respectively, andμk​3\\mu\_\{k3\}denotes angular body momentum\. The total momentum state of theNN\-vehicle ensemble is the concatenated vectorμ=\(μ1⊤,…,μN⊤\)⊤∈ℝ3​N\\mu=\(\\mu\_\{1\}^\{\\top\},\\dots,\\mu\_\{N\}^\{\\top\}\)^\{\\top\}\\in\\mathbb\{R\}^\{3N\}\.

The Poisson structure matrixΛ​\(μ\)\\Lambda\(\\mu\)for this system is block\-diagonal:

Λ​\(μ\)=diag⁡\(Λ1​\(μ1\),…,ΛN​\(μN\)\),\\Lambda\(\\mu\)=\\operatorname\{diag\}\\bigl\(\\Lambda\_\{1\}\(\\mu\_\{1\}\),\\dots,\\Lambda\_\{N\}\(\\mu\_\{N\}\)\\bigr\),\(59\)where each3×33\\times 3blockΛk​\(μk\)\\Lambda\_\{k\}\(\\mu\_\{k\}\)is given by:

Λk​\(μk\)=\(00−μk​200μk​1μk​2−μk​10\)\.\\Lambda\_\{k\}\(\\mu\_\{k\}\)=\\begin\{pmatrix\}0&0&\-\\mu\_\{k2\}\\\\ 0&0&\\mu\_\{k1\}\\\\ \\mu\_\{k2\}&\-\\mu\_\{k1\}&0\\end\{pmatrix\}\.\(60\)Consequently, the dynamics of thekk\-th vehicle satisfy:

μ˙k=Λk​\(μk\)​∇μkh​\(μ\),\\dot\{\\mu\}\_\{k\}=\\Lambda\_\{k\}\(\\mu\_\{k\}\)\\nabla\_\{\\mu\_\{k\}\}h\(\\mu\),\(61\)or, expressed explicitly in scalar components:

μ˙k​1\\displaystyle\\dot\{\\mu\}\_\{k1\}=−μk​2​∂h∂μk​3,\\displaystyle=\-\\mu\_\{k2\}\\frac\{\\partial h\}\{\\partial\\mu\_\{k3\}\},\(62\)μ˙k​2\\displaystyle\\dot\{\\mu\}\_\{k2\}=μk​1​∂h∂μk​3,\\displaystyle=\\mu\_\{k1\}\\frac\{\\partial h\}\{\\partial\\mu\_\{k3\}\},\(63\)μ˙k​3\\displaystyle\\dot\{\\mu\}\_\{k3\}=μk​2​∂h∂μk​1−μk​1​∂h∂μk​2\.\\displaystyle=\\mu\_\{k2\}\\frac\{\\partial h\}\{\\partial\\mu\_\{k1\}\}\-\\mu\_\{k1\}\\frac\{\\partial h\}\{\\partial\\mu\_\{k2\}\}\.\(64\)
The collective multi\-vehicle Hamiltonianh​\(μ\)h\(\\mu\)governing target ensemble dynamics is defined as\[[37](https://arxiv.org/html/2607.28939#bib.bib17)\]:

h​\(μ\)=∑k=1Nμk​1\+12​∑k,m=1NΨk​m​μk​3​μm​3,h\(\\mu\)=\\sum\_\{k=1\}^\{N\}\\mu\_\{k1\}\+\\frac\{1\}\{2\}\\sum\_\{k,m=1\}^\{N\}\\Psi\_\{km\}\\mu\_\{k3\}\\mu\_\{m3\},\(65\)whereΨ∈ℝN×N\\Psi\\in\\mathbb\{R\}^\{N\\times N\}represents a symmetric steering coordination matrix\. We consider a fully coupled network topology \(the*Democracy*topology\)\. For a three\-agent ensemble \(N=3N=3\), the graph LaplacianBBand coordination matrixΨ\\Psi\(evaluated with coupling parameterχ=0\.5\\chi=0\.5\) are:

B=\[2−1−1−12−1−1−12\],Ψ=\(I3\+2​χ​B\)−1=\(I3\+B\)−1=\[121414141214141412\]\.B=\\begin\{bmatrix\}2&\-1&\-1\\\\ \-1&2&\-1\\\\ \-1&\-1&2\\end\{bmatrix\},\\quad\\Psi=\(I\_\{3\}\+2\\chi B\)^\{\-1\}=\(I\_\{3\}\+B\)^\{\-1\}=\\begin\{bmatrix\}\\frac\{1\}\{2\}&\\frac\{1\}\{4\}&\\frac\{1\}\{4\}\\\\\[2\.0pt\] \\frac\{1\}\{4\}&\\frac\{1\}\{2\}&\\frac\{1\}\{4\}\\\\\[2\.0pt\] \\frac\{1\}\{4\}&\\frac\{1\}\{4\}&\\frac\{1\}\{2\}\\end\{bmatrix\}\.\(66\)
A distinguishing property of this Lie–Poisson system is the existence of passive variables in momentum space\. Because networking terms inh​\(μ\)h\(\\mu\)depend exclusively on angular momentum componentsμk​3\\mu\_\{k3\}, translational componentsμk​1\\mu\_\{k1\}andμk​2\\mu\_\{k2\}remain uncoupled from network interactions\. Evaluating energy gradients yields Lie algebra velocitiesξk=∇μkh​\(μ\)∈𝔰​𝔢​\(2\)\\xi\_\{k\}=\\nabla\_\{\\mu\_\{k\}\}h\(\\mu\)\\in\\mathfrak\{se\}\(2\):

ξk=\(ξk​1ξk​2ξk​3\)=\(1\.00\.0∑m=1NΨk​m​μm​3\)\.\\xi\_\{k\}=\\begin\{pmatrix\}\\xi\_\{k1\}\\\\ \\xi\_\{k2\}\\\\ \\xi\_\{k3\}\\end\{pmatrix\}=\\begin\{pmatrix\}1\.0\\\\ 0\.0\\\\ \\sum\_\{m=1\}^\{N\}\\Psi\_\{km\}\\mu\_\{m3\}\\end\{pmatrix\}\.\(67\)The longitudinal momentum coordinatesμk​1\\mu\_\{k1\}generate uniform, constant forward drive \(ξk​1=1\.0\\xi\_\{k1\}=1\.0\), while lateral componentsμk​2\\mu\_\{k2\}receive zero energetic drive \(ξk​2=0\.0\\xi\_\{k2\}=0\.0\)\. Thus, translational terms enter linearly or are absent in the Hamiltonian, isolating active coordination dynamics to the angular momentum subspace\.

##### Non\-Linear Generalized Benchmark Hamiltonian

To make the learning task more demanding and represent complex control costs, we introduce a non\-quadratic modification to \([65](https://arxiv.org/html/2607.28939#S4.E65)\):

Htrue​\(μ\)=∑k=1Nμ1​k\+12​\(∑k,m=13Ψk​m​μk​3​μm​3\)​\(1\+B​∑k=13μk​32\),H\_\{\\text\{true\}\}\(\\mu\)=\\sum\_\{k=1\}^\{N\}\\mu\_\{1k\}\+\\frac\{1\}\{2\}\\left\(\\sum\_\{k,m=1\}^\{3\}\\Psi\_\{km\}\\mu\_\{k3\}\\mu\_\{m3\}\\right\)\\left\(1\+B\\sum\_\{k=1\}^\{3\}\\mu\_\{k3\}^\{2\}\\right\)\\,,\(68\)whereB=0\.01B=0\.01in all numerical experiments\.

#### 4\.4\.2Ground truth data generation for training and testing

For training, we generate 200 trajectories, each containing 101 time points\. Since the system consists ofN=3N=3interacting vehicles onS​E​\(2\)SE\(2\), the latent momentum variable has dimension3​N=93N=9\. The initial momenta are sampled from the standard normal distribution,

μ0train∼𝒩​\(0,I9\),\\mu^\{\\mathrm\{train\}\}\_\{0\}\\sim\\mathcal\{N\}\(0,I\_\{9\}\),whereI9I\_\{9\}is the9×99\\times 9identity matrix\. For testing, we generate 50 trajectories, each containing 2001 time points, from the same initial momentum distribution,

μ0test∼𝒩​\(0,I9\)\.\\mu^\{\\mathrm\{test\}\}\_\{0\}\\sim\\mathcal\{N\}\(0,I\_\{9\}\)\.Just as in the other examples, the time step is fixed atΔ​t=0\.1\\Delta t=0\.1throughout\. For each vehicle, the initial orientation angle is sampled independently from the uniform distribution on\[0,2​π\)\[0,2\\pi\), while the initial position is sampled from the standard normal distribution inℝ2\\mathbb\{R\}^\{2\}\.

#### 4\.4\.3Robustness to Observational Noise

To assess practical applicability under sensory degradation, observation data are corrupted using noise scale parameterη\\eta\. For each vehicle, additive noise is added to Lie algebra velocities:ξk,noisy=ξk\+𝒩​\(𝟎,η2​𝐈3\)\\xi\_\{k,\\text\{noisy\}\}=\\xi\_\{k\}\+\\mathcal\{N\}\(\\mathbf\{0\},\\eta^\{2\}\\mathbf\{I\}\_\{3\}\)\. Concurrently, configuration matrices are perturbed within their body frame via multiplicative intrinsic perturbations:

gk,noisy=gk⋅exp⁡\(ω^k\),ω^k=\(0−d​θk/20d​θk/200000\),g\_\{k,\\text\{noisy\}\}=g\_\{k\}\\cdot\\exp\(\\widehat\{\\omega\}\_\{k\}\),\\quad\\widehat\{\\omega\}\_\{k\}=\\begin\{pmatrix\}0&\-d\\theta\_\{k\}/\\sqrt\{2\}&0\\\\ d\\theta\_\{k\}/\\sqrt\{2\}&0&0\\\\ 0&0&0\\end\{pmatrix\},\(69\)whered​θk∼𝒩​\(0,η2\)d\\theta\_\{k\}\\sim\\mathcal\{N\}\(0,\\eta^\{2\}\)represents stochastic orientation noise\. This noise model evaluates the framework’s ability to recover true underlying mechanics and hidden initial momenta from degraded observations\. We highlight results forη=0\\eta=0\(clean data\) andη=0\.005\\eta=0\.005\(noisy data\); evaluations across intermediate noise levelsη∈\[0\.001,0\.005\]\\eta\\in\[0\.001,0\.005\]yield consistent trends\.

#### 4\.4\.4Experimental Results

The learning procedure processes trajectory data generated onSE​\(2\)N\\mathrm\{SE\}\(2\)^\{N\}under Democracy coupling\[[32](https://arxiv.org/html/2607.28939#bib.bib129)\]\(all particles coupled to each other\) forN=3N=3\. Dynamic variable isolation is performed by identifying coordinates with non\-zero variance across trajectories, partitioning velocity and momentum states into active and passive components:ξ=\(ξact,ξpass\)\\xi=\(\\xi^\{\\text\{act\}\},\\xi^\{\\text\{pass\}\}\)andμ=\(μact,μpass\)\\mu=\(\\mu^\{\\text\{act\}\},\\mu^\{\\text\{pass\}\}\)\.

A global quadratic warm start is executed across all training trajectories to establish an initial quadratic baseline and estimate initial spatial co\-statesp0p\_\{0\}\. This step solves a constrained linear Karush–Kuhn–Tucker \(KKT\) system\[[50](https://arxiv.org/html/2607.28939#bib.bib123)\]to yield a symmetric positive\-definite metric matrix𝔹\\mathbb\{B\}\(subject totr⁡𝔹=1\\operatorname\{tr\}\\mathbb\{B\}=1\) along with initial spatial momentap0p\_\{0\}per trajectory\. As in previous examples, training considers two LLPNN architectures:

1. 1\.Lagrangian LLPNN:The residual Lagrangianℓ​\(ξact\)=12​ξact⋅𝔹​ξact\+ℛL​\(ξact\)\\ell\(\\xi^\{\\text\{act\}\}\)=\\frac\{1\}\{2\}\\xi^\{\\text\{act\}\}\\cdot\\mathbb\{B\}\\xi^\{\\text\{act\}\}\+\\mathcal\{R\}\_\{L\}\(\\xi^\{\\text\{act\}\}\)is optimized via gradient descent by matching predicted coadjoint\-projected spatial momenta against∂ℓ∂ξact\\frac\{\\partial\\ell\}\{\\partial\\xi^\{\\text\{act\}\}\}\.
2. 2\.Hamiltonian LLPNN:The Hamiltonianh​\(μ\)h\(\\mu\)is initialized using warm\-start inverse metric𝕂=𝔹−1\\mathbb\{K\}=\\mathbb\{B\}^\{\-1\}and optimized alongside Lie–Poisson flow parameter networks \(LPNet\) to enforce structure preservation\.

Figure[12](https://arxiv.org/html/2607.28939#S4.F12)compares the reconstructed body velocity componentsξ​\(t\)\\xi\(t\)across evaluated methods against the ground truth for a representative trajectory\. In this system, only the angular velocity components \(ξk​3\\xi\_\{k3\}\) are active and evolve dynamically, whereas the translational velocity components \(ξk​1,ξk​2\\xi\_\{k1\},\\xi\_\{k2\}\) remain constant passive variables\. The Lagrangian LLPNN accurately tracks angular velocity dynamics over a long time horizon \(t=100t=100\), whereas the Hamiltonian LLPNN exhibits minor phase drift aftert≈40t\\approx 40\. As expected from the discussion in Section[2\.2](https://arxiv.org/html/2607.28939#S2.SS2), non\-structure\-preserving baselines \(DeepONet and Neural ODE\) diverge rapidly from the ground truth\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x13.png)Figure 12:Time evolution of body velocity componentsξ=\(ξk​1,ξk​2,ξk​3\)\\xi=\(\\xi\_\{k1\},\\xi\_\{k2\},\\xi\_\{k3\}\)for each vehicle \(N=3N=3\)\. Passive translational velocities \(ξk​1,ξk​2\\xi\_\{k1\},\\xi\_\{k2\}\) remain constant, while active steering velocities \(ξk​3\\xi\_\{k3\}\) evolve dynamically\. Proposed Lagrangian LLPNN \(dotted magenta line\) maintains high\-precision alignment with ground truth \(blue solid line\) over long integration horizons\.Figure[13](https://arxiv.org/html/2607.28939#S4.F13)presents corresponding reconstructed momentum componentsμ​\(t\)\\mu\(t\)\. Unlike velocity space, passive momentum components evolve over time due to geometric cross\-coupling in the Lie–Poisson bracket\. The Lagrangian LLPNN maintains close agreement with true momentum trajectories throughout the evaluated horizon\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x14.png)Figure 13:Time evolution of scaled body momentum componentsμ​\(t\)\\mu\(t\)for each vehicle \(N=3N=3\)\. Both active \(μk​3\\mu\_\{k3\}\) and passive \(μk​1,μk​2\\mu\_\{k1\},\\mu\_\{k2\}\) momentum channels evolve dynamically due to Lie–Poisson bracket coupling\. The Lagrangian LLPNN \(dotted magenta line\) closely tracks the ground\-truth trajectory across the entire evaluation horizon\.Figure[14](https://arxiv.org/html/2607.28939#S4.F14)illustrates reconstructed 2D spatial trajectories\(xk​\(t\),yk​\(t\)\)\(x\_\{k\}\(t\),y\_\{k\}\(t\)\)obtained by integrating predicted velocities for the trajectory shown in Figure[12](https://arxiv.org/html/2607.28939#S4.F12)\. The Lagrangian LLPNN reproduces intricate multi\-vehicle flight paths over long integration windows \(t=100t=100\) with negligible spatial drift, while standard LPNet exhibits trajectory divergence over shorter time intervals\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x15.png)Figure 14:Reconstructed 2D spatial trajectories\(xk​\(t\),yk​\(t\)\)\(x\_\{k\}\(t\),y\_\{k\}\(t\)\)forN=3N=3vehicles on theS​E​\(2\)SE\(2\)plane corresponding to the predictions in Figure[12](https://arxiv.org/html/2607.28939#S4.F12)\. The Lagrangian LLPNN \(dotted magenta line\) remains visually indistinguishable from ground\-truth paths \(solid blue line\) up tot=100t=100\.To assess overall quality of learning, Figure[15](https://arxiv.org/html/2607.28939#S4.F15)presents Mean Absolute Error \(MAE\) for momentumμ\\muand velocityξ\\xiaveraged across5050test trajectories with random initial conditions, under both noiseless \(η=0\\eta=0\) and noisy \(η=0\.005\\eta=0\.005\) conditions\. On average, the Hamiltonian LLPNN achieves the lowest mean error growth across the ensemble, closely followed by the Lagrangian LLPNN\. Non\-geometric baselines struggle on this task due to singular velocity\-space vector fields induced by Hamiltonian degeneracy\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x16.png)Figure 15:Mean Absolute Error \(MAE\) for scaled momentumμ\\mu\(left\) and velocityξ\\xi\(right\) averaged over 50 test trajectories, evaluated under clean \(η=0\\eta=0\) and noisy \(η=0\.005\\eta=0\.005\) observational conditions\. Non\-structure\-preserving baselines, DeepONet \(dashed orange line\) and Neural ODEs \(dash\-dotted cyan line\), diverge rapidly over time\. On average, the Hamiltonian LLPNN \(dashed green line\) achieves the highest predictive accuracy across both latent momenta and observable velocities\.For each vehiclekk, the individual Casimir invariant is given byCk=μk​12\+μk​22C\_\{k\}=\\mu\_\{k1\}^\{2\}\+\\mu\_\{k2\}^\{2\}\. Figure[16](https://arxiv.org/html/2607.28939#S4.F16)confirms that all Lie–Poisson methods preserve individual particle Casimirs to machine precision throughout long\-horizon integration\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x17.png)Figure 16:Casimir invariant conservation error\|Ck​\(t\)−Ck​\(0\)\|\|C\_\{k\}\(t\)\-C\_\{k\}\(0\)\|evaluated across individual vehicle momentum components, confirming machine\-precision preservation for Lie–Poisson formulations\.Figure[17](https://arxiv.org/html/2607.28939#S4.F17)illustrates energy conservation performance for the Hamiltonian LLPNN\. Even under non\-quadratic, degenerate Hamiltonian dynamics \([68](https://arxiv.org/html/2607.28939#S4.E68)\), the Hamiltonian LLPNN maintains long\-term energy preservation with low relative drift across clean and noisy training regimes\. Energy errors for the Lagrangian model are omitted here due to the ill\-posedness of standard Legendre transforms on degenerate momentum directions\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x18.png)Figure 17:Relative Hamiltonian conservation drift\|H​\(t\)−H​\(0\)\|/\|H​\(0\)\|\|H\(t\)\-H\(0\)\|/\|H\(0\)\|for the Hamiltonian LLPNN under clean \(η=0\\eta=0\) and noisy \(η=0\.005\\eta=0\.005\) conditions\.Finally, Figure[18](https://arxiv.org/html/2607.28939#S4.F18)directly compares level sets of learned versus true Hamiltonian functions in active momentum space\. The left panel shows contour lines sliced atμ1,3=0\.5\\mu\_\{1,3\}=0\.5, demonstrating excellent agreement between the predicted and true Hamiltonian\. The right panel displays absolute error contours, confirming that energy function discrepancies across active momentum regions remain below10−310^\{\-3\}\.

![Refer to caption](https://arxiv.org/html/2607.28939v1/x19.png)Figure 18:Hamiltonian level\-set comparison in active momentum space \(μ23\\mu\_\{23\}vs\.μ33\\mu\_\{33\}\) at sliceμ13=0\.5\\mu\_\{13\}=0\.5\. Left: Overlay of true \(solid\) and learned \(dashed\) Hamiltonian level sets\. Right: Absolute approximation error\|Hlearned−Htrue\|\|H\_\{\\text\{learned\}\}\-H\_\{\\text\{true\}\}\|, demonstrating uniform accuracy with errors bounded near∼10−4\\sim 10^\{\-4\}\.

## 5Limitations and Computational Considerations

##### Computational efficiency

The proposed LLPNN framework achieves high sample and parameter efficiency by operating on a compact network architecture\. With each constituent network comprising three hidden layers of width 32, the parameter footprint remains modest \(≈2​K\\approx 2\\,\\text\{K\}parameters per network, supplemented by a few parameters specifying the quadratic core for the warm\-start initialization\)\. Due to the relatively low\-dimensional structure\-preserving parameterization, training remained computationally modest\. For the Kirchhoff problem onS​E​\(3\)SE\(3\), a complete training cycle of20,00020\{,\}000epochs required approximately400400s for the Lagrangian LLPNN and approximately525525s for the Hamiltonian LLPNN on standard consumer hardware \(Apple M2 Max\); the timings for the other examples were comparable\.

Despite these practical advantages, the framework possesses certain theoretical boundary conditions and practical limitations that warrant discussion\.

##### Limitation 1: Assumption of known symmetry

Our formulation explicitly assumes that the underlying Lie group symmetryGGof the physical system is known*a priori*\. While this assumption holds for a broad class of conservative mechanical, fluid, and rigid\-body control systems, it may be restrictive in certain optimal control settings\. For instance, multi\-agent vehicles actively searching for a target fixed in space exhibit symmetry\-breaking cost functions\[[36](https://arxiv.org/html/2607.28939#bib.bib15),[37](https://arxiv.org/html/2607.28939#bib.bib17)\], preventing complete symmetry reduction\. Extending the LLPNN framework to account for such symmetry\-breaking dynamics\[[5](https://arxiv.org/html/2607.28939#bib.bib128)\]remains an important open problem\.

##### Limitation 2: Extension to general Poisson manifolds

While LLPNN is engineered specifically for Lie–Poisson dynamics, one may naturally inquire about its applicability to general Poisson systems\. At present, the method cannot be directly applied to arbitrary Poisson manifolds because generic Poisson brackets lack the explicit Lie\-algebraic coadjoint reconstruction formula \([10](https://arxiv.org/html/2607.28939#S2.E10)\) that forms the core of our latent variable reconstruction procedure\. Developing a generalized geometric momentum\-fitting technique for non\-Lie Poisson manifolds represents an intriguing theoretical challenge\.

## 6Conclusion and Future Work

In this paper, we presented a physics\-informed geometric learning framework designed to infer Lie–Poisson Hamiltonian dynamics directly from observable data, without requiring measurements of latent momentum variables\. The core paradigm relies on the structural assumption that the system possesses underlying Lie–Poisson dynamics generated by a Hamiltonian\. Crucially, our framework explicitly accommodates degenerate Hamiltonians, which lack dual Lagrangian representations or standard Euler–Poincaré formulations in observable velocity space\. For such systems, the observable variables need not form a closed dynamical system and may not admit an autonomous evolution equation\. LLPNNs circumvent this difficulty by reconstructing the latent Lie–Poisson dynamics through Noether invariants and coadjoint reconstruction\. By embedding the Lie\-algebraic structure directly into the neural network architecture, our framework guarantees exact structural preservation of Lie–Poisson brackets and achieves machine\-precision conservation of Casimir invariants\.

We validated our approach across three challenging benchmark systems: generalized rigid\-body motion onSO​\(3\)\\mathrm\{SO\}\(3\), a Kirchhoff underwater vehicle onSE​\(3\)\\mathrm\{SE\}\(3\), and anNN\-agent robotic swarm onSE​\(2\)N\\mathrm\{SE\}\(2\)^\{N\}, all of which exhibit complex non\-linear behavior\. Numerical experiments demonstrate that respecting the intrinsic geometric structure of the Poisson manifold yields superior long\-term energy conservation and trajectory prediction accuracy compared to unconstrained baseline architectures\. Using observable data alone, our framework successfully recovers full dynamical information and enables accurate future forecasting for both observable and latent variables, even under degenerate Hamiltonian conditions\.

The proposed framework reconstructs latent momentum trajectories through Noether invariants and the associated coadjoint action\. Consequently, successful reconstruction requires that the observable variables contain sufficient information to determine the conserved quantityp0p\_\{0\}over a short fitting window\. While this condition is satisfied for the examples considered in this paper, more complex observation operators may require longer fitting windows or additional measurements\. Understanding observability properties of Lie–Poisson systems in the context of latent state reconstruction remains an open question, and an interesting direction for further study\.

Several promising avenues exist for future research\. First, extending the framework to model non\-conservative effects—such as Rayleigh dissipation functions and external fluid disturbances—via structure\-preserving variational integrators\[[18](https://arxiv.org/html/2607.28939#bib.bib139)\]will enable accurate modeling in dissipative physical environments\. Second, incorporating nonholonomic constraints into the Lie–Poisson formulation using structure\-preserving integrators\[[3](https://arxiv.org/html/2607.28939#bib.bib138)\]will generalize the framework to ground and aerial robotic platforms\. Finally, extending the method to accommodate symmetry\-breaking cost functions\[[5](https://arxiv.org/html/2607.28939#bib.bib128)\]represents both an important practical capability and a rich theoretical avenue for geometric machine learning\.

## Acknowledgements

The author acknowledges fruitful and inspiring discussions with Drs\. C\. Eldred, F\. Gay\-Balmaz, A\. Gruber, A\. Sinclair, I\. Tezaur, and D\. V\. Zenkov\. This work was supported in part by the Shelby Foundation at the University of Alabama\.

## CRediT Authorship Contribution Statement

VP: Conceptualization, Methodology, Formal analysis, Software, Validation, Investigation, Data curation, Writing – original draft, Writing – review and editing, Visualization, Project administration\.

## Declaration of Competing Interest

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper\.

## Data Availability Statement

The source code, simulation scripts, and dataset generation routines for the Lie–Poisson Neural Network \(LLPNN\) architectures presented in this study are publicly available at[https://github\.com/vputkaradze/LLPNNs](https://github.com/vputkaradze/LLPNNs)\. All benchmark data can be reproduced using the provided codebase\.

## Declaration of Generative AI and AI\-Assisted Technologies

During the preparation of this work, the author used*Gemini Pro*and*ChatGPT*to assist with code development, improve code readability, and perform language editing of the manuscript\. The author reviewed and edited all generated material and verified the accuracy of the mathematical, scientific, and technical content\. The author takes full responsibility for the content of this publication\.

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## Appendix ALie Groups and their actions

### A\.1General definitions

In this short exposition, we closely follow the notation and definitions of\[[46](https://arxiv.org/html/2607.28939#bib.bib116)\]\. A Lie group is a groupGGthat is also a smooth manifold, such that the group multiplication\(g,h\)↦g​h\(g,h\)\\mapsto ghand inversiong↦g−1g\\mapsto g^\{\-1\}are smooth maps\. The group structure satisfies the following standard properties:

1. 1\.There exists an identity elemente∈Ge\\in Gsuch thate​g=g​e=geg=ge=gfor anyg∈Gg\\in G,
2. 2\.For anyg,h∈Gg,h\\in G, the product satisfiesg​h∈Ggh\\in G,
3. 3\.For anyg∈Gg\\in G, there exists an inverse elementg−1∈Gg^\{\-1\}\\in Gsuch thatg​g−1=g−1​g=egg^\{\-1\}=g^\{\-1\}g=e\.

As shown below, the tangent space at the identity is naturally endowed with a commutator bracket\. To define this bracket, we first introduce the relevant group actions\.

##### Adjoint Group Actions

Suppose we have smooth curvesg​\(s\)∈Gg\(s\)\\in Gandh​\(t\)∈Gh\(t\)\\in Gpassing through the identityeeat parameter zero, such thatg​\(0\)=h​\(0\)=eg\(0\)=h\(0\)=e\. Let their velocity vectors at the identity be denoted byg′​\(0\)=ξg^\{\\prime\}\(0\)=\\xiandh′​\(0\)=ηh^\{\\prime\}\(0\)=\\eta, which are elements of the tangent spaceTe​GT\_\{e\}G\.

##### TheAD\\operatorname\{AD\}Operator

We define the conjugation map \(inner automorphism\) ofGGby a fixed elementg∈Gg\\in Gas:

ADg⁡h=g​h​g−1\.\\operatorname\{AD\}\_\{g\}h=ghg^\{\-1\}\\,\.\(70\)

##### TheAd\\operatorname\{Ad\}Operator

Next, we define the Adjoint action of the group on the Lie algebra\. By keepingggfixed and differentiating \([70](https://arxiv.org/html/2607.28939#A1.E70)\) with respect to the parameterttalong the curveh​\(t\)h\(t\), we obtain:

Adg⁡η=dd​t​\(g​h​\(t\)​g−1\)\|t=0=g​η​g−1,where​η:=h′​\(0\)∈Te​G\.\\operatorname\{Ad\}\_\{g\}\\eta=\\left\.\\frac\{d\}\{dt\}\\left\(gh\(t\)g^\{\-1\}\\right\)\\right\|\_\{t=0\}=g\\eta g^\{\-1\}\\,,\\quad\\text\{where \}\\eta:=h^\{\\prime\}\(0\)\\in T\_\{e\}G\\,\.\(71\)This map,Adg:Te​G→Te​G\\operatorname\{Ad\}\_\{g\}:T\_\{e\}G\\to T\_\{e\}G, represents a linear representation of the group on its tangent space at the identity\.

##### Thead\\operatorname\{ad\}Operator

Now, we letg=g​\(s\)g=g\(s\)vary along its curve while keeping the algebra elementη\\etafixed\. DifferentiatingAdg​\(s\)⁡η\\operatorname\{Ad\}\_\{g\(s\)\}\\etawith respect tossats=0s=0yields the infinitesimal adjoint action, denoted byad\\operatorname\{ad\}:

adξ⁡η=dd​s​\(Adg​\(s\)⁡η\)\|s=0=ξ​η−η​ξ=\[ξ,η\],where​ξ:=g′​\(0\)∈Te​G\.\\operatorname\{ad\}\_\{\\xi\}\\eta=\\left\.\\frac\{d\}\{ds\}\\left\(\\operatorname\{Ad\}\_\{g\(s\)\}\\eta\\right\)\\right\|\_\{s=0\}=\\xi\\eta\-\\eta\\xi=\[\\xi,\\eta\]\\,,\\quad\\text\{where \}\\xi:=g^\{\\prime\}\(0\)\\in T\_\{e\}G\\,\.\(72\)The mapadξ⁡η\\operatorname\{ad\}\_\{\\xi\}\\etanaturally introduces the Lie bracket commutator to the tangent space at the identity\.

###### Definition A\.1\(Lie Algebra\)

The tangent space to the Lie groupGGat the identityg=eg=eis defined as the Lie algebra𝔤≅Te​G\\mathfrak\{g\}\\cong T\_\{e\}Gof the Lie groupGG, equipped with the Lie bracket\[⋅,⋅\]\[\\cdot,\\cdot\]defined in \([72](https://arxiv.org/html/2607.28939#A1.E72)\)\.

##### Coadjoint Actions

To study the dynamics of momenta, we transition to the dual space\. Let⟨⋅,⋅⟩:𝔤∗×𝔤→ℝ\\langle\\cdot,\\cdot\\rangle:\\mathfrak\{g\}^\{\*\}\\times\\mathfrak\{g\}\\to\\mathbb\{R\}denote the natural pairing between the Lie algebra𝔤\\mathfrak\{g\}and its dual space𝔤∗\\mathfrak\{g\}^\{\*\}\. Following\[[46](https://arxiv.org/html/2607.28939#bib.bib116)\], we define the dual operatorAdg∗:𝔤∗→𝔤∗\\operatorname\{Ad\}^\{\*\}\_\{g\}:\\mathfrak\{g\}^\{\*\}\\to\\mathfrak\{g\}^\{\*\}as the algebraic dual ofAdg\\operatorname\{Ad\}\_\{g\}, which satisfies:

⟨Adg∗⁡μ,α⟩=⟨μ,Adg⁡α⟩\\left\\langle\\operatorname\{Ad\}^\{\*\}\_\{g\}\\mu,\\alpha\\right\\rangle=\\left\\langle\\mu,\\operatorname\{Ad\}\_\{g\}\\alpha\\right\\rangle\(73\)for anyμ∈𝔤∗\\mu\\in\\mathfrak\{g\}^\{\*\}andα∈𝔤\\alpha\\in\\mathfrak\{g\}\. Since the assignmentg↦Adg∗g\\mapsto\\operatorname\{Ad\}^\{\*\}\_\{g\}yields a right action, the left coadjoint action ofGGon𝔤∗\\mathfrak\{g\}^\{\*\}is defined using the group inverse:

\(g,μ\)↦Adg−1∗⁡μ\.\(g,\\mu\)\\mapsto\\operatorname\{Ad\}^\{\*\}\_\{g^\{\-1\}\}\\mu\\,\.\(74\)Differentiating this representation yields the infinitesimal coadjoint actionadξ∗\\operatorname\{ad\}^\{\*\}\_\{\\xi\}for anyξ∈𝔤\\xi\\in\\mathfrak\{g\}andμ∈𝔤∗\\mu\\in\\mathfrak\{g\}^\{\*\}:

⟨adξ∗⁡μ,α⟩=⟨μ,adξ⁡α⟩=⟨μ,\[ξ,α\]⟩\.\\left\\langle\\operatorname\{ad\}^\{\*\}\_\{\\xi\}\\mu,\\alpha\\right\\rangle=\\left\\langle\\mu,\\operatorname\{ad\}\_\{\\xi\}\\alpha\\right\\rangle=\\left\\langle\\mu,\[\\xi,\\alpha\]\\right\\rangle\.\(75\)Calculating these operators \(AD\\operatorname\{AD\},Ad\\operatorname\{Ad\},ad\\operatorname\{ad\},Ad∗\\operatorname\{Ad\}^\{\*\},ad∗\\operatorname\{ad\}^\{\*\}\) for concrete physical groups simplifies many aspects of geometric mechanics\. For detailed derivations of these actions on common physical Lie groups, we refer the reader to\[[26](https://arxiv.org/html/2607.28939#bib.bib119)\]\.

### A\.2The Special Orthogonal GroupSO​\(3\)\\mathrm\{SO\}\(3\)

The Special Orthogonal group in three dimensions is defined as:

SO​\(3\)=\{R∈ℝ3×3∣RT​R=𝕀3,det\(R\)=1\}\.\\mathrm\{SO\}\(3\)=\\\{R\\in\\mathbb\{R\}^\{3\\times 3\}\\mid R^\{T\}R=\\mathbb\{I\}\_\{3\}\\,,\\,\\det\(R\)=1\\\}\\,\.\(76\)The Lie algebra𝔰​𝔬​\(3\)\\mathfrak\{so\}\(3\)consists of3×33\\times 3skew\-symmetric matrices\. We identify𝔰​𝔬​\(3\)\\mathfrak\{so\}\(3\)withℝ3\\mathbb\{R\}^\{3\}via the isomorphic “hat” map⋅^:ℝ3→𝔰​𝔬​\(3\)\\hat\{\\cdot\}:\\mathbb\{R\}^\{3\}\\to\\mathfrak\{so\}\(3\):

𝝎=\(ω1ω2ω3\)↦𝝎^=\(0−ω3ω2ω30−ω1−ω2ω10\),\\boldsymbol\{\\omega\}=\\begin\{pmatrix\}\\omega\_\{1\}\\\\ \\omega\_\{2\}\\\\ \\omega\_\{3\}\\end\{pmatrix\}\\ \\mapsto\\ \\hat\{\\boldsymbol\{\\omega\}\}=\\begin\{pmatrix\}0&\-\\omega\_\{3\}&\\omega\_\{2\}\\\\ \\omega\_\{3\}&0&\-\\omega\_\{1\}\\\\ \-\\omega\_\{2\}&\\omega\_\{1\}&0\\end\{pmatrix\}\\,,\(77\)which satisfies𝝎^​𝐱=𝝎×𝐱\\hat\{\\boldsymbol\{\\omega\}\}\\mathbf\{x\}=\\boldsymbol\{\\omega\}\\times\\mathbf\{x\}for any𝐱∈ℝ3\\mathbf\{x\}\\in\\mathbb\{R\}^\{3\}\. We equip𝔰​𝔬​\(3\)\\mathfrak\{so\}\(3\)with the standard inner product matching the Killing form scaling:⟨𝝎^,𝜼^⟩=12​Tr⁡\(𝝎^T​𝜼^\)=𝝎⋅𝜼\\langle\\hat\{\\boldsymbol\{\\omega\}\},\\hat\{\\boldsymbol\{\\eta\}\}\\rangle=\\frac\{1\}\{2\}\\operatorname\{Tr\}\(\\hat\{\\boldsymbol\{\\omega\}\}^\{T\}\\hat\{\\boldsymbol\{\\eta\}\}\)=\\boldsymbol\{\\omega\}\\cdot\\boldsymbol\{\\eta\}\.

##### Adjoint Operators

LetR∈SO​\(3\)R\\in\\mathrm\{SO\}\(3\)and𝝎,𝜼∈ℝ3\\boldsymbol\{\\omega\},\\boldsymbol\{\\eta\}\\in\\mathbb\{R\}^\{3\}\.

- 1\.Conjugation \(AD\\operatorname\{AD\}\):For another group elementQ∈SO​\(3\)Q\\in\\mathrm\{SO\}\(3\), the conjugation action ofRRis explicitly given by: ADR⁡Q=R​Q​RT\.\\operatorname\{AD\}\_\{R\}Q=RQR^\{T\}\\,\.\(78\)Geometrically, ifQQrepresents a rotation by an angleθ\\thetaabout an axis𝐧\\mathbf\{n\}, thenADR⁡Q\\operatorname\{AD\}\_\{R\}Qrepresents a rotation by the same angleθ\\thetaabout the rotated axisR​𝐧R\\mathbf\{n\}\.
- 2\.Adjoint Map \(Ad\\operatorname\{Ad\}\):By identifying𝔰​𝔬​\(3\)\\mathfrak\{so\}\(3\)withℝ3\\mathbb\{R\}^\{3\}, the matrix conjugationR​𝝎^​RT=R​𝝎^R\\hat\{\\boldsymbol\{\\omega\}\}R^\{T\}=\\widehat\{R\\boldsymbol\{\\omega\}\}yields: AdR⁡𝝎=R​𝝎\.\\operatorname\{Ad\}\_\{R\}\\boldsymbol\{\\omega\}=R\\boldsymbol\{\\omega\}\\,\.\(79\)
- 3\.Infinitesimal Adjoint \(ad\\operatorname\{ad\}\):The Lie bracket in𝔰​𝔬​\(3\)\\mathfrak\{so\}\(3\)corresponds to the standard cross product: ad𝝎⁡𝜼=\[𝝎^,𝜼^\]⇔ad𝝎⁡𝜼=𝝎×𝜼\.\\operatorname\{ad\}\_\{\\boldsymbol\{\\omega\}\}\\boldsymbol\{\\eta\}=\[\\hat\{\\boldsymbol\{\\omega\}\},\\hat\{\\boldsymbol\{\\eta\}\}\]\\ \\Leftrightarrow\\ \\operatorname\{ad\}\_\{\\boldsymbol\{\\omega\}\}\\boldsymbol\{\\eta\}=\\boldsymbol\{\\omega\}\\times\\boldsymbol\{\\eta\}\\,\.\(80\)

##### Coadjoint Operators

The dual space𝔰​𝔬​\(3\)∗\\mathfrak\{so\}\(3\)^\{\*\}is identified withℝ3\\mathbb\{R\}^\{3\}\. Let𝝁∈𝔰​𝔬​\(3\)∗\\boldsymbol\{\\mu\}\\in\\mathfrak\{so\}\(3\)^\{\*\}represent the momentum vector\.

- 1\.Dual Adjoint Operator \(Ad∗\\operatorname\{Ad\}^\{\*\}\):Defined by the pairing⟨AdR∗⁡𝝁,𝝎⟩=⟨𝝁,AdR⁡𝝎⟩\\langle\\operatorname\{Ad\}^\{\*\}\_\{R\}\\boldsymbol\{\\mu\},\\boldsymbol\{\\omega\}\\rangle=\\langle\\boldsymbol\{\\mu\},\\operatorname\{Ad\}\_\{R\}\\boldsymbol\{\\omega\}\\rangle, we have: AdR∗⁡𝝁=RT​𝝁\.\\operatorname\{Ad\}^\{\*\}\_\{R\}\\boldsymbol\{\\mu\}=R^\{T\}\\boldsymbol\{\\mu\}\\,\.\(81\)The corresponding left coadjoint action ofGGon𝔤∗\\mathfrak\{g\}^\{\*\}is given byAdR−1∗⁡𝝁=R​𝝁\\operatorname\{Ad\}^\{\*\}\_\{R^\{\-1\}\}\\boldsymbol\{\\mu\}=R\\boldsymbol\{\\mu\}\.
- 2\.Infinitesimal Coadjoint \(ad∗\\operatorname\{ad\}^\{\*\}\):Using the triple product identity𝝁⋅\(𝝎×𝜼\)=\(𝝁×𝝎\)⋅𝜼\\boldsymbol\{\\mu\}\\cdot\(\\boldsymbol\{\\omega\}\\times\\boldsymbol\{\\eta\}\)=\(\\boldsymbol\{\\mu\}\\times\\boldsymbol\{\\omega\}\)\\cdot\\boldsymbol\{\\eta\}, the infinitesimal coadjoint action is: ad𝝎∗⁡𝝁=𝝁×𝝎=−𝝎×𝝁\.\\operatorname\{ad\}^\{\*\}\_\{\\boldsymbol\{\\omega\}\}\\boldsymbol\{\\mu\}=\\boldsymbol\{\\mu\}\\times\\boldsymbol\{\\omega\}=\-\\boldsymbol\{\\omega\}\\times\\boldsymbol\{\\mu\}\\,\.\(82\)

### A\.3The Special Euclidean GroupSE​\(2\)\\mathrm\{SE\}\(2\)

The Special Euclidean group in two dimensions represents planar rigid body motions:

SE​\(2\)=\{g=\(Rθ,𝐱\)=\(Rθ𝐱𝟎T1\)\|Rθ∈SO​\(2\),𝐱∈ℝ2\},\\mathrm\{SE\}\(2\)=\\left\\\{g=\(R\_\{\\theta\},\\mathbf\{x\}\)=\\begin\{pmatrix\}R\_\{\\theta\}&\\mathbf\{x\}\\\\ \\mathbf\{0\}^\{T\}&1\\end\{pmatrix\}\\ \\middle\|\\ R\_\{\\theta\}\\in\\mathrm\{SO\}\(2\),\\ \\mathbf\{x\}\\in\\mathbb\{R\}^\{2\}\\right\\\}\\,,\(83\)whereRθ=\(cos⁡θ−sin⁡θsin⁡θcos⁡θ\)R\_\{\\theta\}=\\begin\{pmatrix\}\\cos\\theta&\-\\sin\\theta\\\\ \\sin\\theta&\\cos\\theta\\end\{pmatrix\}and𝐱=\(xy\)\\mathbf\{x\}=\\begin\{pmatrix\}x\\\\ y\\end\{pmatrix\}\. The Lie algebra𝔰​𝔢​\(2\)\\mathfrak\{se\}\(2\)is isomorphic toℝ3\\mathbb\{R\}^\{3\}, with elements representing planar twist velocitiesξ=\(ω,𝐯\)∈ℝ×ℝ2\\xi=\(\\omega,\\mathbf\{v\}\)\\in\\mathbb\{R\}\\times\\mathbb\{R\}^\{2\}\. The matrix representation is:

ξ^=\(ω​𝕁𝐯𝟎T0\),where​𝕁=\(0−110\)\.\\hat\{\\xi\}=\\begin\{pmatrix\}\\omega\\mathbb\{J\}&\\mathbf\{v\}\\\\ \\mathbf\{0\}^\{T\}&0\\end\{pmatrix\}\\,,\\quad\\text\{where \}\\mathbb\{J\}=\\begin\{pmatrix\}0&\-1\\\\ 1&0\\end\{pmatrix\}\\,\.\(84\)The pairing between dual algebra elementμ=\(M,𝐩\)∈𝔰​𝔢​\(2\)∗\\mu=\(M,\\mathbf\{p\}\)\\in\\mathfrak\{se\}\(2\)^\{\*\}andξ=\(ω,𝐯\)∈𝔰​𝔢​\(2\)\\xi=\(\\omega,\\mathbf\{v\}\)\\in\\mathfrak\{se\}\(2\)is⟨μ,ξ⟩=M​ω\+𝐩⋅𝐯\\langle\\mu,\\xi\\rangle=M\\omega\+\\mathbf\{p\}\\cdot\\mathbf\{v\}\.

##### Adjoint Operators

Letg=\(Rθ,𝐱\)∈SE​\(2\)g=\(R\_\{\\theta\},\\mathbf\{x\}\)\\in\\mathrm\{SE\}\(2\)andη=\(ω,𝐯\)∈𝔰​𝔢​\(2\)\\eta=\(\\omega,\\mathbf\{v\}\)\\in\\mathfrak\{se\}\(2\)\.

- 1\.Conjugation \(AD\\operatorname\{AD\}\):For another group elementh=\(Rϕ,𝐲\)∈SE​\(2\)h=\(R\_\{\\phi\},\\mathbf\{y\}\)\\in\\mathrm\{SE\}\(2\), conjugation yields: ADg⁡h=g​h​g−1=\(Rϕ,Rθ​𝐲\+\(𝕀2−Rϕ\)​𝐱\)\.\\operatorname\{AD\}\_\{g\}h=ghg^\{\-1\}=\\left\(R\_\{\\phi\},\\,R\_\{\\theta\}\\mathbf\{y\}\+\(\\mathbb\{I\}\_\{2\}\-R\_\{\\phi\}\)\\mathbf\{x\}\\right\)\\,\.\(85\)BecauseSO​\(2\)\\mathrm\{SO\}\(2\)is abelian, the rotational componentRθ​Rϕ​RθTR\_\{\\theta\}R\_\{\\phi\}R\_\{\\theta\}^\{T\}simplifies directly toRϕR\_\{\\phi\}\.
- 2\.Adjoint Map \(Ad\\operatorname\{Ad\}\):Differentiating \([85](https://arxiv.org/html/2607.28939#A1.E85)\) along a curveh​\(t\)=\(Rt​ω,𝐲​\(t\)\)h\(t\)=\(R\_\{t\\omega\},\\mathbf\{y\}\(t\)\)passing through the identity with velocityη=\(ω,𝐯\)\\eta=\(\\omega,\\mathbf\{v\}\)att=0t=0yields: Adg⁡η=\(ω,Rθ​𝐯−ω​𝕁​𝐱\)\.\\operatorname\{Ad\}\_\{g\}\\eta=\\left\(\\omega,\\,R\_\{\\theta\}\\mathbf\{v\}\-\\omega\\mathbb\{J\}\\mathbf\{x\}\\right\)\\,\.\(86\)
- 3\.Infinitesimal Adjoint \(ad\\operatorname\{ad\}\):For two algebra elementsξ=\(ω1,𝐯1\)\\xi=\(\\omega\_\{1\},\\mathbf\{v\}\_\{1\}\)andη=\(ω2,𝐯2\)\\eta=\(\\omega\_\{2\},\\mathbf\{v\}\_\{2\}\), the Lie bracket is: adξ⁡η=\[ξ^,η^\]⇔adξ⁡η=\(0,ω1​𝕁​𝐯2−ω2​𝕁​𝐯1\)\.\\operatorname\{ad\}\_\{\\xi\}\\eta=\[\\hat\{\\xi\},\\hat\{\\eta\}\]\\ \\Leftrightarrow\\ \\operatorname\{ad\}\_\{\\xi\}\\eta=\\left\(0,\\,\\omega\_\{1\}\\mathbb\{J\}\\mathbf\{v\}\_\{2\}\-\\omega\_\{2\}\\mathbb\{J\}\\mathbf\{v\}\_\{1\}\\right\)\\,\.\(87\)

##### Coadjoint Operators

Letμ=\(M,𝐩\)∈𝔰​𝔢​\(2\)∗\\mu=\(M,\\mathbf\{p\}\)\\in\\mathfrak\{se\}\(2\)^\{\*\}andξ=\(ω,𝐯\)∈𝔰​𝔢​\(2\)\\xi=\(\\omega,\\mathbf\{v\}\)\\in\\mathfrak\{se\}\(2\)\.

- 1\.Dual Adjoint Operator \(Ad∗\\operatorname\{Ad\}^\{\*\}\):Computing the dual operator via the pairing relation yields: Adg∗⁡μ=\(M−\(𝐱×𝐩\)⋅𝐞3,RθT​𝐩\),\\operatorname\{Ad\}^\{\*\}\_\{g\}\\mu=\\left\(M\-\(\\mathbf\{x\}\\times\\mathbf\{p\}\)\\cdot\\mathbf\{e\}\_\{3\},\\,R\_\{\\theta\}^\{T\}\\mathbf\{p\}\\right\)\\,,\(88\)where\(𝐱×𝐩\)⋅𝐞3:=𝐩⋅𝕁​𝐱=x​py−y​px\(\\mathbf\{x\}\\times\\mathbf\{p\}\)\\cdot\\mathbf\{e\}\_\{3\}:=\\mathbf\{p\}\\cdot\\mathbb\{J\}\\mathbf\{x\}=xp\_\{y\}\-yp\_\{x\}is the 2D scalar cross product\.
- 2\.Infinitesimal Coadjoint \(ad∗\\operatorname\{ad\}^\{\*\}\):Under the duality pairing, we find: adξ∗⁡μ=\(−\(𝐯×𝐩\)⋅𝐞3,−ω​𝕁​𝐩\),\\operatorname\{ad\}^\{\*\}\_\{\\xi\}\\mu=\\left\(\-\(\\mathbf\{v\}\\times\\mathbf\{p\}\)\\cdot\\mathbf\{e\}\_\{3\},\\,\-\\omega\\mathbb\{J\}\\mathbf\{p\}\\right\)\\,,\(89\)where\(𝐯×𝐩\)⋅𝐞3=vx​py−vy​px\(\\mathbf\{v\}\\times\\mathbf\{p\}\)\\cdot\\mathbf\{e\}\_\{3\}=v\_\{x\}p\_\{y\}\-v\_\{y\}p\_\{x\}\.

### A\.4The Special Euclidean GroupSE​\(3\)\\mathrm\{SE\}\(3\)

The Special Euclidean group in three dimensions describes general spatial motions:

SE​\(3\)=\{g=\(R,𝐱\)=\(R𝐱𝟎T1\)\|R∈SO​\(3\),𝐱∈ℝ3\}\.\\mathrm\{SE\}\(3\)=\\left\\\{g=\(R,\\mathbf\{x\}\)=\\begin\{pmatrix\}R&\\mathbf\{x\}\\\\ \\mathbf\{0\}^\{T\}&1\\end\{pmatrix\}\\ \\middle\|\\ R\\in\\mathrm\{SO\}\(3\),\\ \\mathbf\{x\}\\in\\mathbb\{R\}^\{3\}\\right\\\}\\,\.\(90\)The Lie algebra𝔰​𝔢​\(3\)\\mathfrak\{se\}\(3\)is isomorphic toℝ6\\mathbb\{R\}^\{6\}\. We identify any twistξ=\(𝝎,𝐯\)∈ℝ3×ℝ3\\xi=\(\\boldsymbol\{\\omega\},\\mathbf\{v\}\)\\in\\mathbb\{R\}^\{3\}\\times\\mathbb\{R\}^\{3\}with the matrix:

ξ^=\(𝝎^𝐯𝟎T0\)\.\\hat\{\\xi\}=\\begin\{pmatrix\}\\hat\{\\boldsymbol\{\\omega\}\}&\\mathbf\{v\}\\\\ \\mathbf\{0\}^\{T\}&0\\end\{pmatrix\}\\,\.\(91\)The dual space𝔰​𝔢​\(3\)∗\\mathfrak\{se\}\(3\)^\{\*\}is isomorphic toℝ6\\mathbb\{R\}^\{6\}\. For a wrenchμ=\(𝐌,𝐩\)∈𝔰​𝔢​\(3\)∗\\mu=\(\\mathbf\{M\},\\mathbf\{p\}\)\\in\\mathfrak\{se\}\(3\)^\{\*\}, the pairing is given by⟨μ,ξ⟩=𝐌⋅𝝎\+𝐩⋅𝐯\\langle\\mu,\\xi\\rangle=\\mathbf\{M\}\\cdot\\boldsymbol\{\\omega\}\+\\mathbf\{p\}\\cdot\\mathbf\{v\}\.

##### Adjoint Operators

Letg=\(R,𝐱\)∈SE​\(3\)g=\(R,\\mathbf\{x\}\)\\in\\mathrm\{SE\}\(3\)andη=\(𝝎,𝐯\)∈𝔰​𝔢​\(3\)\\eta=\(\\boldsymbol\{\\omega\},\\mathbf\{v\}\)\\in\\mathfrak\{se\}\(3\)\.

- 1\.Conjugation \(AD\\operatorname\{AD\}\):For another group elementh=\(Q,𝐲\)∈SE​\(3\)h=\(Q,\\mathbf\{y\}\)\\in\\mathrm\{SE\}\(3\), conjugation yields: ADg⁡h=g​h​g−1=\(R​Q​RT,R​𝐲\+\(𝕀3−R​Q​RT\)​𝐱\)\.\\operatorname\{AD\}\_\{g\}h=ghg^\{\-1\}=\\left\(RQR^\{T\},\\,R\\mathbf\{y\}\+\(\\mathbb\{I\}\_\{3\}\-RQR^\{T\}\)\\mathbf\{x\}\\right\)\\,\.\(92\)
- 2\.Adjoint Map \(Ad\\operatorname\{Ad\}\):Differentiating \([92](https://arxiv.org/html/2607.28939#A1.E92)\) along a curveh​\(t\)=\(Q​\(t\),𝐲​\(t\)\)h\(t\)=\(Q\(t\),\\mathbf\{y\}\(t\)\)passing through the identity with velocityη=\(𝝎,𝐯\)\\eta=\(\\boldsymbol\{\\omega\},\\mathbf\{v\}\)att=0t=0yields: Adg⁡η=\(R​𝝎,R​𝐯\+𝐱×\(R​𝝎\)\),\\operatorname\{Ad\}\_\{g\}\\eta=\\left\(R\\boldsymbol\{\\omega\},\\,R\\mathbf\{v\}\+\\mathbf\{x\}\\times\(R\\boldsymbol\{\\omega\}\)\\right\)\\,,\(93\)where we use the identityR​𝝎^​RT=R​𝝎^R\\hat\{\\boldsymbol\{\\omega\}\}R^\{T\}=\\widehat\{R\\boldsymbol\{\\omega\}\}and−\(R​𝝎\)×𝐱=𝐱×\(R​𝝎\)\-\(R\\boldsymbol\{\\omega\}\)\\times\\mathbf\{x\}=\\mathbf\{x\}\\times\(R\\boldsymbol\{\\omega\}\)\.
- 3\.Infinitesimal Adjoint \(ad\\operatorname\{ad\}\):For two elementsξ=\(𝝎1,𝐯1\)\\xi=\(\\boldsymbol\{\\omega\}\_\{1\},\\mathbf\{v\}\_\{1\}\)andη=\(𝝎2,𝐯2\)\\eta=\(\\boldsymbol\{\\omega\}\_\{2\},\\mathbf\{v\}\_\{2\}\), the Lie bracket is: adξ⁡η=\(𝝎1×𝝎2,𝝎1×𝐯2\+𝐯1×𝝎2\)\.\\operatorname\{ad\}\_\{\\xi\}\\eta=\\left\(\\boldsymbol\{\\omega\}\_\{1\}\\times\\boldsymbol\{\\omega\}\_\{2\},\\,\\boldsymbol\{\\omega\}\_\{1\}\\times\\mathbf\{v\}\_\{2\}\+\\mathbf\{v\}\_\{1\}\\times\\boldsymbol\{\\omega\}\_\{2\}\\right\)\\,\.\(94\)

##### Coadjoint Operators

Letμ=\(𝐌,𝐩\)∈𝔰​𝔢​\(3\)∗\\mu=\(\\mathbf\{M\},\\mathbf\{p\}\)\\in\\mathfrak\{se\}\(3\)^\{\*\}andξ=\(𝝎,𝐯\)∈𝔰​𝔢​\(3\)\\xi=\(\\boldsymbol\{\\omega\},\\mathbf\{v\}\)\\in\\mathfrak\{se\}\(3\)\.

- 1\.Dual Adjoint Operator \(Ad∗\\operatorname\{Ad\}^\{\*\}\):Resolving the pairing equation gives: Adg∗⁡μ=\(RT​𝐌−RT​𝐱×RT​𝐩,RT​𝐩\),\\operatorname\{Ad\}^\{\*\}\_\{g\}\\mu=\\left\(R^\{T\}\\mathbf\{M\}\-R^\{T\}\\mathbf\{x\}\\times R^\{T\}\\mathbf\{p\},\\,R^\{T\}\\mathbf\{p\}\\right\)\\,,\(95\)using the rotation\-invariance of cross productsRT​\(𝐩×𝐱\)=RT​𝐩×RT​𝐱R^\{T\}\(\\mathbf\{p\}\\times\\mathbf\{x\}\)=R^\{T\}\\mathbf\{p\}\\times R^\{T\}\\mathbf\{x\}\.
- 2\.Infinitesimal Coadjoint \(ad∗\\operatorname\{ad\}^\{\*\}\):Splitting the dual pairing across angular and linear coordinates yields: adξ∗⁡μ=\(𝐌×𝝎−𝐯×𝐩,𝐩×𝝎\)=\(−𝝎×𝐌−𝐯×𝐩,−𝝎×𝐩\)\.\\hskip\-28\.45274pt\\operatorname\{ad\}^\{\*\}\_\{\\xi\}\\mu=\\left\(\\mathbf\{M\}\\times\\boldsymbol\{\\omega\}\-\\mathbf\{v\}\\times\\mathbf\{p\},\\,\\mathbf\{p\}\\times\\boldsymbol\{\\omega\}\\right\)=\\left\(\-\\boldsymbol\{\\omega\}\\times\\mathbf\{M\}\-\\mathbf\{v\}\\times\\mathbf\{p\},\\,\-\\boldsymbol\{\\omega\}\\times\\mathbf\{p\}\\right\)\\,\.\(96\)

## Appendix BMathematical Foundations of Lie\-Poisson Systems

##### Dynamics on Lie Groups

Let the configuration space of a mechanical system be modeled by a Lie groupGG\. The trajectories of the system under a LagrangianL​\(g,g˙\)L\(g,\\dot\{g\}\)are determined by Hamilton’s variational principle\. Provided that the fiber derivative \(the mapping from velocities to momenta\)g˙↦p​\(g,g˙\)=∂L∂g˙​\(g,g˙\)\\dot\{g\}\\mapsto p\(g,\\dot\{g\}\)=\\frac\{\\partial L\}\{\\partial\\dot\{g\}\}\(g,\\dot\{g\}\)is a diffeomorphism for allg∈Gg\\in G, the system can be represented equivalently in terms of canonical Hamiltonian dynamics\. This yields the Hamiltonian functionH​\(g,p\)=⟨p,g˙​\(g,p\)⟩−L​\(g,g˙​\(g,p\)\)H\(g,p\)=\\left\\langle p,\\dot\{g\}\(g,p\)\\right\\rangle\-L\(g,\\dot\{g\}\(g,p\)\)defined on the cotangent bundleT∗​GT^\{\*\}G\.

##### Symmetries on Lie Groups

We assume the Lagrangian exhibits left\-invariance under the action of the group on itself, such thatL​\(h​g,Tg​Lh​g˙\)=L​\(g,g˙\)L\(hg,T\_\{g\}L\_\{h\}\\dot\{g\}\)=L\(g,\\dot\{g\}\)for everyh∈Gh\\in G\. This invariance immediately translates to the Hamiltonian, satisfyingH​\(h​g,h​p\)=H​\(g,p\)H\(hg,hp\)=H\(g,p\)for allh∈Gh\\in G\. Consequently,HHis entirely characterized by its values evaluated at the group identityg=eg=e, allowing us to express the Hamiltonian asH\(g,p\)=H\(e,g−1p\)=:h\(g−1p\)H\(g,p\)=H\(e,g^\{\-1\}p\)=:h\(g^\{\-1\}p\)\. Here, the quantityμ=g−1​p:=Te∗​Lg​\(p\)\\mu=g^\{\-1\}p:=T^\{\*\}\_\{e\}L\_\{g\}\(p\)defines the reduced \(or body\-fixed\) momentum, which resides in the dual space𝔤∗\\mathfrak\{g\}^\{\*\}of the Lie algebra𝔤\\mathfrak\{g\}associated withGG\. The mappingh:𝔤∗→ℝh:\\mathfrak\{g\}^\{\*\}\\rightarrow\\mathbb\{R\}represents the reduced Hamiltonian function\. Given this symmetry, the evolution of the full phase\-space variables\(g​\(t\),p​\(t\)\)\(g\(t\),p\(t\)\)can be projected onto the lower\-dimensional dual algebra space𝔤∗\\mathfrak\{g\}^\{\*\}solely in terms of the reduced momentumμ​\(t\)=g​\(t\)−1​p​\(t\)\\mu\(t\)=g\(t\)^\{\-1\}p\(t\)\. This reduction yields the Lie\-Poisson equations for the evolution ofμ​\(t\)\\mu\(t\)\.

##### The Lie\-Poisson Equations

Formulated in coordinate\-free notation, the Lie\-Poisson equations for a left\-invariant system read:

μ˙=ad∂h∂μ∗⁡μ,\\dot\{\\mu\}=\\operatorname\{ad\}^\{\*\}\_\{\\frac\{\\partial h\}\{\\partial\\mu\}\}\\mu\\,,\(97\)whereadξ∗:𝔤∗→𝔤∗\\operatorname\{ad\}^\{\*\}\_\{\\xi\}:\\mathfrak\{g\}^\{\*\}\\rightarrow\\mathfrak\{g\}^\{\*\}denotes the infinitesimal coadjoint operator\. It is defined through the pairing with the adjoint operatoradξ⁡η=\[ξ,η\]\\operatorname\{ad\}\_\{\\xi\}\\eta=\[\\xi,\\eta\]via⟨adξ∗⁡μ,η⟩=⟨μ,adξ⁡η⟩\\langle\\operatorname\{ad\}^\{\*\}\_\{\\xi\}\\mu,\\eta\\rangle=\\langle\\mu,\\operatorname\{ad\}\_\{\\xi\}\\eta\\ranglefor anyξ,η∈𝔤\\xi,\\eta\\in\\mathfrak\{g\}andμ∈𝔤∗\\mu\\in\\mathfrak\{g\}^\{\*\}\. The temporal evolution of an arbitrary smooth functionf​\(μ\)f\(\\mu\)along the trajectories of \([97](https://arxiv.org/html/2607.28939#A2.E97)\) is governed byf˙=\{f,h\}\\dot\{f\}=\\\{f,h\\\}, where the underlying Poisson structure is determined by the minus Lie\-Poisson bracket:

\{f,h\}=−⟨μ,\[∂f∂μ,∂h∂μ\]⟩\.\\left\\\{f,h\\right\\\}=\-\\left\\langle\\mu,\\left\[\\frac\{\\partial f\}\{\\partial\\mu\},\\frac\{\\partial h\}\{\\partial\\mu\}\\right\]\\right\\rangle\\,\.\(98\)While this bracket is non\-canonical, it inherits its properties directly from the canonical Poisson bracket\{⋅,⋅\}can\\\{\\cdot,\\cdot\\\}\_\{\\rm can\}onT∗​GT^\{\*\}G\. Specifically, they are linked by:

\{f∘π,h∘π\}can=\{f,h\}∘π\\left\\\{f\\circ\\pi,h\\circ\\pi\\right\\\}\_\{\\rm can\}=\\left\\\{f,h\\right\\\}\\circ\\piwhereπ:T∗​G→𝔤∗\\pi:T^\{\*\}G\\rightarrow\\mathfrak\{g\}^\{\*\}is the projection mapπ​\(g,p\)=g−1​p=μ\\pi\(g,p\)=g^\{\-1\}p=\\muthat maps the full momentum to its body\-fixed representation\. Thus,π\\piacts as a Poisson map connecting\{⋅,⋅\}can\\\{\\cdot,\\cdot\\\}\_\{\\rm can\}and\{⋅,⋅\}\\\{\\cdot,\\cdot\\\}\. This reduction scheme, which is a classic demonstration of Poisson reduction, explains the source of the Lie\-Poisson brackets across diverse physical frameworks\.

For systems that exhibit right invariance instead of left invariance, the reduced momentum is defined asμ=p​g−1\\mu=pg^\{\-1\}, which introduces a sign change on the right\-hand sides of both \([7](https://arxiv.org/html/2607.28939#S2.E7)\) and \([98](https://arxiv.org/html/2607.28939#A2.E98)\) \(leading to a plus Lie\-Poisson bracket and a minus sign in the dynamical equation\)\.

##### The Flow of Lie\-Poisson Systems

LetAdg:𝔤→𝔤\\operatorname\{Ad\}\_\{g\}:\\mathfrak\{g\}\\rightarrow\\mathfrak\{g\}denote the adjoint action ofGGon its Lie algebra, defined byAdg⁡ξ:=dd​ε\|ε=0​g​cε​g−1\\operatorname\{Ad\}\_\{g\}\\xi:=\\left\.\\frac\{d\}\{d\\varepsilon\}\\right\|\_\{\\varepsilon=0\}gc\_\{\\varepsilon\}g^\{\-1\}, wherecε∈Gc\_\{\\varepsilon\}\\in Gis a curve satisfyingc0=ec\_\{0\}=eandc˙0=ξ\\dot\{c\}\_\{0\}=\\xi\. LetAdg∗:𝔤∗→𝔤∗\\operatorname\{Ad\}^\{\*\}\_\{g\}:\\mathfrak\{g\}^\{\*\}\\rightarrow\\mathfrak\{g\}^\{\*\}represent the dual of the adjoint operatorAdg\\operatorname\{Ad\}\_\{g\}, defined via the pairing⟨Adg∗⁡μ,ξ⟩=⟨μ,Adg⁡ξ⟩\\langle\\operatorname\{Ad\}^\{\*\}\_\{g\}\\mu,\\xi\\rangle=\\langle\\mu,\\operatorname\{Ad\}\_\{g\}\\xi\\rangle\. To ensure a proper left group action on the dual space, the left coadjoint action is defined by\(g,μ\)↦Adg−1∗⁡μ\(g,\\mu\)\\mapsto\\operatorname\{Ad\}^\{\*\}\_\{g^\{\-1\}\}\\mu\. For any curveg​\(t\)∈Gg\(t\)\\in G, and for any constantp0∈𝔤∗p\_\{0\}\\in\\mathfrak\{g\}^\{\*\}, the time derivative of the dual adjoint operator satisfies:

dd​t​Adg​\(t\)∗⁡p0=adg​\(t\)−1​g˙​\(t\)∗⁡Adg​\(t\)∗⁡p0\.\\frac\{d\}\{dt\}\\operatorname\{Ad\}^\{\*\}\_\{g\(t\)\}p\_\{0\}=\\operatorname\{ad\}^\{\*\}\_\{g\(t\)^\{\-1\}\\dot\{g\}\(t\)\}\\operatorname\{Ad\}^\{\*\}\_\{g\(t\)\}p\_\{0\}\\,\.A direct consequence of this identity is that a solution with conserved spatial momentump0p\_\{0\}of the Lie\-Poisson system \([97](https://arxiv.org/html/2607.28939#A2.E97)\) can be explicitly parameterized as:

ϕt​\(p0\)=Adg​\(t\)∗⁡p0,whereg​\(t\)−1​g˙​\(t\)=∂h∂μ​\(t\)\.\\phi\_\{t\}\(p\_\{0\}\)=\\operatorname\{Ad\}^\{\*\}\_\{g\(t\)\}p\_\{0\}\\,,\\quad\\text\{where\}\\quad g\(t\)^\{\-1\}\\dot\{g\}\(t\)=\\frac\{\\partial h\}\{\\partial\\mu\}\(t\)\\,\.This expression guarantees that the dynamical flow preserves the coadjoint orbits𝒪=\{Adg∗⁡p0∣g∈G\}⊂𝔤∗\\mathcal\{O\}=\\\{\\operatorname\{Ad\}^\{\*\}\_\{g\}p\_\{0\}\\mid g\\in G\\\}\\subset\\mathfrak\{g\}^\{\*\}at all times\. Conversely, suppose\(μ​\(t\),g​\(t\)\)\(\\mu\(t\),g\(t\)\)is a solution of the extended Lie\-Poisson system\. Then, the spatial momentump0:=Adg​\(t\)−1∗⁡μ​\(t\)p\_\{0\}:=\\operatorname\{Ad\}^\{\*\}\_\{g\(t\)^\{\-1\}\}\\mu\(t\)is conserved for all times, satisfying:

dd​t​Adg​\(t\)−1∗⁡μ​\(t\)=0\.\\frac\{d\}\{dt\}\\operatorname\{Ad\}^\{\*\}\_\{g\(t\)^\{\-1\}\}\\mu\(t\)=0\\,\.\(99\)This conservation ofp0p\_\{0\}is a direct consequence of Noether’s theorem associated with the left\-invariance of the system\. In practice, the coadjoint\-dual relationp0=Adg−1∗⁡μp\_\{0\}=\\operatorname\{Ad\}^\{\*\}\_\{g^\{\-1\}\}\\mumaps the body\-fixed momentumμ\\muto the conserved spatial momentump0p\_\{0\}, while its inverseμ=Adg∗⁡p0\\mu=\\operatorname\{Ad\}^\{\*\}\_\{g\}p\_\{0\}reconstructs the body\-fixed momentum from this conserved quantity\.

## Appendix CReference Formulas for Specific Lie Groups

To facilitate practical implementation, we compile the explicit forms of the conserved spatial momentump0=p0​\(g,μ\)p\_\{0\}=p\_\{0\}\(g,\\mu\)and its corresponding inversionμ=μ​\(g,p0\)\\mu=\\mu\(g,p\_\{0\}\)for three standard physical Lie groups\.

### C\.1The Special Orthogonal GroupSO​\(3\)\\mathrm\{SO\}\(3\)

For rotational dynamics, letg=R∈SO​\(3\)g=R\\in\\mathrm\{SO\}\(3\)be the rotation matrix satisfyingRT​R=𝕀3R^\{T\}R=\\mathbb\{I\}\_\{3\}anddet\(R\)=1\\det\(R\)=1\. The body momentum is represented by𝝁∈ℝ3\\boldsymbol\{\\mu\}\\in\\mathbb\{R\}^\{3\}\(body angular momentum\), and the conserved spatial momentum is𝒑0∈ℝ3\\boldsymbol\{p\}\_\{0\}\\in\\mathbb\{R\}^\{3\}\(spatial angular momentum\)\.

- 1\.Conserved Spatial Momentump0​\(R,μ\)\\boldsymbol\{p\}\_\{0\}\(R,\\boldsymbol\{\\mu\}\): 𝒑0=R​𝝁\\boldsymbol\{p\}\_\{0\}=R\\boldsymbol\{\\mu\}\(100\)
- 2\.Inversion to Body Momentumμ​\(R,p0\)\\boldsymbol\{\\mu\}\(R,\\boldsymbol\{p\}\_\{0\}\): 𝝁=RT​𝒑0\\boldsymbol\{\\mu\}=R^\{T\}\\boldsymbol\{p\}\_\{0\}\(101\)

### C\.2The Special Euclidean GroupSE​\(2\)\\mathrm\{SE\}\(2\)

For planar rigid motions, letg=\(Rθ,𝐱\)∈SE​\(2\)g=\(R\_\{\\theta\},\\mathbf\{x\}\)\\in\\mathrm\{SE\}\(2\), whereRθ∈SO​\(2\)R\_\{\\theta\}\\in\\mathrm\{SO\}\(2\)is the rotation matrix of angleθ\\theta, and𝐱∈ℝ2\\mathbf\{x\}\\in\\mathbb\{R\}^\{2\}is the translation vector\. The body momentum isμ=\(J,𝐏\)∈ℝ×ℝ2\\mu=\(J,\\mathbf\{P\}\)\\in\\mathbb\{R\}\\times\\mathbb\{R\}^\{2\}\(body angular and linear momentum\), and the conserved spatial momentum isp0=\(J0,𝐏0\)∈ℝ×ℝ2p\_\{0\}=\(J\_\{0\},\\mathbf\{P\}\_\{0\}\)\\in\\mathbb\{R\}\\times\\mathbb\{R\}^\{2\}\.

- 1\.Conserved Spatial Momentump0​\(g,μ\)p\_\{0\}\(g,\\mu\): 𝐏0\\displaystyle\\mathbf\{P\}\_\{0\}=Rθ​𝐏\\displaystyle=R\_\{\\theta\}\\mathbf\{P\}\(102\)J0\\displaystyle J\_\{0\}=J\+\(𝐱×𝐏0\)⋅𝐞z,\\displaystyle=J\+\(\\mathbf\{x\}\\times\\mathbf\{P\}\_\{0\}\)\\cdot\\mathbf\{e\}\_\{z\}\\,,\(103\)where\(𝐱×𝐏0\)⋅𝐞z:=x​P0,y−y​P0,x\(\\mathbf\{x\}\\times\\mathbf\{P\}\_\{0\}\)\\cdot\\mathbf\{e\}\_\{z\}:=xP\_\{0,y\}\-yP\_\{0,x\}represents the standard 2D cross product\.
- 2\.Inversion to Body Momentumμ​\(g,p0\)\\mu\(g,p\_\{0\}\): 𝐏\\displaystyle\\mathbf\{P\}=RθT​𝐏0\\displaystyle=R\_\{\\theta\}^\{T\}\\mathbf\{P\}\_\{0\}\(104\)J\\displaystyle J=J0−\(𝐱×𝐏0\)⋅𝐞z\.\\displaystyle=J\_\{0\}\-\(\\mathbf\{x\}\\times\\mathbf\{P\}\_\{0\}\)\\cdot\\mathbf\{e\}\_\{z\}\\,\.\(105\)

### C\.3The Special Euclidean GroupSE​\(3\)\\mathrm\{SE\}\(3\)

For full 3D rigid motions, letg=\(R,𝐱\)∈SE​\(3\)g=\(R,\\mathbf\{x\}\)\\in\\mathrm\{SE\}\(3\), whereR∈SO​\(3\)R\\in\\mathrm\{SO\}\(3\)is the rotation matrix and𝐱∈ℝ3\\mathbf\{x\}\\in\\mathbb\{R\}^\{3\}is the translation vector\. The body momentum isμ=\(𝐉,𝐏\)∈ℝ3×ℝ3\\mu=\(\\mathbf\{J\},\\mathbf\{P\}\)\\in\\mathbb\{R\}^\{3\}\\times\\mathbb\{R\}^\{3\}\(body angular and linear momentum\), and the conserved spatial momentum isp0=\(𝐉0,𝐏0\)∈ℝ3×ℝ3p\_\{0\}=\(\\mathbf\{J\}\_\{0\},\\mathbf\{P\}\_\{0\}\)\\in\\mathbb\{R\}^\{3\}\\times\\mathbb\{R\}^\{3\}\.

- 1\.Conserved Spatial Momentump0​\(g,μ\)p\_\{0\}\(g,\\mu\): 𝐏0\\displaystyle\\mathbf\{P\}\_\{0\}=R​𝐏\\displaystyle=R\\mathbf\{P\}\(106\)𝐉0\\displaystyle\\mathbf\{J\}\_\{0\}=R​𝐉\+𝐱×𝐏0\.\\displaystyle=R\\mathbf\{J\}\+\\mathbf\{x\}\\times\\mathbf\{P\}\_\{0\}\\,\.\(107\)
- 2\.Inversion to Body Momentumμ​\(g,p0\)\\mu\(g,p\_\{0\}\): 𝐏\\displaystyle\\mathbf\{P\}=RT​𝐏0\\displaystyle=R^\{T\}\\mathbf\{P\}\_\{0\}\(108\)𝐉\\displaystyle\\mathbf\{J\}=RT​\(𝐉0−𝐱×𝐏0\)\.\\displaystyle=R^\{T\}\(\\mathbf\{J\}\_\{0\}\-\\mathbf\{x\}\\times\\mathbf\{P\}\_\{0\}\)\\,\.\(109\)