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An energy-momentum method for ordinary differential equations with an underlying $k$-polysymplectic manifold

Leonardo Colombo, Javier de Lucas, Xavier Rivas, Bartosz M. Zawora

TL;DR

This work extends geometric mechanics to ordinary differential equations by developing a comprehensive energy--momentum method for Hamiltonian systems on $k$-polysymplectic manifolds. It first cleans up and broadens the $k$-polysymplectic Marsden--Weinstein reduction theory, highlighting weak regular values and affine momentum action, then defines relative equilibria and a formal stability framework via second variations of an ${oldsymbol{ om{oldsymbol{oldsymbol{ullet}}}}}$-valued energy function. The paper provides rigorous reduction criteria, fixes previous literature errors, and demonstrates the approach through multiple physically and mathematically meaningful examples, including complex Schwarz equations, products of oscillators, affine Lie systems, quantum harmonic oscillators with dissipation, and polynomial dynamical systems. The results broaden the applicability of polysymplectic geometry to ODEs, offering practical tools for stability analysis and reduced dynamics with potential impact in control theory, mathematical physics, and applied dynamical systems.

Abstract

This work presents a comprehensive review of the $k$-polysymplectic Marsden-Weinstein reduction theory, rectifying prior errors and inaccuracies in the literature while introducing novel findings. It also emphasises the genuine practical significance of seemingly minor technical details. On this basis, we introduce a novel $k$-polysymplectic energy-momentum method, new related stability analysis techniques, and apply them to Hamiltonian systems of ordinary differential equations relative to a $k$-polysymplectic manifold. We provide detailed examples of both physical and mathematical significance, including the study of complex Schwarz equations related to the Schwarz derivative, a series of isotropic oscillators, integrable Hamiltonian systems, quantum oscillators with dissipation, affine systems of differential equations, and polynomial dynamical systems.

An energy-momentum method for ordinary differential equations with an underlying $k$-polysymplectic manifold

TL;DR

This work extends geometric mechanics to ordinary differential equations by developing a comprehensive energy--momentum method for Hamiltonian systems on -polysymplectic manifolds. It first cleans up and broadens the -polysymplectic Marsden--Weinstein reduction theory, highlighting weak regular values and affine momentum action, then defines relative equilibria and a formal stability framework via second variations of an -valued energy function. The paper provides rigorous reduction criteria, fixes previous literature errors, and demonstrates the approach through multiple physically and mathematically meaningful examples, including complex Schwarz equations, products of oscillators, affine Lie systems, quantum harmonic oscillators with dissipation, and polynomial dynamical systems. The results broaden the applicability of polysymplectic geometry to ODEs, offering practical tools for stability analysis and reduced dynamics with potential impact in control theory, mathematical physics, and applied dynamical systems.

Abstract

This work presents a comprehensive review of the -polysymplectic Marsden-Weinstein reduction theory, rectifying prior errors and inaccuracies in the literature while introducing novel findings. It also emphasises the genuine practical significance of seemingly minor technical details. On this basis, we introduce a novel -polysymplectic energy-momentum method, new related stability analysis techniques, and apply them to Hamiltonian systems of ordinary differential equations relative to a -polysymplectic manifold. We provide detailed examples of both physical and mathematical significance, including the study of complex Schwarz equations related to the Schwarz derivative, a series of isotropic oscillators, integrable Hamiltonian systems, quantum oscillators with dissipation, affine systems of differential equations, and polynomial dynamical systems.
Paper Structure (21 sections, 12 theorems, 192 equations)

This paper contains 21 sections, 12 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Fundamentals
  3. Lyapunov stability
  4. On -polysymplectic manifolds
  5. On -Hamiltonian functions and vector fields
  6. -Polysymplectic momentum maps
  7. -Polysymplectic Marsden--Weinstein reduction
  8. A review on the -polysymplectic Marsden--Weinstein reduction
  9. On the conditions for the --polysymplectic Marsden--Weinstein reduction
  10. On the -polysymplectic manifold given by the product of symplectic manifolds
  11. The -polysymplectic energy momentum-method
  12. -polysymplectic relative equilibrium points
  13. Stability in the -polysymplectic energy momentum-method
  14. Applications and examples
  15. Complex Schwarz equations
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $x_e$ be an equilibrium point of Eq::NonAutDyn and let $\mathcal{M}:P\rightarrow \mathbb{R}$ be a continuous function such that $\mathcal{M}(x_e)=0$, $\mathcal{M}(x)>0$, and $\dot{\mathcal{M}}(x)\leq 0$ for every $x\in B_{x_e,r}$ and some $r\in \mathbb{R}^+$. Then, $x_e$ is stable.

Theorems & Definitions (30)

  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 20 more