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<p>This work presents a comprehensive review of the <i>k</i>-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 <i>k</i>-polysymplectic energy–momentum method, new related stability analysis techniques, and apply them to Hamiltonian systems of ordinary differential equations relative to a <i>k</i>-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.</p>

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An Energy–Momentum Method for Ordinary Differential Equations with an Underlying k-Polysymplectic Manifold

  • 期刊论文

  • 发表日期：

    2025年2月12日

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

摘要

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.