标题： 关于具有基本$k$-多辛流形的常微分方程的能量-动量方法 显示英文标题

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

摘要： 本文对$k$-多辛 Marsden-Weinstein约化理论进行了全面回顾，纠正了文献中的先前错误和不准确之处，并引入了新的研究成果。同时，它还强调了一些看似技术细节的小问题的实际重要性。在此基础上，我们提出了一个新的$k$-多辛能量-动量方法，新的相关稳定性分析技术，并将其应用于与$k$-多辛流形相关的常微分方程的哈密顿系统。我们提供了具有物理和数学意义的详细例子，包括与施瓦茨导数相关的复杂数值Schwarz方程的研究、一系列各向同性振荡器、可积哈密顿系统、耗散量子振荡器、微分方程的仿射系统以及多项式动力学系统。 显示英文摘要

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.