Abstract

This paper considers the problem of representing a sufficiently smooth non-linear dynamical [system] as a structured potential-driven system. The proposed method is based on a decomposition of a differential one-form associated to a given vector field into its exact and anti-exact components, and into its co -exact and anti-coexact components. The decomposition method, based on the Hodge decomposition theorem, is rendered constructive by introducing a dual operator to the standard homotopy operator. The dual operator inverts locally the co-differential operator, and is used in the present paper to identify the symplectic structure of the dynamics. Applications of the proposed approach to gradient systems, Hamiltonian systems and generalized Hamiltonian systems are given to illustrate the proposed approach. Finally, integrability conditions for generalized Hamiltonian systems are established using the proposed decomposition.

中文翻译：

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非线性系统的几何分解、基于势的表示和可积性

摘要

本文考虑将足够平滑的非线性动态[系统]表示为结构化势驱动系统的问题。所提出的方法基于将与给定向量场相关联的微分单形式分解为其精确和反精确分量，以及其共精确和反共精确分量。基于霍奇分解定理的分解方法通过在标准同伦算子中引入对偶算子而变得具有建设性。对偶算子在局部反演协微分算子，在本文中用于识别动力学的辛结构。给出了所提出的方法在梯度系统、哈密顿系统和广义哈密顿系统中的应用，以说明所提出的方法。最后，