标题： 雅可比哈密顿积分方法 显示英文标题

标题： Jacobi Hamiltonian Integrators

摘要： 我们开发了一种在雅可比流形中构造保持结构的积分方法。 哈密顿力学源于辛几何和泊松几何，长期以来为经典物理中的保守系统建模提供了基础。 雅可比流形，作为接触流形和泊松流形的推广，扩展了这一理论，并适用于包含时间依赖、耗散和热力学现象的情况。 基于几何积分方法的最新进展——特别是保持泊松系统关键特征的泊松哈密顿积分方法（PHI）——我们提出了雅可比哈密顿积分方法的构造。 我们的方法探索雅可比流形与齐次泊松流形之间的对应关系，旨在扩展PHI技术的同时确保保持齐次结构。 这项工作开发了这种推广所需的理论工具，并概述了一种与雅可比动力学兼容的数值积分技术。 通过专注于齐次泊松视角而非直接接触实现，我们提供了一条清晰的路径，在雅可比框架内对时间依赖和耗散系统进行结构保持积分。 显示英文摘要

摘要： We develop a method of constructing structure-preserving integrators for Hamiltonian systems in Jacobi manifolds. Hamiltonian mechanics, rooted in symplectic and Poisson geometry, has long provided a foundation for modelling conservative systems in classical physics. Jacobi manifolds, generalizing both contact and Poisson manifolds, extend this theory and are suitable for incorporating time-dependent, dissipative and thermodynamic phenomena. Building on recent advances in geometric integrators - specifically Poisson Hamiltonian Integrators (PHI), which preserve key features of Poisson systems - we propose a construction of Jacobi Hamiltonian Integrators. Our approach explores the correspondence between Jacobi and homogeneous Poisson manifolds, with the aim of extending the PHI techniques while ensuring preservation of the homogeneity structure. This work develops the theoretical tools required for this generalization and outlines a numerical integration technique compatible with Jacobi dynamics. By focusing on the homogeneous Poisson perspective rather than on direct contact realizations, we provide a clear pathway for structure-preserving integration of time-dependent and dissipative systems within the Jacobi framework.