非自治系统的离散时间广义正则变换
Discrete-time generalized canonical transformations for non-autonomous systems
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中文总结 AI 辅助
研究非自治哈密顿系统离散化问题，基于广义正则变换提出几何方法，在扩展相空间构造辛同胚，其投影定义保结构离散流，保证关键不变量守恒。
中文摘要 AI 辅助
当描述其演化的微分方程明确依赖于时间时，动力系统被称为非自治的。在研究此类系统的各种几何方法中，余辛形式提供了一个将辛几何扩展到含时哈密顿系统的自然框架。然而，在数值离散化下保持相关几何结构仍是一个具有挑战性的问题：标准积分器，如显式欧拉格式，通常无法守恒余辛体积或底层泊松结构。在这项工作中，我们基于广义正则变换提出了一种用于非自治哈密顿系统离散化的几何方法。该方法在扩展相空间$T^*(Q \times \mathbb{R})$上构造一个辛同胚，其在$T^*Q \times \mathbb{R}$上的投影定义了一个保结构的离散流。我们表明这种表述保证了关键不变量的守恒，包括体积形式、泊松括号以及每个时间纤维上的辛结构。
英文摘要
A dynamical system is said to be \emph{non-autonomous} when the differential equations describing its evolution depends explicitly on time. Among the various geometric approaches to investigate such systems, the cosymplectic formulation provides a natural framework that extends symplectic geometry to time-dependent Hamiltonians systems. However, preserving the associated geometric structures under numerical discretization remains a challenging problem: standard integrators, such as explicit Euler schemes, generally fail to conserve the cosymplectic volume or the underlying Poisson structure. In this work we propose a geometric method for the discretization of non-autonomous Hamiltonian systems based on \emph{generalized canonical transformations}. The approach constructs a symplectomorphism on the extended phase space $T^*(Q \times \mathbb{R})$ whose projection onto $T^*Q \times \mathbb{R}$ defines a structure-preserving discrete flow. We show that this formulation guarantees the preservation of key invariants, including the volume form, the Poisson bracket, and the symplectic structure on each time fiber.
