标题： 可解结构用于哈密顿系统 显示英文标题

标题： Solvable Structures for Hamiltonian Systems

摘要： 在本文中，我们研究了与哈密顿方程相关的可解结构。 对于具有$n$自由度的完全可积哈密顿系统，我们构建了一个由$2n$个哈密顿向量场组成的规范可解结构。 我们推导出相应的显式表达式的普法夫形式，它们的积分提供了哈密顿方程的解。 我们表明，上部$n$形式给出了作用变量，而下部$n$形式则给出了系统的角度变量。 这为阿诺尔德-刘维尔定理提供了一种新的可解结构解释。 我们通过推导显式解和$n$谐振子以及 Calogero-Moser 系统的作用-角度变量来说明这一理论。 显示英文摘要

摘要： In this paper, we investigate solvable structures associated to Hamiltonian equations. For a completely integrable Hamiltonian system with $n$ degrees of freedom, we construct a canonical solvable structure consisting of $2n$ Hamiltonian vector fields. We derive explicit expressions for the corresponding Pfaffian forms, whose integration provides solutions to the Hamiltonian equations. We show that the upper $n$ forms give the action varibles, while the lower $n$ forms yield the angle variables of the system. This offers a novel interpretation of the Arnold--Liouville theorem in terms of solvable structures. We ilustrate the theory by deriving explicit solutions and action--angle variables for $n$ harmonic oscillators and the Calogero--Moser system.