构造微分方程组的线性哈密顿结构

Constructing Linear Hamiltonian Structure for System of Differential Equations

经典力学系统的任意守恒量都可以作为系统的哈密顿量,只需在定义的相空间上增加一个适当的辛结构或泊松括号,就可以把原运动方程表示为哈密顿系统的正则方程。本文分析了辛结构与哈密顿函数、泊松括号的关系,研究了几例二自由度的微分方程,并推导了这些系统的哈密顿量和泊松括号。同时构造了此辛结构下的正则坐标,推导了系统的拉格朗日量,显式地给出了几个(超)可积系统的额外守恒量在正则坐标下的表达式。

Any conserved quantity of a classical mechanical system can be regarded as the Hamiltonian of the system,and the original equation of motion can be expressed as the canonical Hamiltonian equations of the system by redefining an ap-propriate symplectic structure or Poisson bracket in the phase space.In this paper,the relationship between symplectic struc-ture and Hamiltonian function and Poisson's bracket is analyzed,several differential equations with two degrees of freedom are studied,and the Hamiltonians and Poisson's brackets of these systems are derived.The corresponding canonical coordi-nates are constructed,the Lagrangians of the systems are derived,and the explicit forms of the additional first integrals of some(super-)integrable systems in canonical coordinates are given.