对称控制Hamilton系统约化理论的研究进展

Some developments of reduction theory for controlled Hamiltonian system with symmetry

一个正则控制Hamilton (regular controlled Hamiltonian, RCH)系统是一个具有外力和控制的Hamilton系统,在外力和控制作用下,一般来讲,这个RCH系统不是Hamilton系统.然而,它是一个与Hamilton系统密切相关的动力系统,我们可以在Hamilton系统的研究中,通过对外力和控制扩展的方法研究这种RCH系统.本文从具有对称的Hamilton系统的Marsden-Weinstein辛约化完备性和系统相空间几何结构变化出发,简要介绍了最近几年控制Hamilton系统约化理论及其应用研究工作的研究进展情况.

A regular controlled Hamiltonian (RCH)system is a Hamiltonian system with external force and control.In general,an RCH system under the action of external force and control is not Hamiltonian,however,it is a dynamical system closely related to a Hamiltonian system,and it can be explored and studied by extending the methods for external force and control in the study of Hamiltonian systems.In this paper,we introduce briefly some recent developments in the study of reduction theory for the RCH systems with symmetries and applications, in terms of the completeness of Marsden-Weinstein reduction and the change of geometrical structures.