Birkhoff力学的研究进展

Birkhoff力学是Hamilton力学的一个自然发展,是分析力学发展的一个新阶段,它广泛应用于力学、物理学和工程.本文总结Birkhoff力学的形成和发展,特别是近二十年所取得的成就.首先,从Birkhoff的《动力系统》中的有关段落开始,叙述Birkhoff力学的起源.其次,叙述这个力学的基本原理——Pfaff--Birkhoff原理以及这个力学的基本方程——Birkhoff方程的形成和发展.第三,简述Birkhoff力学的一些专门问题,包括约束Birkhoff系统,Birkhoff方程的积分方法,Birkhoff动力学逆问题,Birkhoff方程的运动稳定性,Birkhoff系统的几何方法,Birkhoff系统的全局分析等.最后,对Birkhoff力学的未来研究提出一些建议.

ADVANCES IN RESEARCH ON BIRKHOFFIAN MECHANICS

The Birkhoffian mechanics is a natural development of Hamiltonian mechanics, and is a new development stage of analytical mechanics.This mechanics can be widely applied to mechanics, physics and engineering.In this paper, the formation and development of Birkhoffian mechanics, especially the achievements in the last twenty years are summarized.First, to start with the relevant paragraphs of Birkhoff's "Dynamical Systems", the origin of the Birkhoffian mechanics is narrated.Next, the formations and developments of the basic principle, the Pfaff-Birkhoff principle, and the basic equations, Birkhoff equations for the mechanics, are described.After that, a brief introduction to some special problems of the mechanics is given, including constrained Birkhoffian systems, integration methods for Birkhoff equations, inverse problems of Birkhoffian dynamics, stability of motion of Birkhoff equations, geometric methods for Birkhoffian systems, global analysis of Birkhoffian systems and so on.Finally, some suggestions are put forward for the future research of Birkhoffian mechanics.