非完整约束系统几何动力学研究进展:Lagrange理论及其它

近10年来, 非完整力学的发展主要集中在两个相互关联的方向上, 一个是非完整运动规划, 另一个则是非完整约束系统的几何动力学, 这两个研究方向都充分地利用了现代几何学, 如纤维丛理论、辛流形和Poisson流形结构等等.本文主要综述非完整约束系统几何动力学的外附型和内禀型Lagrange理论, 包括非定常力学系统所需要的射丛几何学的基本概念、射丛按约束的直和分解、约束流形上的水平分布、D'Alembert-Lagrange方程与Chaplygin方程的整体描述、以及Riemann-Cartan流形上的非完整力学, 文中对Chetaev条件和d-δ交换关系的几何意义作了深入讨论.除此之外, 简要评述非完整力学的Hamilton理论与赝Poisson结构、Noether对称性和Lie对称性、动量映射与对称约化、Vakonomic动力学等几个非常重要专题的研究进展. 

PROGRESS OF GEOMETRIC DYNAMICS OF NON-HOLONOMIC CONSTRAINED MECHANICAL SYSTEMS:LAGRANGE THEORY AND OTHERS

In recent ten years,nonholonomic mechanics develops mainly in two associated directions. One isnonholonomic motion planning, the other is geometric dynamics ofnonholonomic constrained systems, which both amply use themodern differential geometry, such as fibre bundle theory, the structureof symplectic and Poisson manifolds. The two kinds of Lagrange theories of geometric dynamics for nonholonomic constrained systems,i.e., the extrinsic and the intrinsic ones, are summarized specially. They include the fundamentalconcepts of jet bundle geometry needed in describing time-dependent mechanical systems,decomposition of jet bundles into a direct sum according to the constraints, horizontal distributions on constraintmainifolds, the global formulation of D'Alembert-Lagrange's equations and Chaplygin's equations, and nonholonomic mechanicson a Riemann-Cartan manifold. Meanwhile, the geometric significance ofChetaev's conditions and $rd$-$delta$ commutation relation are discussed in depth. Finally some other important topics,such as Hamiltonian framework and pseudo-Poisson structure, Noether'ssymmetries and Lie's symmetries, momentum maps and reduction theory of nonholonomicmechanics, and Vakonomic dynamics are briefly reviewed.