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 is nonholonomic motion planning, the other is geometric dynamics of nonholonomic constrained systems, which both amply use the modern differential geometry, such as fibre bundle theory, the structure of 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 fundamental concepts 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 constraint mainifolds, the global formulation of D'Alembert-Lagrange's equations and Chaplygin's equations, and nonholonomic mechanics on a Riemann-Cartan manifold. Meanwhile, the geometric significance of Chetaev'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's symmetries and Lie's symmetries, momentum maps and reduction theory of nonholonomic mechanics, and Vakonomic dynamics are briefly reviewed.