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.