Reduction of Constrained Mechanical Systems and Stability of Relative Equilibria

A mechanical system with perfect constraints can be described, under some mild assumptions , as a constrained Hamiltonian system (M; ; H; D; W): (M;) (the phase space) is a symplectic manifold, H (the Hamiltonian) a smooth function on M, D (the constraint submanifold) a submanifold of M, and W (the projection bundle) a vector sub-bundle of T D M, the reduced tangent bundle along D. We prove that when these data satisfy some suitable conditions, the time evolution of the system is governed by a well deened diieren-tial equation on D. We deene constrained Hamiltonian systems with symmetry, and prove a reduction theorem. Application of that theorem is illustrated on the example of a convex heavy body rolling without slipping on a horizontal plane. Two other simple examples show that constrained mechanical systems with symmetry may have an attractive (or repulsive) set of relative equilibria.