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,Omega,H,D,W): (M,Omega) (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 defined differential equation on D. We define 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.