Reduction of constrained mechanical systems and stability of relative equilibria
Abstract
A mechanical system with perfect constraints can be described, under some mild assumptions, as a constrained Hamiltonian system (M, Ω, H, D, W): (M, Ω) (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.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- December 1995
- DOI:
- 10.1007/BF02099604
- Bibcode:
- 1995CMaPh.174..295M