Communications in Mathematical Physics

, Volume 174, Issue 2, pp 295–318 | Cite as

Reduction of constrained mechanical systems and stability of relative equilibria

  • Charles-Michel Marle


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) (thephase space) is a symplectic manifold,H (theHamiltonian) a smooth function onM, D (theconstraint submanifold) a submanifold ofM, andW (theprojection bundle) a vector sub-bundle ofT D M, the reduced tangent bundle alongD. We prove that when these data satisfy some suitable conditions, the time evolution of the system is governed by a well defined differential equation onD. 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.


Neural Network Manifold Complex System Time Evolution Nonlinear Dynamics 
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Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Charles-Michel Marle
    • 1
  1. 1.Institut de Mathématiques de JussieuUniversité Pierre et Marie CurieParis cedex 05France

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