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Reduction of constrained mechanical systems and stability of relative equilibria. (English) Zbl 0859.70012
The author develops in detail an intrinsic notion of a mechanical system with constraints, which may be holonomic or nonholonomic, and nonlinear in the velocities, based on the Lagrangian mechanics. Special and most important constraints are the regular and perfect ones, for which a Hamiltonian formulation is possible, i.e. the motions of the constrained system are described as integral curves of a vector field on a certain submanifold of the cotangent bundle of the configuration space, the constraint force being given by an intrinsic formula. These results lead to a generalized version of constrained Hamiltonian system with symmetry, for which a reduction theorem is proved. An application and motivation for the theory is the system of a rolling stone. Other simple examples are discussed.

MSC:
70H05 Hamilton’s equations
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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