- 03-293 Thomas Chen
- Critical Manifolds and Stability in Hamiltonian Systems with Non-Holonomic Constraints
(140K, AMS LaTeX)
Jun 20, 03
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Abstract. We explore a particular approach to the analysis of dynamical and
geometrical properties of autonomous, Pfaffian
non-holonomic systems in classical mechanics.
The method is based on the construction of
a certain auxiliary constrained Hamiltonian system, which comprises the
non-holonomic mechanical system as a dynamical subsystem on an invariant manifold.
The embedding system possesses a completely natural structure in the context
of symplectic geometry, and using it in order to understand
properties of the subsystem has compelling advantages.
We discuss generic geometric and topological properties of the critical sets of both
embedding and physical system, using Conley-Zehnder theory and by relating
the Morse-Witten complexes of the 'free' and constrained system to one another.
Furthermore, we give a qualitative discussion of the stability of motion in
the vicinity of the critical set. We point out key relations
to sub-Riemannian geometry, and a potential computational application.