Critical manifolds and stability in Hamiltonian systems with non-holonomic constraints

TL;DR

This paper introduces a symplectic geometric approach to analyze the stability and critical manifolds of non-holonomic Hamiltonian systems, linking geometric properties with dynamical stability in classical mechanics.

Contribution

It develops an auxiliary constrained Hamiltonian framework for non-holonomic systems, enabling geometric and topological analysis of critical sets and stability properties.

Findings

Characterization of critical sets using Conley-Zehnder theory

Relation of Morse-Witten complexes between free and constrained systems

Insights into stability near critical manifolds

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.