Mechanical Hamiltonization of unreduced $\phi$-simple Chaplygin systems

TL;DR

This paper demonstrates that trajectories of unreduced $$-simple Chaplygin systems can be viewed as reparametrized geodesics under a modified metric, providing explicit constructions and extending results beyond purely kinetic systems.

Contribution

It establishes a constructive proof linking system trajectories to geodesics with explicit metrics, extending to non-kinetic $$-simple Chaplygin systems.

Findings

Trajectories are reparametrizations of horizontal geodesics under a modified metric.

Explicit construction of these metrics in specific examples.

Extension of results to non-kinetic $$-simple Chaplygin systems.

Abstract

In this paper, we prove that the trajectories of unreduced $\phi$-simple Chaplygin kinetic systems are reparametrizations of horizontal geodesics with respect to a modified Riemannian metric. Furthermore, our proof is constructive and these Riemannian metrics, which are not unique, are obtained explicitly in interesting examples. We also extend these results to $\phi$-simple Chaplygin mechanical systems (not necessarily kinetic).