Geometric description of vakonomic and nonholonomic dynamics. Comparison of solutions

TL;DR

This paper introduces a new geometric framework for vakonomic and nonholonomic dynamics, enabling comparison of solutions and application to optimal control problems with illustrative examples.

Contribution

It develops a novel geometric approach to vakonomic and nonholonomic mechanics, facilitating solution comparison and advancing understanding of constrained dynamical systems.

Findings

New geometric description for vakonomic dynamics

Algorithm for comparing vakonomic and nonholonomic solutions

Illustrative examples demonstrating the theory

Abstract

We treat the vakonomic dynamics with general constraints within a new geometric framework which will be appropriate to study optimal control problems. We compare our formulation with Vershik-Gershkovich one in the case of linear constraints. We show how nonholonomic mechanics also admits a new geometrical description wich enables us to develop an algorithm of comparison between the solutions of both dynamics. Some examples illustrating the theory are treated.