Kinematic reduction and the Hamilton-Jacobi equation

TL;DR

This paper explores the connection between Hamilton-Jacobi theory and control system reduction using skew-symmetric algebroids, providing new geometric insights into nonholonomic mechanics with controls and forces.

Contribution

It introduces a geometric framework linking Hamilton-Jacobi theory to kinematic reduction via skew-symmetric algebroids, enhancing understanding of nonholonomic control systems.

Findings

Established a geometric interpretation of control reduction

Applied techniques to nonholonomic systems with external forces

Provided a new mathematical approach for mechanics on algebroids

Abstract

A close relationship between the classical Hamilton-Jacobi theory and the kinematic reduction of control systems by decoupling vector fields is shown in this paper. The geometric interpretation of this relationship relies on new mathematical techniques for mechanics defined on a skew-symmetric algebroid. This geometric structure allows us to describe in a simplified way the mechanics of nonholonomic systems with both control and external forces.