Reduction of a Hamilton-Jacobi equation for nonholonomic systems

TL;DR

This paper develops a reduction method for Hamilton-Jacobi equations in nonholonomic systems with symmetries, enabling simplified analysis and solution reconstruction for complex mechanical systems.

Contribution

It introduces a generalized reduction procedure for Hamilton-Jacobi theory applicable to various nonholonomic systems with different constraints and symmetry configurations.

Findings

Reduction simplifies solving Hamilton-Jacobi equations in nonholonomic systems.

Reconstruction of solutions from reduced equations is feasible.

Applicable to diverse scenarios like vehicle motion and free particles.

Abstract

Nonholonomic mechanical systems have been attracting more interest in recent years because of their rich geometric properties and their applications in Engineering. In all generality, we discuss the reduction of a Hamilton-Jacobi theory for systems subject to nonholonomic constraints and that are invariant under the action of a group of symmetries. We consider nonholonomic systems subject to linear or nonlinear constraints, with different positioning with respect to the symmetries. We describe the reduction procedure first, to later reconstruct solutions in the unreduced picture, by starting from a reduced Hamilton-Jacobi equation. Examples can be depicted in a wide range of scenarios: from free particles with linear constraints, to vehicle motion.