Geometric Hamilton-Jacobi Theory for Nonholonomic Dynamical Systems

TL;DR

This paper develops a geometric Hamilton-Jacobi framework for nonholonomic systems, linking solutions to constraints and symmetries, with explicit local expressions and an illustrative example.

Contribution

It introduces a geometric Hamilton-Jacobi theory tailored for nonholonomic systems, connecting solutions with constraints and constants of motion.

Findings

Established a geometric formulation for nonholonomic Hamilton-Jacobi theory

Analyzed the relationship between solutions, constraints, and symmetries

Provided explicit local expressions and an example with a nonholonomic free particle

Abstract

The geometric formulation of Hamilton--Jacobi theory for systems with nonholonomic constraints is developed, following the ideas of the authors in previous papers. The relation between the solutions of the Hamilton--Jacobi problem with the symplectic structure defined from the Lagrangian function and the constraints is studied. The concept of complete solutions and their relationship with constants of motion, are also studied in detail. Local expressions using quasivelocities are provided. As an example, the nonholonomic free particle is considered.