Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints

TL;DR

This paper extends Hamilton-Jacobi theory to degenerate Lagrangian systems with holonomic and nonholonomic constraints, introducing the Dirac-Hamilton-Jacobi equation and demonstrating its application to integrate equations of motion.

Contribution

It develops a generalized Hamilton-Jacobi framework for Lagrange-Dirac systems, including degenerate and constrained cases, and applies it to weakly degenerate Chaplygin systems.

Findings

Reduction to non-degenerate almost Hamiltonian systems

Derivation of the Dirac-Hamilton-Jacobi equation for constrained systems

Exact integration of equations of motion using the new theory

Abstract

We extend Hamilton-Jacobi theory to Lagrange-Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton-Jacobi equation as the Dirac-Hamilton-Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the theory specializes to the recently developed nonholonomic Hamilton-Jacobi theory. We are particularly interested in applications to a certain class of degenerate nonholonomic Lagrangian systems with symmetries, which we refer to as weakly degenerate Chaplygin systems, that arise as simplified models of nonholonomic mechanical systems; these systems are shown to reduce to non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian systems defined with non-closed two-forms. Accordingly, the Dirac-Hamilton-Jacobi equation reduces to a variant of the nonholonomic Hamilton-Jacobi equation associated with the reduced system. We illustrate through a few examples how the Dirac-Hamilton-Jacobi equation can be used to exactly integrate the equations of motion.