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

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.

44 pages, 3 figures

References in corpus (9)

• Geometric Hamilton-Jacobi Theory
• Geometric Hamilton-Jacobi Theory for Nonholonomic Dynamical Systems
• Nonholonomic Hamilton-Jacobi Theory via Chaplygin Hamiltonization
• Variational calculus with constraints on general algebroids
• Variational and Geometric Structures of Discrete Dirac Mechanics
• Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic Mechanics
• Nonholonomic Hamilton-Jacobi equation and Integrability
• Non-holonomic constrained systems as implicit differential equations
• On the Hamilton-Jacobi equation for second class constrained systems

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