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

摘要

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