  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.
