Dirac structures and the Legendre transformation for implicit Lagrangian and Hamiltonian systems

Yoshimura, H. and J. E. Marsden

Lecture Notes in Control and Inform. Sci., 366, (2007), 233-247

Abstract:

This paper begins by recalling how a constraint distribution on a configuration manifold induces a Dirac structure together with an implicit Lagrangian system, a construction that is valid even for degenerate Lagrangians. In such degenerate cases, it is shown in this paper that an implicit Hamiltonian system can be constructed by using a generalized Legendre transformation, where the primary constraints are incorporated into a generalized Hamiltonian on the Pontryagin bundle. Some examples of degenerate Lagrangians for L-C circuits, nonholonomic systems, and point vortices illustrate the theory.