Dirac Structures in Lagrangian Mechanics Part II: Variational Structures

Part I of this paper introduced the notion of implicit Lagrangian systems and their geometric structure was explored in the context of Dirac structures. In this part, we develop the variational structure of implicit Lagrangian systems. Specifically, we show that the implicit Euler-Lagrange equations can be formulated using an extended variational principle of Hamilton called the Hamilton-Pontryagin principle. This variational formulation incorporates, in a natural way, the generalized Legendre transformation, which enables one to treat degenerate Lagrangian systems. The definition of this generalized Legendre transformation makes use of natural maps between iterated tangent and cotangent spaces. Then, we develop an extension of the classical Lagranged’Alembert principle called the Lagrange-d’Alembert-Pontryagin principle for implicit Lagrangian systems with constraints and external forces. A particularly interesting case is that of nonholonomic mechanical systems that can have both constraints and external forces. In addition, we define a constrained Dirac structure on the constraint momentum space, namely the image of the Legendre transformation (which, in the degenerate case, need not equal the whole cotangent bundle). We construct an implicit constrained Lagrangian system associated with this constrained Dirac structure by making use of an Ehresmann connection. Two examples, namely a vertical rolling disk on a plane and an L-C circuit are given to illustrate the results.