Dirac Structures, Variational Principles, and Implicit Lagrangian Systems

Dr. Hiroaki Yoshimura, Department of Mechanical Engineering, Waseda University, Tokyo, Japan

We develop an idea of implicit Lagrangian systems in the context of an induced Dirac structure $D_{\Delta_{Q}}$ on $T^{\ast}Q$. First, we illustrate that an induced Dirac structure can be defined from a constraint distribution $\Delta_Q$ on a configuration manifold $Q$ and also that the Dirac differential $\mathfrak{D}L$ of a (possibly degenerate) Lagrangian $L$ can be defined by employing the natural symplectomorphisms between the spaces $T^{\ast}TQ$, $TT^{\ast}Q$ and $T^{\ast}T^{\ast}Q$, which include the generalized Legendre transform. Then, an implicit Lagrangian system $(L, \Delta_{Q}, X)$ is defined associated to a vector filed $X$ on $T^{\ast}Q$ and the induced Dirac structure $D_{\Delta_{Q}}$ such that $(X,\mathfrak{D}L) \in D_{\Delta_{Q}}$. Further, we demonstrate variational structures by a generalized variational principle called the Hamilton-Pontryagin principle, and illustrative examples of nonholonomic constrained systems, as well as an L-C circuit that is a typical degenerate Lagrangian system with holonomic constraints, are shown to be represented by implicit Lagrangian systems.

http://www.cds.caltech.edu/research/seminars/pdf/2005-10-04_yoshimura_abstract.pdf