Representations of Dirac Structures and Implicit Port-Controlled Lagrangian Systems

Yoshimura, H. and J. E. Marsden

Proceedings of the 18th International Symposium on Mathematical Theory of Networks and Systems (MTNS), (2008)

Abstract:

In this paper, we will develop two different representations for induced Dirac structures and their associated IPCL systems; namely, (1) a standard representation with using Lagrange multipliers; and (2) a representation without using Lagrange multipliers. Those representations are consistent with those developed by Courant and Weinstein. Specifically, the second representation without using Lagrange multipliers may be crucial in formulation of constrained mechanical systems since it systematically enables one to eliminate unnecessary constraint forces. In mechanics, it is known that the elimination of constraint forces can be done by the orthogonal complement method or the null space method, although the link with Dirac structures has not been clarified. The present paper fills this gap to show that the orthogonal complement method can be incorporated into the context of Dirac structures and the associated IPCL systems and we will further show the link with the topological method in electrical network theory using the so-called fundamental cut-set and loop matrices. In the paper, we shall illustrate out ideas by an example of L-C circuits.