Yoshimura, Hiroaki (Waseda Univ.), Marsden, Jerrold (California Inst. of Tech.)

Dirac Structures and Implicit Lagrangian Systems in Electric Networks

Scheduled for presentation during the Regular Session "Physics and Control" (WeA08), Wednesday, July 26, 2006, 12:05−12:30, Room I

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on April 25, 2024

Abstract

The design of devices such as L-C transmission lines for extremely wideband signal shaping has been highly exquisite and hence sophisticated mathematical modeling may be required for further developments. In conjunction with a formulation of conservative electric circuits, a notion of implicit Hamiltonian systems was developed by van der Schaft and Maschke and Bloch and Crouch, where interconnections of circuit elements were modeled by Dirac structures and then incorporated into the Hamiltonian formalism. However, from the viewpoint of the Lagrangian side, conservative electric circuits such as L-C circuits, while more fundamental, are generally degenerate Lagrangian systems and hence one requires a special technique, such as the generalized Legendre transform, to establish a link to the Hamiltonian side. Recently, a notion of implicit Lagrangian system, that is, a Lagrangian analogue of implicit Hamiltonian systems, has been developed by Yoshimura and Marsden, where nonholonomic mechanical systems and degenerate Lagrangian systems such as L-C circuits can be systematically formulated in the implicit Lagrangian context in which Dirac structures are also used. In the paper, we illustrate how electric circuits such as L-C circuits can be formulated in the context of implicit Lagrangian systems, along with some illustrative examples. First, we employ ideas of geometric mechanics for circuits by a mechanical analogy with electric systems, where a configuration space for circuits can be defined as the space of charges. Then, we demonstrate that an induced Dirac structure on the cotangent bundle of the configuration space can be defined from given KCL constraints and that electric circuits with degenerate Lagrangians can be formulated in the context of implicit Lagrangian systems and associated induced Dirac structures. We shall also investigate how implicit Lagrangian systems can be related to implicit Hamiltonian systems in the context of the generalized Legendre transformation together with illustrative examples such as one-dimensional lossless transmission lines.