Representations of Dirac Structures and Implicit Port-Controlled Lagrangian Systems

The idea of multiport systems has been known as a useful tool when regarding a system as an interconnection of physical elements throughout principle of power invariance, which has been widely used in electrical circuits and networks (see, for instance, [4, 14]), where the principle of power invariance is known as Tellegen’s theorem in electrical network theory (see [6]). From the viewpoint of the analogy between mechanical and electrical systems, much effort has been done to develop a network-theoretic approach to nonlinear mechanical systems such as multibody systems in the context of interconnected systems (see, for instance, [15]). Recently, it was shown by [12] and [3] that such interconnections can be represented by Dirac structures, which may be a generalization of both symplectic as well as Poisson structures (see [8, 7]) and also that interconnections of L-C circuits can be modeled by Dirac structures and then incorporated into the context of implicit Hamiltonian systems (see also [1, 2]). On the Lagrangian side, a notion of implicit Lagrangian systems, namely, a Lagrangian analogue of implicit Hamiltonian systems, was developed by [16, 17], where nonholonomic mechanical systems and L-C circuits as degenerate Lagrangian systems were shown to be formulated in the context of implicit Lagrangian systems in which induced Dirac structures were systematically introduced. Furthermore, it was shown by [19] that even for the case in which a given Lagrangian is degenerate, an implicit Hamiltonian system can be constructed