Dirac Structures in Pseudo-Gradient Systems With an Emphasis on Electrical Networks

The relationship between implicitly defined Hamiltonian systems and pseudo-gradient systems is investigated. It is shown that under certain conditions a Hamiltonian system, implicitly defined with respect to a Dirac structure on the state space manifold, can be used to generate a pseudo-gradient system on the Lagrangian submanifold of the cotangent bundle of the state space. This elucidates the relationship between the Smale pseudo-gradient and the Bloch-Couch Hamiltonian formulations of lossless circuit dynamics. A more general situation is also considered where the implicit Hamiltonian system gives rise to a foliation of the Lagrangian submanifold in which each leaf is a pseudo-Riemannian manifold which in turn gives rise to a pseudo-gradient system.