Reduction of Stokes-Dirac structures and gauge symmetry in port-Hamiltonian systems

TL;DR

This paper presents a unified Poisson reduction method for simplifying Stokes-Dirac and simplicial Dirac structures in port-Hamiltonian systems, highlighting gauge symmetries and their physical implications.

Contribution

It introduces a Poisson reduction framework that unifies the symmetry reduction of infinite-dimensional and finite-dimensional Dirac structures in port-Hamiltonian systems.

Findings

Unified reduction technique for Dirac structures

Application to vibrating string example

Insight into gauge symmetry in port-Hamiltonian systems

Abstract

Stokes-Dirac structures are infinite-dimensional Dirac structures defined in terms of differential forms on a smooth manifold with boundary. These Dirac structures lay down a geometric framework for the formulation of Hamiltonian systems with a nonzero boundary energy flow. Simplicial triangulation of the underlaying manifold leads to the so-called simplicial Dirac structures, discrete analogues of Stokes-Dirac structures, and thus provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The port-Hamiltonian systems defined with respect to Stokes-Dirac and simplicial Dirac structures exhibit gauge and a discrete gauge symmetry, respectively. In this paper, employing Poisson reduction we offer a unified technique for the symmetry reduction of a generalized canonical infinite-dimensional Dirac structure to the Poisson structure associated with Stokes-Dirac structures and of a fine-dimensional Dirac structure to simplicial Dirac structures. We demonstrate this Poisson scheme on a physical example of the vibrating string.