Stokes-Dirac Structures through Reduction of Infinite-Dimensional Dirac Structures

TL;DR

This paper introduces a method to derive Stokes-Dirac structures via symmetry reduction of canonical Dirac structures, unifying boundary control theory with geometric reduction techniques and recovering key physical features.

Contribution

It presents a novel Poisson reduction approach to obtain Stokes-Dirac structures from infinite-dimensional Dirac structures, linking geometric reduction with boundary control models.

Findings

Derivation of Stokes-Dirac structures through symmetry reduction

Recovery of standard structure matrices and advection terms

Connection between boundary control and geometric reduction techniques

Abstract

We consider the concept of Stokes-Dirac structures in boundary control theory proposed by van der Schaft and Maschke. We introduce Poisson reduction in this context and show how Stokes-Dirac structures can be derived through symmetry reduction from a canonical Dirac structure on the unreduced phase space. In this way, we recover not only the standard structure matrix of Stokes-Dirac structures, but also the typical non-canonical advection terms in (for instance) the Euler equation.