Differential operator Dirac structures

TL;DR

This paper extends the theory of Dirac structures on bounded domains by using pairs of skew-adjoint differential operators and polynomial calculus, also exploring Lagrangian subspaces within the same framework.

Contribution

It introduces a new framework for Dirac structures using pairs of differential operators and polynomial calculus, unifying effort constraints and Lagrangian subspaces.

Findings

Extended Dirac structures to pairs of skew-adjoint operators.

Streamlined boundary variable handling via polynomial calculus.

Unified treatment of effort constraints and Lagrangian subspaces.

Abstract

As shown in earlier work, skew-adjoint linear differential operators, mapping efforts into flows, give rise to Dirac structures on a bounded spatial domain by a proper definition of boundary variables. In the present paper this is extended to pairs of linear differential operators defining a formally skew-adjoint relation between flows and efforts. Furthermore it is shown how the underlying repeated integration by parts operation can be streamlined by the use of two-variable polynomial calculus. Dirac structures defined by formally skew adjoint operators and differential operator effort constraints are treated within the same framework. Finally it is sketched how the approach can be also used for Lagrangian subspaces on bounded domains.