Infinite-dimensional Lagrange-Dirac systems with boundary energy flow I: Foundations

TL;DR

This paper introduces a geometric framework for infinite-dimensional systems with boundary energy flow using Dirac structures, extending classical mechanics principles to systems like wave and Maxwell equations.

Contribution

It develops a new infinite-dimensional Dirac structure approach that incorporates boundary energy flow within the Lagrangian formalism, extending finite-dimensional geometric mechanics.

Findings

Framework applies to systems with boundary energy flow

Derives evolution equations from variational and Dirac perspectives

Includes examples like wave, telegraph, and Maxwell equations

Abstract

A new geometric approach to systems with boundary energy flow is developed using infinite-dimensional Dirac structures within the Lagrangian formalism. This framework satisfies a list of consistency criteria with the geometric setting of finite-dimensional mechanics. In particular, the infinite-dimensional Dirac structure can be constructed from the canonical symplectic form on the system's phase space; the system's evolution equations can be derived equivalently from either a variational perspective or a Dirac structure perspective; the variational principle employed is a direct extension of Hamilton's principle in classical mechanics; and the approach allows for a process of system interconnection within its formulation. This is achieved by developing an appropriate infinite dimensional version of the previously developed Lagrange-Dirac dynamical systems. A key step in this construction is the careful choice of a suitable dual space to the configuration space, specifically, a subspace of the topological dual that captures the system's behavior in both the interior and the boundary, while allowing for a natural extension of the canonical geometric structures of mechanics. This paper focuses on systems where the configuration space consists of differential forms on a smooth manifold with a boundary. To illustrate our theory, several examples, including nonlinear wave equations, the telegraph equation, and the Maxwell equations are presented.