Infinite Dimensional Lagrange–Dirac Mechanics with Boundary Conditions

The Lagrange–Dirac theory is extended to systems defined on the family of smooth functions on a manifold with boundary, which provides an instance of systems with a Fréchet space as a configuration space. To that end, we introduce the restricted cotangent bundle, a vector subbundle of the topological cotangent bundle which contains the partial derivatives of Lagrangian functions defined through a density. The main achievement of our proposal is that the Lagrange–Dirac equations on the restricted cotangent bundle properly account for the boundary value problem, i.e., the boundary conditions do not need to be imposed ad hoc, but they arise naturally from the Lagrange–Dirac formulation. After giving the main theoretical results, and showing how boundary forces can be naturally included in the Lagrange–Dirac formulation, we illustrate our framework with the dynamical equations of a vibrating membrane.