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

François Gay-Balmaz, Álvaro Rodríguez Abella, Hiroaki Yoshimura

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.

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

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.

Paper Structure

This paper contains 26 sections, 18 theorems, 146 equations, 1 table.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Preliminaries
  4. Dirac structures on manifolds
  5. Finite-dimensional Lagrange--Dirac systems in mechanics
  6. Fréchet spaces and dual systems
  7. Lagrange--Dirac dynamical systems on the space of smooth functions
  8. Restricted dual and restricted cotangent bundle
  9. Canonical forms, Tulczyjew triple and the canonical Dirac structure
  10. Infinite-dimensional Lagrange--Dirac dynamical systems
  11. Variational structures for infinite-dimensional Lagrange--Dirac dynamical systems
  12. Lagrange--d'Alembert principle for infinite-dimensional Lagrangian systems
  13. Lagrange--d'Alembert--Pontryagin principle for infinite-dimensional Lagrange--Dirac dynamical systems
  14. Examples
  15. Vibrating membrane and nonlinear wave equations
  16. ...and 11 more sections

Key Result

Theorem 2.4

Consider the curves $(q,v,p):[t_0,t_1]\to TQ\oplus T^* Q$ and $q=\rho_Q\circ(q,v,p): [t_{0},t_{1}] \to Q$, where $\rho_Q:TQ\oplus T^*Q\to Q$ is the natural projection. The following statements are equivalent:

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3
  • Definition 3.4
  • Proposition 3.5
  • Remark 3.6: $\Omega _{T^\star V}$ as a strong form on the restricted duals
  • ...and 37 more