Generating Functionals and Lagrangian PDEs

Abstract

We introduce the concept of Type-I/II generating functionals defined on the space of boundary data of a Lagrangian field theory. On the Lagrangian side, we define an analogue of Jacobi's solution to the Hamilton-Jacobi equation for field theories, and we show that by taking variational derivatives of this functional, we obtain an isotropic submanifold of the space of Cauchy data, described by the so-called multisymplectic form formula. We also define a Hamiltonian analogue of Jacobi's solution, and we show that this functional is a Type-II generating functional. We finish the paper by defining a similar framework of generating functions for discrete field theories, and we show that for the linear wave equation, we recover the multisymplectic conservation law of Bridges.

31 pages; 1 figure -- v2: minor changes

References in corpus (6)

• Finite element exterior calculus: from Hodge theory to numerical stability
• Momentum Maps and Classical Relativistic Fields. Part II: Canonical Analysis of Field Theories
• Discrete Hamilton-Jacobi Theory
• Discrete Lagrangian field theories on Lie groupoids
• Variational Integrators for Maxwell's Equations with Sources
• $G$-Strands

Cited by in corpus (4)

• Multisymplecticity of hybridizable discontinuous Galerkin methods
• Multisymplectic Hamiltonian Variational Integrators
• Variational Principles for Hamiltonian Systems
• Discrete Formulation for the dynamics of rods deforming in space