Generating Functionals and Lagrangian PDEs

TL;DR

This paper develops a novel framework of generating functionals for Lagrangian and Hamiltonian field theories, connecting boundary data, multisymplectic structures, and conservation laws, including discrete cases like the linear wave equation.

Contribution

It introduces Type-I/II generating functionals for boundary data in Lagrangian field theories and establishes their relation to multisymplectic structures and conservation laws.

Findings

Defined a Lagrangian analogue of Jacobi's solution for field theories.

Showed that variational derivatives yield isotropic submanifolds related to multisymplectic form.

Recovered multisymplectic conservation law for the linear wave equation.

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.