• DOI:10.1063/1.4817391
• Corpus ID: 14797823

Generating Functionals and Lagrangian PDEs

@inproceedings{Vankerschaver2011GeneratingFA,
  title={Generating Functionals and Lagrangian PDEs},
  author={Joris Vankerschaver and Cuicui Liao and Melvin Leok},
  year={2011}
}

• J. Vankerschaver, Cuicui Liao, M. Leok
• Published 1 November 2011
• Mathematics

The main goal of this paper is to derive an alternative characterization of the multisymplectic form formula for classical field theories using the geometry of the space of boundary values. We review 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. As an example of the latter, we show that Lorentz’s reciprocity principle in electromagnetism is a particular instance of the 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. ∗Current Address: Imperial College London, London SW7 2AZ, UK. Email: Joris.Vankerschaver@ gmail.com. †Current Address: Department of Mathematics, Jiangnan University. No.1800 Lihu Avenue, Wuxi, Jiangsu, 214122, China. Email: [e-mail]. ‡Email: [e-mail]. 1 ar X iv :1 11 1. 02 80 v2 [ m at hph ] 2 0 Ju n 20 13