# Generating Functionals and Lagrangian PDEs

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

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…

## 8 Citations

### Space-Time Finite-Element Exterior Calculus and Variational Discretizations of Gauge Field Theories

- Mathematics
- 2014

Many gauge field theories can be described using a multisymplectic Lagrangian formulation, where the Lagrangian density involves space-time differential forms. While there has been much work on…

### Multisymplectic Hamiltonian variational integrators

- MathematicsInt. J. Comput. Math.
- 2022

It is demonstrated that one can use the notion of Type II generating functionals for Hamiltonian partial differential equations as the basis for systematically constructing Galerkin Hamiltonian variational integrators that automatically satisfy a discrete multisymplectic conservation law.

### Spectral variational integrators

- Mathematics, Computer ScienceNumerische Mathematik
- 2015

This paper constructs a scheme for numerical integration that converges geometrically, and is symplectic and momentum preserving, and proves that under appropriate assumptions, variational integrators constructed using Galerkin techniques will yield numerical methods that are arbitrarily high-order.

### Spectral variational integrators

- Mathematics, Computer ScienceNumerische Mathematik
- 2014

This paper constructs a scheme for numerical integration that converges geometrically, and is symplectic and momentum preserving, and proves that under appropriate assumptions, variational integrators constructed using Galerkin techniques will yield numerical methods that are arbitrarily high-order.

### R-Adaptive Multisymplectic and Variational Integrators

- Computer ScienceMathematics
- 2019

Two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories are presented.

### Geometric integration applied to moving mesh methods and degenerate Lagrangians

- Computer Science
- 2014

Two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories are presented and an attempt to extend the approach to degenerate field theories to higher-order variational integrators is made.

### Nonstandard finite difference variational integrators for nonlinear Schrödinger equation with variable coefficients

- MathematicsAdvances in Difference Equations
- 2013

In this paper, the idea of nonstandard finite difference discretization is employed to develop two variational integrators for the nonlinear Schrödinger equation with variable coefficients. These…

### Nonstandard Finite Difference Variational Integrators for Multisymplectic PDEs

- Mathematics, Computer ScienceJ. Appl. Math.
- 2012

A non standard finite difference variational integrator for linear wave equation with a triangle discretization and two nonstandard finite difference Variational integrators for the nonlinear Klein-Gordon equation withA triangle discretizations and a square discretized, respectively are obtained.

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