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… 

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References

SHOWING 1-10 OF 38 REFERENCES

Momentum Maps and Classical Relativistic Fields. Part II: Canonical Analysis of Field Theories

With the covariant formulation in hand from the first paper of this series (physics/9801019), we begin in this second paper to study the canonical (or ``instantaneous'') formulation of classical

Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

Abstract:This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental

Parametrization and stress–energy–momentum tensors in metric field theories

We give an exposition of the 1972 parametrization method of Kuchař in the context of the multisymplectic approach to field theory. The purpose of the formalism developed here is to make any classical

Multi-symplectic structures and wave propagation

  • T. Bridges
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1997
A Hamiltonian structure is presented, which generalizes classical Hamiltonian structure, by assigning a distinct symplectic operator for each unbounded space direction and time, of a Hamiltonian

Discrete exterior calculus

This thesis presents the beginnings of a theory of discrete exterior calculus (DEC). Our approach is to develop DEC using only discrete combinatorial and geometric operations on a simplicial complex

Finite element exterior calculus: From hodge theory to numerical stability

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we

Foundations of computational geometric mechanics

Geometric mechanics involves the study of Lagrangian and Hamiltonian mechanics using geometric and symmetry techniques. Computational algorithms obtained from a discrete Hamilton's principle yield a

General techniques for constructing variational integrators

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact