Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

TL;DR

This paper reviews the multisymplectic geometric frameworks for classical field theories, covering both Lagrangian and Hamiltonian formalisms, and introduces a unified approach inspired by Rusk and Skinner.

Contribution

It provides a comprehensive overview of the multisymplectic formulations for classical field theories and introduces a recent unified formalism combining Lagrangian and Hamiltonian approaches.

Findings

Revisits the Lagrangian formalism and variational principles.

Constructs the Hamiltonian formalism using Hamiltonian sections.

Unifies both formalisms in a new geometric framework.

Abstract

This review paper is devoted to presenting the standard multisymplectic formulation for describing geometrically classical field theories, both the regular and singular cases. First, the main features of the Lagrangian formalism are revisited and, second, the Hamiltonian formalism is constructed using Hamiltonian sections. In both cases, the variational principles leading to the Euler-Lagrange and the Hamilton-De Donder-Weyl equations, respectively, are stated, and these field equations are given in different but equivalent geometrical ways in each formalism. Finally, both are unified in a new formulation (which has been developed in the last years), following the original ideas of Rusk and Skinner for mechanical systems.