Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 100, 25 pages      math-ph/0506022      http://dx.doi.org/10.3842/SIGMA.2009.100

Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

Narciso Román-Roy
Dept. Matemática Aplicada IV, Edificio C-3, Campus Norte UPC, C/ Jordi Girona 1, E-08034 Barcelona, Spain

Received July 02, 2009, in final form October 30, 2009; Published online November 06, 2009

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.

Key words: classical field theories; Lagrangian and Hamiltonian formalisms; fiber bundles; multisymplectic manifolds.

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