Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

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.

v1: 17 pages, Talk presented in the "MAT.ES2005: 1st Joint Meeting of Mathematics RSME-SCM-SEIO-SEMA" (Valencia, Spain 2005, http://www.uv.es/mat.es2005/); v3: published version

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• Extended Hamiltonian systems in multisymplectic field theories
• Geometric aspects of nonholonomic field theories
• Unified formalism for non-autonomous mechanical systems
• Classical Field theory on Lie algebroids: multisymplectic formalism
• The momentum map for nonholonomic field theories with symmetry
• K-symplectic formalism on Lie algebroids
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