Multisymplectic Lagrangian and Hamiltonian formalisms of First-order Classical Field theories

This review paper is devoted to presenting the standard multisymplectic formulation for describing geometrically first-order 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 recently developed), following the original ideas of Rusk and Skinner for mechanical systems