Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

Abstract

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.

A contact geometry framework for field theories with dissipation

“…In the last decades, the methods of geometric mechanics and field theory have been widely used in order to give a geometrical description of a large variety of systems in physics and applied mathematics; in particular, those of symplectic and multisymplectic or k-symplectic (polysymplectic) geometry (see, for instance, [1,2,8,13,21,24,25] and references therein). All these methods are developed, in general, to model systems of variational type without dissipation or damping, both in the Lagrangian and Hamiltonian formalisms.…”

Some Properties of Multisymplectic Manifolds

“…As an interesting example, if π : E → M is a fiber bundle (where M is an m-dimensional oriented manifold), J 1 π is the corresponding first-order jet bundle, and L is a first-order hyperregular Lagrangian density, then the Poincaré-Cartan form Ω L ∈ Ω m+1 (J 1 π) is a multisymplectic form and (J 1 π, Ω L ) is a special multisymplectic manifold [4,15,31].…”

“…Although there are several geometrical models for describing classical field theories, namely, polysymplectic, k-symplectic and k-cosymplectic manifolds [12,17,23,29,30]; multisymplectic manifolds are the most general and complete tool for describing geometrically (covariant) first and higher-order field theories (see, for instance, [1,4,8,14,16,18,20,24,28,31,33] and the references quoted on them). All of these kinds of manifolds are generalizations of the concept of symplectic manifold, which are used to describe geometrically mechanical (autonomous) systems.…”

Constraint algorithm for singular field theories in the <i>k</i>-cosymplectic framework

“…These two formulations admit straightforward generalizations to first order classical field theory using k-symplectic and k-cosymplectic structures, which are the generalization to field theories of the autonomous and nonautonomous cases in mechanics [3,8,22]. A more general framework for classical field theories can be built up by using multisymplectic geometry (see [27] and references therein; see also [26] for an analysis of the relationship among these formulations).…”