Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory

Abstract

-In a first part we propose an introduction to multisymplectic formalisms, which are generalisations of Hamilton's formulation of Mechanics to the calculus of variations with several variables: we give some physical motivations, related to the quantum field theory, and expound the simplest example, based on a theory due to T. de Donder and H. Weyl. In a second part we explain quickly a work in collaboration with J. Kouneiher on generalizations of the de Donder-Weyl theory (known as Lepage theories). Lastly we show that in this framework a perturbative classical field theory (analog of the perturbative quantum field theory) can be constructed.

The Notion of Observable in the Covariant Hamiltonian Formalism for the Calculus of Variations with Several Variables

“…Moreover an alternative definition of the pseudobracket (see Definition 3.5) can be given using the tensor field ξ F defined by (13).…”

“…If the variational problem can be seen as a deformation of a linear one (i.e., of a free field theory) then it could be possible to construct a perturbation theory, leading to Feynman type expansions for classical fields. For an example of such a theory, see [13] and [12]. Another interesting direction would be to explore completely integrable systems.…”

“…This example illustrates the fact that gauge symmetry helps strongly in constructing dynamical observable functionals. Another possibility in order to enlarge the number of dynamical functionals is when the nonlinear variational problem can be approximated by a linear one: this gives rise to observable functionals defined by expansions [12], [13].…”

Planar binary trees and perturbative calculus of observables in classical field theory

“…If the time variable is replaced by several spacetime variables, the multisymplectic formalism is based on an analogue to the canonical symplectic structure on the cotangent bundle, a manifold equipped with a multisymplectic form. For an introduction to the multisymplectic geometry one can refer to [11] and for more complete informations one can read the papers of F. Hélein and J. Kouneiher [12,13]. Starting from a Lagrangian density which describes the dynamics of the field, one can construct a Hamiltonian function through a Legendre transform and obtain a geometric formulation of the problem.…”

“…When λ = 0, the identity ( * ) gives us a way to do this manipulation, but when λ = 0 this is no longer the case. So F. Hélein proposed an approach based on perturbation; the reader will find more details on this subject in his paper [11].…”