Fock Space Methods and the Lagrangian Formalism on Finite Jet Bundle Extensions

D. R. Grigore


Abstract. The geometric Lagrangian theory is formulated in the language of jet bundle extensions. The use of finite jet bundle extensions corresponds more closely to spirit of the original Lagrangian theory as studied in physical literature corresponding to a Lagrangian theory of arbitrary, but finite order. The underlying structure is based on the analysis of some basic mathematical objects such as: the contact ideal and the (exact) variational sequence. In this paper we give new and much simpler proofs for the whole theory using Fock space methods. For instance, in a given chart, one finds out quite naturally that some global differential forms involved in the description of the variational sequence (the so-called contact forms) can be locally written in terms of expressions having some symmetry and/or antisymmetry properties. These expressions can be regarded as tensors in some Fock space and some very complicated identities become very simple if one uses the creation and annihilation operators. Using these results we give the most general expression for a variationally trivial Lagrangian.

AMSclassification. 49S05, 70G50, 70H35

Keywords. Lagrangian Formalism, Variational Sequence