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

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*D. R. Grigore*

**E-mail:** grigore@theory.nipne.ro, grigore@roifa.ifa.ro
**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