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

Abstract

This papers is concerned with multisymplectic formalisms which are the frameworks for Hamiltonian theories for fields theory. Our main purpose is to study the observable (n − 1)-forms which allows one to construct observable functionals on the set of solutions of the Hamilton equations by integration. We develop here two different points of view: generalizing the law {p, q} = 1 or the law dF/dt = {H, F }. This leads to two possible definitions; we explore the relationships and the differences between these two concepts. We show that -in contrast with the de Donder-Weyl theory -the two definitions coincides in the Lepage-Dedecker theory.e-print archive:

Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl

“…And second because one is then led to impose conditions on F in such a way that the resulting functional carries only dynamical information. The analysis of these conditions is the subject of our companion paper [11]. And we believe that the conditions required on these forms are connected with the definitions of r-regular functions given in this paper, although we have not completely elucidated this point.…”

“…We can further assume restrictions on the choice of F in order to guarantee the fact that the resulting functional is physically observable. Such a situation is achieved if for example F is so that dF |TmΓ depends only on dH m (see [11] for details).…”

“…Note that beside the the Lepage-Dedecker multisymplectic manifold (Λ n T * N , Ω) studied in the previous Section, other examples of multisymplectic manifolds arises naturally as for example a multisymplectic structure associated to the Palatini formulation of pure gravity in 4-dimensional space-time (see [10], [11], [17]). …”

“…Beside these properties in this paper and in its companion paper [11] we wish to address other kind of questions related to the physical gain of these theories: the main advantage of multisymplectic formalisms is to offer us a Hamiltonian theory which is consistent with the principles of Relativity, i.e. being covariant.…”

Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory

“…From this point of view, as we will see, the LepageDedecker theory seems to be much more appropriate than the de Donder-Weyl one. This is the philosophy that we have followed in [21]. Here a caveat is in order, in the classical one-dimensional Hamiltonian formalism: we start with a Lagrangian action functional where (q, p) −→ (q, V (t, q, p)) is the inverse mapping of the Legendre mapping.…”