On the construction of K-operators in field theories as sections along Legendre maps

Abstract

The ``time-evolution operator'' in mechanics is a powerful tool which can be geometrically defined as a vector field along the Legendre map. It has been extensively used by several authors for studying the structure and properties of the dynamical systems (mainly the non-regular ones), such as the relation between the Lagrangian and Hamiltonian formalisms, constraints, and higher-order mechanics. This paper is devoted to defining a generalization of this operator for field theories, in a covariant formulation. In order to do this, we also use sections along maps, in particular multivector fields (skew-symmetric contravariant tensor fields of order greater than 1), jet fields and connection forms along the Legendre map. As a first relevant property, we use these geometrical objects to obtain the solutions of the Lagrangian and Hamiltonian field equations, and the equivalence among them (specially for non-regular field theories).

35 pages, LaTeX. Replaced with the edited version. The title has been changed. Minor details are corrected

References in corpus (6)

• Geometry of multisymplectic Hamiltonian first-order field theories
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• Finite dimesional Hamiltonian formalism for gauge and field theories
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Cited by in corpus (1)

• Gunther's formalism (k-symplectic formalism) in classical field theory: Skinner-Rusk approach and the evolution operator