On the Construction of K-Operators in Field Theories as Sections Along Legendre Maps |
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Authors: | Arturo Echeverría-Enríquez Jesús Marín-Solano Miguel C Muñoz-Lecanda Narciso Román-Roy |
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Institution: | (1) Departamento de Matemática Aplicada y Telemática, Campus Norte UPC, Módulo C-3, C/ Jordi Girona 1-3, E-08034 Barcelona, Spain;(2) Departamento de Matemática Económica, Financiera y Actuarial, UB Av. Diagonal 690, E-08034 Barcelona, Spain |
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Abstract: | The time-evolution K-operator (or relative Hamiltonian vector field) 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 nonregular 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 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 relevant result, we use these geometrical objects to obtain the solutions of the Lagrangian and Hamiltonian field equations, and the equivalence among them (specially for nonregular field theories). |
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Keywords: | jet bundles multivector fields connections jet fields sections along maps first-order field theories Lagrangian and Hamiltonian formalisms |
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