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28 March 2024
 
  » arxiv » math-ph/0103019

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On the construction of K-operators in field theories as sections along Legendre maps
A. Echeverría-Enrí quez ; J. Marín-Solano ; M.C. Mu~{n}oz-Lecanda ; N. Román-Roy ;
Date 15 Mar 2001
Journal Acta Applicandae Mathematicae {f 77}(1) (2003) 1-40
Subject Mathematical Physics MSC-class: 51P05; 43C05; 53C80; 55R10; 58A20; 58A30; 70S05 | math-ph math.MP
AbstractThe ``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).
Source arXiv, math-ph/0103019
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