关于场论中K算子沿Legendre映射截面的构造
On the construction of K-operators in field theories as sections along Legendre maps
摘要：
力学中的“时间演化算子”是一个强有力的工具，它可以几何地定义为沿着勒让德映射的矢量场。它已经被一些作者广泛用于研究动力系统（主要是非正则系统）的结构和性质，例如拉格朗日和哈密顿形式主义，约束和高阶力学之间的关系。本文致力于定义一个推广的场论，在协变公式。为了做到这一点，我们还使用沿地图的部分，特别是多向量场（阶数大于1的反对称逆变张量场），喷流场和沿勒让德地图的连接形式。作为第一个相关性质，我们使用这些几何对象来获得拉格朗日场方程和汉密尔顿场方程的解，以及它们之间的等价性（特别是对于非正则场论）。
英文摘要：
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).
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