复变流形中变分问题的若干几何性质
Some geometric aspects of variational problems in fibred manifolds
摘要：
这项工作包含了纤维流形的变分学的基础的阐述。重点放在几何方面的理论。特别研究了纤维流形上第一喷流延拓上的实函数（Lagrange函数）或微分形式（Lagrange形式）定义的泛函。临界点（临界截面）的泛函进行检查，并为他们的欧拉方程推导出一个完全不变的方式。第一个变分公式是由所谓的勒巴吉形式导出的。理论中出现的所有变分都是由矢量场产生的。定义了可投影向量场的喷流分解。欧拉形式，与一个给定的拉格朗日函数（拉格朗日形式）的介绍通过欧拉方程的变分法。用微分形式及其外微分的形式给出了欧拉形式消失的充分必要条件。给出了拉格朗日函数使欧拉方程同零的相应条件。研究了一些特殊的Lepagian形式。定义了变分问题的对称类。不变，广义不变，对称变换的特点是在李导数。给出了具有预定对称变换的变分问题，并研究了其解的充要条件。几何方面的所谓的一般协变变分理论进行了研究。定义和定理很好地适用于物理场论中的情况。
英文摘要：
This work contains an exposition of foundations of the variational calculus in fibered manifolds. The emphasis is laid on the geometric aspects of the theory. Especially functionals defined by real functions (Lagrange functions) or differential forms (Lagrangian forms) on the first jet prolongation of a given fibered manifold are studied. Critical points (critical cross sections) of the functionals are examined and the Euler equations for them are derived in a completely invariant manner. The first variation formula is derived by means of the so-called Lepagian forms. All variations appearing in the theory are generated by vector fields. Jet prolongations of projectable vector fields are defined. The Euler form, associated with a given Lagrange function (of Lagrangian form) is introduced by means of the Euler equations of the calculus of variations. Necessary and sufficient conditions for the vanishing of the Euler form are stated in terms of differential forms and their exterior differential. The corresponding conditions for a Lagrange function leading to identically vanishing Euler equations are given. Some special Lepagian forms are studied. Classes of symmetries of a variational problem are defined. Invariant, generalized invariant, and symmetry transformations are characterized in terms of the Lie derivatives. The variational problem with prescribed symmetry transformations is formulated, and necessary and sufficient conditions for its solutions are studied. The geometrical aspects of the so-called generally covariant variational theories are studied. Definitions and theorems are well adapted to the situation in physical field theories.
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