Some geometric aspects of variational problems in fibred manifolds

TL;DR

This paper explores the geometric foundations of variational calculus in fibered manifolds, emphasizing invariant formulations, symmetries, and applications to physical field theories.

Contribution

It provides a geometric and invariant approach to variational problems in fibered manifolds, including derivation of Euler equations and symmetry analysis.

Findings

Derived invariant Euler equations for variational problems

Characterized symmetries using Lie derivatives

Formulated conditions for invariant solutions in covariant theories

Abstract

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.