Lagrangian Theory of Tensor Fields over Spaces with Contravariant and Covariant Affine Connections and Metrics and Its Application to Einstein's Theory of Gravitation in V 4 Spaces

Abstract

The Lagrangian formalism for tensor fields over differentiable manifolds with contravariant and covariant affine connections (whose components differ not only by sign) and a metric \([(\bar L_n ,g) - spaces]\) is considered. The functional, the Lie, the covariant, and the total variations of a Lagrangian density, depending on components of tensor fields (with finite rank) and their first and second covariant derivatives, are established. A variation operator is determined and the corollaries of its commutation relations with the covariant and the Lie differential operators are found. The canonical (common) method of Lagrangians with partial derivatives (MLPD) and the method of Lagrangians with covariant derivatives (MLCD) are outlined. They differ each other by the commutation relations the variation operator has to obey with the covariant and the Lie differential operator. The covariant Euler–Lagrange equations are found on the basis of the MLCD. The energy-momentum tensors are considered on the basis of the Lie variation and the covariant Noether identities.