Variational principle for the Finslerian extension of general relativity

The Lagrangian density for formulating the Finslerian gravitational field equations is constructed by replacing the tangent vectors entering a direction-dependent density by the auxiliary vector field. The Lagrangian derivative is represented in terms of the tensor densities associated with an initial direction-dependent density. A particular case, where the direction-dependent density is chosen in the form of the contraction of the FinslerianK-tensor of curvature multiplied by the Jacobian, is treated in detail.