Equations of Motion and Navier–Stokes Equations

In this research, we present the analogies between variational calculations in cosmology and in classical mechanics. Our approach is based on the invariants for transformations of affine connections defined on N-dimensional manifolds (special cases are the 8-dimensional, 5-dimensional, and 4-dimensional manifolds used in cosmology and 2-dimensional manifolds used in classical mechanics). Any of these transformations represents a class of curves on initial manifolds, which transmits to an another class of curves on the current manifolds. The main results of this paper are general equations of motion, which are obtained from the invariants caused by the transformation rule of an initial affine connection to the current one and the corresponding Navier–Stokes equations, recognized in transformations of curves along which moves a fluid particle.