Abstract
The Lagrange-Jacobi equation is one of the significant tools for the qualitative analysis of the n-body problem. In this paper, we present the modified Lagrange-Jacobi equation by introducing a new formal parameter of n-body problem and propose its application to the dynamical study of clusters of galaxies which are large-scale structures of Universe. We put forward and study a new dynamical problem which is related to the stage of relaxation of observed stationary clusters of galaxies which are considered as a non-equilibrium systems of point masses. We also received the analytical form of the potential energy of such galaxy clusters. One of the applications of this analytical form is the analytical relation between the time of setting up the virial equilibrium in relaxing clusters of galaxies and the cosmological epoch T.