Dynamic equations for time-averaged coordinates at thermal equilibrium

The Hamilton variational principle with coarse variation (over a limited number of parameters introduced into the coordinates) is used to find relations for the time-averages of defined functions of the coordinates and velocities. The dynamic equations for the parameters characterizing the average behavior of the coordinates are then found by orthogonalization of the coordinates. It is shown that the entropy of the dynamic system is expressed in terms of these averaged parameters, and the dynamic equations and stability conditions may be found by minimization of the defined function of the motion, which is constructed like the thermodynamic potential. A perturbation procedure is suggested for solving the dynamic equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 87–94, April, 1991.