Derivation of Jacobi’s Virial Equation for Description of Dynamics of Natural Systems

Abstract

Let us begin by deriving Jacobi’s virial equation from the equations of Newton, Euler, Hamilton, Einstein and also from equations of quantum mechanics. By doing so we will show that Jacobi’s virial equation appears to be a unified instrument for the description of dynamics of natural systems using volumetric (integral) characteristics in the framework of the various physical models of the matter interaction employed. The assumptions under which this equation is derived put only one restriction on the potential energy function to be homogeneous in the co-ordinates. But it will be seen that even this single restriction does not have to be always obligatory. The limitations following from any concrete physical model used for describing dynamics of systems in classical mechanics, hydrodynamics, statistical physics, or the theory of relativity, become unimportant.