Universality of Jacobi’s Virial Equation for Description of Dynamics of Natural Systems in Terms of Integral Characteristics

Abstract

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