Derivation of Jacobi’s Virial Equation for Description of Dynamics of a Self-Gravitating Body

Abstract

In order to demonstrate the universality of Jacobi’s virial equation for the description of the dynamics of natural systems, including their origin and evolution, it was derived from the main existing equations, describing a wide range of physical models of the systems. In particular, Jacobi’s virial equation was derived from the equations of motion of Newton, Euler, Hamilton, Einstein, and quantum mechanics.

The derived equation represents not only formal mathematical transformation of the initial equations of motion. Physical quintessence of the mathematical transformation of the equations of motion involves changes in the vector forces and moment of momentums by the volumetric forces or pressure and the oscillation of the interacted mass particles (inner energy) expressed through the energy of oscillation of the polar moment of inertia of a body. Here the potential (kinetic) energy and the polar moment of inertia of a body have a functional relationship and within the period of oscillation are inversely changed by the same law. Moreover, the virial oscillations of a body represent the main part of the body’s kinetic energy, which is lost in the hydrostatic equilibrium model.

The change in the vector forces and moment of momentums by the force pressure and the oscillation of the interacting mass particles disclose the physical meaning of the gravitation and mechanism of generation of the gravitational and electromagnetic energy and their common nature. The most important advantage given by Jacobi’s virial equation is its independence from the choice of the coordinate system, the transformation of which, as a rule, creates many mathematical difficulties.