Derivation of Unified Jacobi’s Equation for Different Types of Physical Interactions

Abstract

In this chapter, we derive a universal Jacobi’s virial equation for description of the gravitation and dynamics of natural systems. It is derived from the main existing equations, describing a wide range of physical models of the systems. In particular, Jacobi’s virial equation is derived from equations of motion of Newton, Euler, Hamilton, Einstein and quantum mechanics. The obtained equation appears to be a scalar-quantum mathematical model of matter’s motion and gravitation. The derived equation represents not only a formal mathematical transformation of the initial equations of motion. The physical quintessence of mathematical transformation of the equations of motion involves changes of the vector forces and moment of momentums by the volumetric forces or pressure and oscillation of interacting mass particles (inner energy) expressed through the energy of oscillation of the polar moment of inertia of a body. Here, the potential and kinetic energy and the polar moment of inertia of a body have a functional relationship and within periods of oscillation are inversely changed by the same law. Moreover, the virial oscillations of a body represent the main part of a body’s kinetic energy, which is lost in the hydrostatic equilibrium model. The change of vector forces and moment of momentums by force pressure and 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 a coordinate system, transformation of which, as a rule creates many mathematical difficulties.