Solution of Jacobi’s Virial Equation for Self-gravitating Systems

Abstract

It is shown in this chapter that Jacobi’s virial equation provides, first of all, a solution for the models of natural systems, which have explicit solutions in the framework of the classical many-body problem. The particular example of this is the unperturbed problem of Keplerian motion, when the system consists of only two material points interacting by Newtonian law. The parallel solutions for both the classical and dynamical approaches are given, and in doing so we show that, with the dynamical approach, the solution acquires a new physical meaning, namely, oscillating motion. That solution appeared to be possible because of an existing relationship between the gravitational energy and the polar moment of inertia in the form \( \left| U \right|\sqrt\Phi = B = {\text{const}}. \) It is also done for the solution of Jacobi’s virial equation in hydrodynamics, in quantum mechanics for dissipative systems, for systems with friction and in the framework of the theory of relativity. The above solutions acquire a new physical meaning because the dynamics of a system is considered with respect to new parameters, i.e. its Jacobi function (polar moment of inertia) and potential (kinetic) energy. The solution, with respect to the Jacobi function and the potential energy, identifies the evolutionary processes of the structure or redistribution of the mass density of the system. Moreover, the main difference of the two approaches is that the classical problem considers motion of a body in the outer central force field. The virial approach considers motion of a body both in the outer and in its own force field applying, instead of linear forces and moments, the volumetric forces (pressure) and moments (oscillations). Finally, analytical solution of the generalized equation of perturbed virial oscillations in the form \( {\ddot{\Phi }} = - {A} + {B}/\sqrt\Phi + {X}\left( {{t},\Phi ,{\dot{\Phi }}} \right) \) was done. Derivation of the equation of dynamical equilibrium and its solution for conservative and dissipative systems shows that dynamics of celestial bodies in their own force field puts forward a wide class of geophysical, astrophysical and geodetic problems which can be solved by the methods of celestial mechanics and with new physical concepts.