Virial oscillations of celestial bodies IV: The Lyapunov stability of motion
Abstract
It is shown that with a virial approach to the solution of the manybody problem the integral characteristics of a system (Jacobi's function and total energy), being present in Jacobi's equation, are immanent to its own integrals. Estimating the Lyapunov stability of motion of a system they play the role of Lyapunov functions. Studying Lyapunov stability of the virial oscillations of celestial bodies we used the Duboshin criterion applicable when permanent perturbations are present. In the case of conservative systems the potential energy of the system plays the role of such a perturbation. Thus, the nature of the virial oscillations can be understood as an effect of nonlinear resonance between the kinetic and the potential energies. It is shown that the stability of virial oscillations of conservative systems relative to variations of the formfactors αβ product is only a necessary condition in the proof of the hypothesis that αβ=const. for celestial bodies. The sufficient condition for the proof of this equality consists of the given direct derivation of the equation of virial oscillations of celestial bodies from Einstein's equation, as well as of the equivalence of Schwarzschild's solution and the solution of Jacobi's equation atddot φ = 0. The stability of virial oscillations for dissipative systems is studied. It is shown that the stability is limited by the period of time of its bifurcation.
 Publication:

Celestial Mechanics
 Pub Date:
 January 1985
 DOI:
 10.1007/BF01229112
 Bibcode:
 1985CeMec..35...23F
 Keywords:

 Celestial Bodies;
 Jacobi Integral;
 Liapunov Functions;
 Many Body Problem;
 Motion Stability;
 Virial Theorem;
 Branching (Mathematics);
 Cosmology;
 Einstein Equations;
 Electromagnetic Radiation;
 Pendulums;
 Schwarzschild Metric;
 Astronomy