Abstract
It is shown that with a virial approach to the solution of the many-body 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 non-linear 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 form-factors 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 at\(\ddot \phi = 0\).
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.
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References
Chetaev, N. G.: 1962,The Stability of Motion. The Works on Analytical Mechanics, Acad. Sci. of USSR, Moscow.
Duboshin, G.N.: 1952,Foundations of the Stability of Motion, Moscow State University, Moscow.
Duboshin, G.B.: 1978,Celestial Mechanics. Analytical and Qualitative Methods Nauka, Moscow.
Ferronsky, S.V.: 1983,Celest. Mech. 30, 71.
Ferronsky, S.V.: 1984a,Physics of Atmosphere and Oceans 20, 922.
Ferronsky, S.V.: 1984b, inProc. Higher Hydrometeorol. Observ. 20(1), 11.
Ferronsky, V.I., Denisik, S.A., and Ferronsky, S.V.: 1978,Celest. Mech.,18, 113.
Ferronsky, V.I., Denisik, S.A., and Ferronsky, S.V.: 1979a,Celest. Mech. 19, 173.
Ferronsky, V.I., Denisik, S.A., and Ferronsky, S.V.: 1979b,Celest. Mech. 20, 69.
Ferronsky, V.I., Denisik, S.A., and Ferronsky, S.V.: 1979c,Celest. Mech. 20, 143.
Ferronsky, V.I., Denisik, S.A. and Ferronsky, S.V.: 1981,Celest. Mech. 23, 243.
Ferronsky, V.I., Denisik, S.A., and Ferronsky, S.V.: 1982,Celest. Mech. 27, 285.
Ferronsky, V.I., Denisik, S.A., and Ferronsky, S.V.: 1984,Celest. Mech. 32, 173.
Frank-Kamenetsky, D.A.: 1959,Physical Processes in Stars Interior, Physmatgiz, Moscow.
Jacobi, C. G. J.: 1884,Vorlesungen über Dynamik, Berlin.
Kapitza, P.Z.: 1951,J. Theoret. and Exp. Phys. 21, 5.
Landau, L.D. and Lifschitz, E.M.: 1963,The Quantum Mechanics, Nauka, Moscow.
Landau, L.D. and Lifschitz, E.M.: 1973,The Field Theory, Nauka, Moscow.
Misner, Ch.W., Thorne, K.S., and Wheeler, J.A.: 1973,Gravitation, Freeman, San Francisco.
Weinberg, S.: 1972,Gravitation and Cosmology, John Wiley, New York.
Wintner, A.: 1941,The Analytical Foundations of Celestial Mechanics, Princeton Univ. Press, Princeton.
Zeldovich, Ya. B. and Novikov, I.D.: 1967,Relativistic Astrophysics, Nauka, Moscow.
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Ferronsky, V.I., Denisik, S.A. & Ferronsky, S.V. Virial oscillations of celestial bodies IV: The Lyapunov stability of motion. Celestial Mechanics 35, 23–43 (1985). https://doi.org/10.1007/BF01229112
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DOI: https://doi.org/10.1007/BF01229112