Abstract
A relation between nonintegrability of nonlinear dynamical systems with a continuous Fourier spectrum and irreversibility is investigated in terms of the Liealgebraic formalism. Résibois and Prigogine's singular invariants of motion play an essential role. As an application of the formalism, we solve the restricted three-body problem for the case of nearly parabolic motion of the third body. This gives a model of the motion of a comet in the solar system. The results indicate that there is (deterministic) chaos in the motion of a comet in a nearly parabolic orbit. A possible physical implication of the chaotic motion is the existence of a cometary cloud surrounding the solar system. The theoretical results are compared with numerical results, and show good agreement.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. Prigogine,Non-Equilibrium Statistical Mechanics (Interscience, New York, 1962).
I. Prigogine and C. George,Proc. Natl. Acad. Sci. USA 80:4590 (1983).
R. Résibois and I. Prigogine,Bull. Classe Sci. Acad. Roy. Belg. 46:53 (1960).
H. Poincaré,Les Méthodes Nouvelles de la Mechanique Celeste (Dover, New York, 1957) [English translation: NASA Technical Translation F-451].
E. T. Whittaker,A Treatise on the Analytical Dynamics of Particles and Rigid Bodies 4th ed. (Cambridge University Press, Cambridge, 1959).
J. Moser,Stable and Random Motions in Dynamical Systems (Princeton University Press, Princeton, New Jersey, 1973).
A. N. Kolmogorov,Dokl. Akad. Nauk. SSSR 98:527 (1954) [English translation: Los Alamos Scientific Laboratory Translation LA-TR-71-67].
J. Moser,Nach. Akad. Wiss. Göttingen Math. Phys. Kl. 2 1:1 (1962).
V. I. Arnold,Usp. Mat. Nauk 18:9 (1963) [Russ. Math. Surv. 18(6):85 (1963)].
S. Yabushita,Q. J. R. Astron. Soc. 24:430 (1983).
G. Hori,Publ. Astron. Soc. Japan 18:287 (1966).
J. R. Cary,Phys. Rep. 79:129 (1981).
I. Prigogine, A. Grecos, and C. George,Celestial Mechanics 16:489 (1977).
T. Y. Petrosky and I. Prigogine, Poincaré's Theorem and Unitary Transformations for Classical and Quantum Systems,Physica, submitted.
R. Balescu,Equilibrium and Non-Equilibrium Statistical Mechanics (Wiley, New York, 1975).
T. Y. Petrosky,Phys. Lett. A 117:328 (1986).
V. Szebehely,Theory of Orbits (Academic Press, New York, 1967).
T. Y. Petrosky and R. Broucke, Chaotic motion of comets in nearly parabolic orbits,Celestial Mechanics, submitted.
B. V. Chirikov,Phys. Rep. 52:263 (1979).
J. M. Greene,J. Math. Phys. 20:1183 (1980).
R. A. Lyttleton and J. M. Hammersley,Mon. Not. R. Astron. Soc. 127:257 (1964).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Petrosky, T.Y. The cometary cloud in the solar system and the Résibois-Prigogine singular invariants of motion. J Stat Phys 48, 1363–1372 (1987). https://doi.org/10.1007/BF01009550
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01009550