Abstract
About half a century ago, A. M. Molchanov conjectured the existence of a resonance structure of the solar system, stating that as the solar system evolves, its planets inevitably drift to orbits for which the ratios of their orbital periods become close to ratios of relatively small natural numbers. In Molchanov’s opinion, this resonance property is ensured by small dissipative forces. We offer another possible explanation of this phenomenon without taking dissipation into account.
Similar content being viewed by others
References
E. A. Grebenikov and Yu. A. Ryabov, Resonances and Small Denominators in Celestial Mechanics [in Russian], Nauka, Moscow (1978).
A. N. Kolmogorov, Dokl. Akad. Nauk SSSR, 98, 527–530 (1954).
V.I. Arnol’d, Mathematical Methods of Classical Mechanics [in Russian], Éditorial URSS, Moscow (2003); English transl., Springer, New York (1978).
V.I. Arnol’d, Sov. Math. Dokl., 5, 581–585 (1964).
N. N. Nekhoroshev, Funct. Anal. Appl., 5, 338–339 (1971).
A. Bounemoura and V. Kaloshin, Moscow Math. J., 14, 181–203 (2014).
A. M. Molchanov, “A hypothesis on the resonance structure of the solar system [in Russian],” NTsBI, Acad. Sci. USSR, Dubna (1974).
A. E. Roy, Orbital Motion, IOP, London (2005).
N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Non-linear Oscillations [in Russian], Fizmatlit, Moscow (1958); English transl., Gordon and Breach, New York (1961).
W. F. Sierpinski, On the Solution of Equations in Integers [in Russian], Fizmatlit, Moscow (1961).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 187, No. 1, pp. 139–154, April, 2016.
Rights and permissions
About this article
Cite this article
Gubaidullin, M.B. Commensurability and Molchanov’s hypothesis. Theor Math Phys 187, 570–582 (2016). https://doi.org/10.1134/S0040577916040115
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577916040115