Research article

Five new methods of celestial mechanics

  • Received: 06 September 2019 Accepted: 12 June 2020 Published: 22 June 2020
  • MSC : 37J40, 37J45

  • The last volume of the book "Les méthods nouvelles de la Mécanique céleste" by Poincaré [28] was published more than 120 years ago. Since then, the following methods have arisen. 1. Method of normal forms, allowing to study regular perturbations near a stationary solution, near a periodic solution and so on. 2. Method of truncated systems, which are found with a help of the Newton polyhedrons, allowing to study singular perturbations. 3. Method of generating families of periodic solutions (regular and singular). 4. Method of generalized problems, allowing bodies with negative masses. 5. Computation of a net of families of periodic solutions as a "skeleton" of a part of the phase space.

    Citation: Alexander Bruno. Five new methods of celestial mechanics[J]. AIMS Mathematics, 2020, 5(5): 5309-5319. doi: 10.3934/math.2020340

    Related Papers:

  • The last volume of the book "Les méthods nouvelles de la Mécanique céleste" by Poincaré [28] was published more than 120 years ago. Since then, the following methods have arisen. 1. Method of normal forms, allowing to study regular perturbations near a stationary solution, near a periodic solution and so on. 2. Method of truncated systems, which are found with a help of the Newton polyhedrons, allowing to study singular perturbations. 3. Method of generating families of periodic solutions (regular and singular). 4. Method of generalized problems, allowing bodies with negative masses. 5. Computation of a net of families of periodic solutions as a "skeleton" of a part of the phase space.
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    [1] A. B. Batkhin, Web of families of periodic orbits of the generalized Hill problem, Doklady Math., 90 (2014), 539-544. doi: 10.1134/S1064562414060064
    [2] A. B. Batkhin, Bifurcations of periodic solutions of a Hamiltonian system with a discrete symmetry group, Programm. Comput. Software, 46:2 (2020), 84-97.
    [3] D. Benest, Libration effects for retrograde satellites in the restricted three-body problem, I: Circular plane Hill's case, Celest. Mech., 13 (1976), 203-215.
    [4] A. D. Bruno, Analytical form of differential equations (II), Trans. Moscow Math. Soc., 26 (1972), 199-239.
    [5] A. D. Bruno, On periodic flybys of the Moon, Celest. Mech., 24 (1981), 255-268. doi: 10.1007/BF01229557
    [6] A. D. Bruno, Local Methods in Nonlinear Differential Equations, Springer-Verlag, BerlinHeidelberg-New York-London-Paris-Tokyo, 1989.
    [7] A. D. Bruno, The Restricted 3-body Problem: Plane Periodic Orbits, Walter de Gruyter, Berlin, 1994.
    [8] A. D. Bruno, Power Geometry in Algebraic and Differential Equations. Elsevier Science, Amsterdam, 2000.
    [9] A. D. Bruno, Normal form of a Hamiltonian system with a periodic perturbation, Comput. math. Math. Phys., 60 (2020), 36-52. doi: 10.1134/S0965542520010066
    [10] A. D. Bruno, Normalization of the periodic Hamiltonian system, Program. Comput. Software, 46:2 (2020), 76-83.
    [11] A. D. Bruno, V. P. Varin, The limit problems for the equation of oscillations of a satellite, Celest. Mechan. Dyn. Astron., 67, 1997.
    [12] A. D. Bruno, V. P. Varin, Periodic solutions of the restricted three-body problem for small mass ratio, J. Appl. Math. Mech., 71 (2007), 933-960. doi: 10.1016/j.jappmathmech.2007.12.012
    [13] A. D. Bruno, V. P. Varin, Closed families of periodic solutions of the restricted problem, Solar System Res., 43 (2009), 253-276. doi: 10.1134/S0038094609030071
    [14] A. D. Bruno, V. P. Varin, Families c and i of periodic solutions of the restricted problem for µ = 5 . 10−5, Solar Syst. Res., 43 (2009), 26-40. doi: 10.1134/S0038094609010031
    [15] A. D. Bruno, V. P. Varin, Family h of periodic solutions of the restricted problem for big µ, Solar Syst. Res., 43 (2009), 158-177. doi: 10.1134/S0038094609020099
    [16] A. D. Bruno, V. P. Varin, Family h of periodic solutions of the restricted problem for small µ, Solar Syst. Res., 43 (2009), 2-25. doi: 10.1134/S003809460901002X
    [17] A. D. Bruno, V. P. Varin, On asteroid distribution, Solar Syst. Res., 45 (2011), 323-329. doi: 10.1134/S0038094611040010
    [18] A. D. Bruno, V. P. Varin, Periodic solutions of the restricted three body problem for small µ and the motion of small bodies of the solar system, Astron. Astrophys. Trans. (AApTr), 27 (2012), 479-488.
    [19] M. Hénon, Sur les orbites interplanetaires qui rencontrent deux fois la terre, Bull. astron. Ser. 3, 3 (1968), 377-402.
    [20] M. Hénon, Numerical exploration of the restricted problem. V. Hill's case: Periodic orbits and their stability, Astron. & Astrophys., 1 (1969), 223-238.
    [21] M. Hénon, Generating Families in the Restricted Three-Body problem, Number 52 in Lecture Note in Physics. Monographs, Springer, Berlin, Heidelberg, New York, 1997.
    [22] M. Hénon, Generating Families in the Restricted Three-Body Problem. II. Quantitative Study of Bifurcations, Number 65 in Lecture Note in Physics. Monographs. Springer-Verlag, Berlin, Heidelberg, New York, 2001.
    [23] G. W. Hill, Researches in the lunar theory, Amer. J. Math., 1 (1878), 5-26, 129-147, 245-260. doi: 10.2307/2369430
    [24] A. Y. Kogan, Distant satellite orbits in the restricted circular three-body problem, Cosmic Res., 26 (1989), 705-710.
    [25] M. L. Lidov, M. A. Vashkov'yak, Analytical Celestial Mechanics, Chapter Quasisatellite periodic orbits, 53-57, Kasan' University, Kazan', 1990. (in Russian).
    [26] M. L. Lidov, M. A. Vashkov'yak, Perturbation theory and analysis of evolution of quasisatellite orbits in the restricted three-body problem, Cosmic Res., 31 (1993), 187-207.
    [27] M. L. Lidov, M. A. Vashkov'yak, On quasi-satellite orbits in a restricted elliptic three-body problem, Astron. Lett., 20 (1994), 676-690.
    [28] H. Poincaré, Les méhtods nouvelles de la mécanique céleste, volume 3, Gauthier-Villars, Paris, 1899.
    [29] V. F. Zhuravlev, A. G. Petrov, M. M. Shunderyuk, Selected Problems of Hamiltonian Mechanics, (in Russian).

    © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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