Analytical model of the Earth - Moon system gravitational potential in the form of a general solution to the Newtonian three-body problem | Engineering Journal: Science and Innovation

Article

Analytical model of the Earth - Moon system gravitational potential in the form of a general solution to the Newtonian three-body problem

Published: 05.03.2018

Published in issue: #2(74)/2018

DOI: 10.18698/2308-6033-2018-2-1734

Category: Aviation and Rocket-Space Engineering | Chapter: Aircraft Dynamics, Ballistics, Motion Control

The article presents a mathematical model of the gravitational potential of the Earth — Moon system, which allows analytical calculating the optimal space vehicle trajectories in this celestial mechanical system. This model represents the canonical quaternion generalization of the classical mathematical Aksenov — Grebenikov — Demin model of the Earth gravitational potential. It is shown that the proposed model also realizes the complete separation of the variables in the Hamiltonian of the classical Newtonian three-body problem and corresponds to its analytic solvability, which is the unexpected fact from the classical point of view. Thus, the resulting formulas for an exact general solution of the three-body problem simulate equipotential surfaces of the gravitational potential of the Earth — Moon system. Equipotential lines, in particular, simulate the orbits of space vehicles in the Earth — Moon system. Optimum control of space vehicles is mathematically simulated by a corresponding change in the parameters of the general solution to the Newtonian three-body problem. It is shown that such parametric deformations geometrically correspond to the isometries of the analytic complex three-dimensional Lobachevsky space.

References
[1] Abrashkin V.I., Bogoyavlensky N.L., Voronov K.E., Kazakova A.E., Puzin Yu.Ya., Sazonov V.V., Semkin N.D., Chebukov S.Yu. Kosmicheskie issledovaniya — Cosmic Research, 2007, vol. 45, no. 5, pp. 450–470.
[2] Putin G.F., Glukhov A.F., Babushkin I.A., Zavalishin D.A., Belyaev M.Yu., Ivanov A.A., Sazonov V.V. Kosmicheskie issledovaniya — Cosmic Research, 2012, vol. 50, no. 5, pp. 373–379.
[3] Solovyov V.A., Lyubinsky V.E., Mishurova N.V. Inzhenernyy zhurnal: nauka i innovatsii — Engineering Journal: Science and Innovation, 2017, issue 5. Available at: http://dx.doi.org/10.18698/2308-6033-2017-5-1614
[4] Sukhanov A.A. Astrodinamika [Astrodynamics]. Moscow, IKI RAN Publ., 2010, 204 p.
[5] Aksenov E.P., Grebenikov E.A., Demin V.G. Obshchee reshenie zadachi o dvizhenii iskusstvennogo sputnika v normalnom pole prityazheniya Zemli [General solution of the problem of an artificial satellite motion in the normal gravitational field of the Earth]. In: Iskusstvennye sputniki Zemli [Earth Artificial Satellites]. 1961, issue 8, pp. 64–71.
[6] Aksenov E.P., Grebenikov E.A., Demin V.G. Astronomicheskiy zhurnal — Astronomy Reports, 1963, vol. 40, no. 2, pр. 363–372.
[7] Aksenov E.P., Grebenikov E.A., Demin V.G. Primenenie obobshchennoy zadachi dvukh nepodvizhnykh tsentrov v teorii dvizheniya iskusstvennykh sputnikov Zemli [Application of the generalized problem of two fixed centers in the theory of artificial Earth satellite motion]. In: Problemy dvizheniya iskusstvennykh sputnikov nebesnykh tel [The problems of motion of artificial satellites of celestial bodies]. Moscow, AN SSSR Publ., 1963, pр. 92–98.
[8] Aksenov E.P., Grebenikov E.A., Demin V.G. Kachestvennyy analiz form dvizheniya v zadache o dvizhenii iskusstvennogo sputnika v normalnom pole prityazheniya Zemli [Qualitative analysis of the forms of motion in the problem of the artificial satellite motion in the normal gravitational field of the Earth]. In: Iskusstvennye sputniki Zemli [Earth Artificial Satellites]. 1963, issue 16, pр. 173–197.
[9] Demin V.G. Dvizhenie iskusstvennogo sputnika v netsentralnom pole tyagoteniya [The motion of an artificial satellite in an off-center gravitational field]. Moscow – Izhevsk, NITs Regulyarnaya i khaoticheskaya dinamika. Institut kompyuternykh issledovaniy Publ., 2010, 420 p.
[10] Sidorenkov N.S. The nature seasonal and interannual variations of the Earth’s rotation. Figure and Dynamics of the Earth, Moon and Planets. Prague, 1987, рp. 947–960.
[11] Murray C.D., Dermott S.F. Solar System Dynamics. Cambridge University Press Publ., 1999, 606 p. [In Russ.: Murray C., Dermott S. Dinamika Solnechnoy sistemy. Moscow, Fizmatlit Publ., 2010, 588 p.].
[12] Poincaré H. New Methods of Celestial Mechanics. D. Goroff, ed. Springer Science & Business Media Publ., 1967. [In Russ.: Poincaré H. Izbrannye Trudy. V trekh tomakh. Tom 2. Novye metody nebesnoy mekhaniki. Topologiya. Teoriya chisel. Moscow, Nauka Publ., 1972, pp. 445–452].
[13] Poincaré H. New Methods of Celestial Mechanics. D. Goroff, ed. Springer Science & Business Media Publ., 1967 [In Russ.: Poincaré H. Izbrannye Trudy. V trekh tomakh. Tom 2. Novye metody nebesnoy mekhaniki. Topologiya. Teoriya chisel. Moscow, Nauka Publ., 1972, pp. 325–356].
[14] Poincaré H. New Methods of Celestial Mechanics. D. Goroff, ed. Springer Science & Business Media Publ., 1967 [In Russ.: Poincaré H. Izbrannye Trudy. V trekh tomakh. Tom 2. Novye metody nebesnoy mekhaniki. Topologiya. Teoriya chisel. Moscow, Nauka Publ., 1972, pp. 356–441].
[15] Abrarov D.L. Dzeta-funktsii i L-funktsii v gamiltonovoy dinamike [Zeta functions and L-functions in the Hamiltonian dynamics]. Moscow, VTs RAN Publ., 2010, 225 p.
[16] Abrarov D.L. Dzeta-model klassicheskoy mekhaniki. Teoreticheskie i prikladnye aspekty [The Zeta model of classical mechanics. Theoretical and applied aspects]. Saarbruecken, LAP Lambert Academic Publishing, 2016, 276 с.
[17] Abrarov D.L. Exact solvability of model problems of classical mechanics in global L-function and its mechanical and physical meaning. Sbornik tezisov dokladov Mezhdunarodnoy konferentsii po matematicheskoy teorii upravleniya v mekhanike [International Conference on mathematical theory of control and mechanics. Abstracts]. Suzdal, 2017, pp. 149–150.
[18] Abrarov D.L. Tochnaya matematicheskaya model nyutonovoy zadachi trekh tel [Exact mathematical model of the Newtonian three-body problem]. Tezisy dokladov Mezhdunarodnoy konferentsii, Posvyashchennoy 170- letiyu velikogo russkogo uchenogo N.E. Zhukovskogo “Fundamentalnye i prikladnye zadachi mekhaniki”, Moskva, 24–27 oktyabrya 2017 g. [International scientific conference Fundamental and Applied Problems of Mechanics (FAPM–2017) dedicated to the 170th anniversary of a distinguished Russian scientist Nikolay Egorovich Zhukovsky. Moscow, October 24–27, 2017. Abstracts]. Moscow, BMSTU, 2017, pp. 103–104.
[19] Abrarov D.L. Tochnaya matematicheskaya model nyutonovoy zadachi trekh tel. [Exact mathematical model of the Newtonian three-body problem]. Petrovskie chteniya 2017. Tezisy dokladov 3-y Mezhdunarodnoy zimney shkoly-seminara po gravitatsii, kosmologii i astrofizike [Petrovsky readings 2017. 3rd International winter school-seminar on gravity, cosmology and astrophysics. Abstracts]. Kazan, KFU, 2017, p. 40.
[20] Sosinsky A.B. Geometrii [Geometries]. Moscow, MTsNMO Publ., 2017, 263 p.
[21] Modular Forms and Fermat's Last Theorem. Springer-Verlag, New York, Inc., 1997, 582 p.
[22] Rubin K., Silverberg A. Families of elliptic curves with constant mod p representations. In: Elliptic curves, modular forms & Fermat’s last theorem. Proc. of the Conf. on Elliptic Curves and Modular Forms. Hong Kong, December 18–21, 1994. Cambridge, MA, International Press, 1995, pp. 148–161.
[23] Darmon H. Andrew Wiles's Marvelous Proof. Notices of the AMS, 2017, рp. 209–216.
[24] Hatcher A. Algebraic Topology. Cambridge, Cambridge University Press, 2002 [In Russ.: Hatcher A. Algebraicheskaya topologiya. Moscow, MTsNMO Publ., 2011, 688 p.].