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Euler’s problem and its applications in celestial mechanics and space dynamics. (English. Russian original) Zbl 1272.70063
J. Appl. Math. Mech. 75, No. 6, 660-666 (2011); translation from Prikl. Mat. Mekh. 75, No. 6, 940-950 (2011).
Summary: The numerous generalizations of the classical problem of two fixed centres are analysed, starting from the formulation of the problem and its solution by Euler in 1760 to the present day. The role played by numerous researchers in analysing this problem is noted. The publications cited indicate conclusively that the main results of generalizations of the problem and analytical and qualitative investigations had already been obtained in the nineteenth century and at the beginning of the twentieth century. Present-day researchers can only lay claim to a few occasionally productive and at the same time effective applications of individual generalizations (the Gredeaks problem, for example).
MSC:
70F15 Celestial mechanics
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[1] Vinti, J.P., A new method of solution for unretarded satellite orbits, Nat bur standards J res math and math phys, 63B, 3, 105-116, (1959)
[2] Vinti, J.P., Theory of an accurate intermediate orbit for satellite astronomy, Nat bur standards J res math and math phys, 65B, 2, 169-201, (1961) · Zbl 0148.44701
[3] Izsak IG. The theory of satellite motion about an oblate planet. 1. A second order solution of Vinti’s dynamical problem. Special Report No. 52 of the Smithsonian Institution Astrophysical Observatory, 1960.
[4] Izsak, I.G., On satellite orbits with very small eccentricities, Astron J, 66, 3, 129-131, (1961)
[5] Izsak, I.G., The theory of satellite motion about an oblate planet. A second order solution of Vinti’s dynamical problem, Smithsonian contribut to astrophys, 6, 81-107, (1963)
[6] Aksenov YeP, Grebenikov YeA, Demin VG. The general solution of the problem of the motion of an artificial satellite in the Earth’s normal gravitational field. In: Artificial Satellites. No. 8. Moscow: Izd Akad Nauk SSSR. 1961: 64-71.
[7] Aksenov YeP, Grebenikov YeA, Demin VG. Application of the generalized problem of two fixed centres in the theory of the motion of an artificial satellite. In: Problems of the Motion of Artificial Celestial Bodies. Moscow: Izd Akad Nauk SSSR. 1963: 92-8
[8] Euler, L., Un corps étant attiré en raison réciproque quarreé des distances vers deux points fixes donnés, trouver LES cas ou la curbe decrite par ce corps sera algebraique, In: Mémoirs de l’academie de Berlin for, 1760, 228-247, (1767)
[9] Euler, L., De motu corporis ad duo centra virium fixa attracti, In: st Petersburg memoirs, 1764/1765, 10, 207-242, (1766)
[10] Euler, L., Du motu corporis ad duo centra virium fixa attracti, Novi comment acad scient imper petropol st Petersburg memoirs, 1765/1767, 11, 152-184, (1767)
[11] Hiltebeitel, A.M., Note on a problem in mechanics, Amer J math (Read before amer math soc, 25 February 1905), 3, 433-436, (1911) · JFM 36.0768.03
[12] Hiltebeitel, A.M., On the problem of two fixed centers and certain of its generalizations, Amer J math, 33, 337-362, (1911) · JFM 42.0764.03
[13] Tallqvist, Über die bewegung eines punktes, welcher von zwei festen zentren nach dem newtonschen gesetze angezogen wird, Acta soc sci fennicae nova ser A helsingfors, 1, 1, (1927) · JFM 53.0738.01
[14] Tallqvist, Die bewegung eines massepunktes unter dem einfluss den schwere und einer newtonschen zentralkraft, Acta soc sci fennicae nova ser A helsingfors, 1, 2, (1927) · JFM 53.0738.02
[15] Tallqvist, Zum zweizentrenproblem in raume, Acta soc sci fennicae nova ser A helsingfors, 1, 3, (1927) · JFM 53.0739.01
[16] Tallqvist, Zum zweizentrenproblem beim vorhandensein auch abstossender kräfter, Acta soc sci fennicae nova ser A helsingfors, 1, 4, (1928) · JFM 54.0835.02
[17] Tallqvist, H.I., Zum zweizentrenproblem im raume beim vorhandsein auch abstossender kräfte, Acta soc sci fennicae nova ser A helsingfors, 1, 5, (1928) · JFM 54.0835.03
[18] Tallqvist, H.I., Die bewegung eines massepunktes unter dem schwere und einer abstossenden newtonschen zentralkraft, Acta soc sci fennicae nova ser A helsingfors, 1, 6, (1929) · JFM 55.1099.01
[19] Tallqvist, H.I., Über ein spezielles dreizentrenproblem, Acta soc sci fennicae nova ser A helsingfors, 1, 8, (1929) · JFM 55.1099.02
[20] Luk’yanov, L.G.; Yemel’yanov, N.V.; Shirmin, G.I., The generalized problem of two fixed centres or the darboux – gredeaks problem, Kosmich issled, 43, 3, 194-200, (2005)
[21] Desboves, A.-H., Sur le mouvement d’un point matériel attiré en raison inverse du carré des distances par deux centres mobiles, J math pures et appl (Liouville’s J), 13, 369-396, (1848)
[22] Jacobi CGJ. Anziehung eines Punktes nach zwei festen Centren. Vorlesungen über Dynamik No. 29 (read 1842-1843). Berlin: A. Clebsh; 1843.(second edition 1884).
[23] Jacobi, C.G.J., Vorlesungen über dynamik, (1884), G. Reimer Berlin
[24] Lagrange JL. Recherches sur le mouvement d’un corps qui est attiré vers deux centres fixes. Ouvres 1766-1769. Vol. 2. I Mémoir.
[25] Lagrange JL. Ou l’on suppose que, l’attraction est en reson, inverse des carrées de distances (II Mémoir, 94-121). Tourin Mémoirs for 1766-69\bf4:215-43. L Ouvres 1868;\bf2:65-121. (Discussed in Méchanique Analitique, 2nd edition. Paris: 1811-1815.).
[26] Lagrange, J.L., Mécanique analytique Paris;, (1815), Ve Coureier
[27] Zhuravlev SG, Naniyev VS. Some generalizations of the problem of two fixed centres. In: Theory and Applied Problems of Non-linear Analysis. Moscow: A.A. Dorodnitsyn Computing Centre, Russian Acad. Sci.; 2009. 201-12.
[28] Bolotin, S.V., The non-integrability of the problem of n centres for n > 2, Vestn MGU ser 1 matematika mekhanika, 3, 65-68, (1984) · Zbl 0551.70008
[29] Kozlov, V.V., Topological obstacles to the integrability of natural mechanical systems, Dokl akad nauk SSSR, 249, 6, 1299-1302, (1979)
[30] Kozlov, V.V., Integrability and non-integrability in Hamiltonian mechanics, Uspekhi mat nauk, 38, 1, 3-67, (1983) · Zbl 0525.70023
[31] Perelomov, A.M., Integrable systems of classical mechanics and Lie algebra, (1990), Nauka Moscow · Zbl 0717.70003
[32] Legendre A-M. Exersices der Calcul Intégral sur Divers Orders de Transcendantes et sur les Quadratures. Vol. 1. Paris: 1811; Vol. 2. Paris: 1816; Vol. 3. Paris: 1817.
[33] Legendre A-M. Traité des Functions Elliptiques. Paris: Imprimerie de Huzard-Coucier; Vol. 1 1825. 2 1826. Vol. 3 1828.
[34] Legendre, A.-M., Traité des functions elliptiques et des intégrals eulériennes, J math pures et appl (Liouville’s J), 9, 113-115, (1825)
[35] Liouville J. Sur quelques cas particuliers où les équations du movement d’un point matériele peuvent s’intégrer. J Math Pures et Appl (Liouville’s J). Premier Mémoir 1846; \bf11:345-68. Second Mémoir 1847;\bf12:410-44.
[36] Liouville J. Mémoir sur l’intégration des équations différentielles du movement d’un nombre quelconque des points matériels. Connaisance de Temps 1850\bf1847: 1-40: J Math Pures et Appl (Liouville’s J) 1849: \bf14;257-99.
[37] Vasquez, B.J.A., Potential of the force field in a model problem of celestial mechanics and space geodesy, Izv VUZ geodez i aerofotos’emka, 5, 107-112, (2006)
[38] Königsberger, W.I., De motu puncti versus dou fixa centra attracti. dissertation, (1860), Berolini Berlin
[39] Velde W. Über einen Specialfall der Bewegung eines Punktes, welcher von festen Centren angezogen wird. Wissenschaftliche Beilage zum Programm der Ersten Städtischen Höheren Bürgerschule in Berlin, Programm No. 104. Berlin: Gärtners; 1889. · JFM 20.0936.04
[40] Velde, W., Short exposition, Bull des sciences mathematiques, 14, 1, 125-126, (1890)
[41] Darboux, G., Sur un problème de méchanique, Arch Néelandaises des sci exactes et naturéeles la haye, Ser. 2, 6, 371-376, (1901) · JFM 32.0725.02
[42] Kaisin, V.K., A case of the generalization of the problem of two fixed centres, Byull inst teor astron, 12, 2, 163-171, (1970)
[43] Kaisin, V.K., The motion of a spacecraft in the Earth’s normal gravitational field under additional forces, Kosmich issled, 7, 5, 686-693, (1969)
[44] Kozlov, I.S., The problem of four fixed centres and its applications to the theory of motion of celestial bodies, Astron zh, 51, 1, 191-198, (1974) · Zbl 0278.70011
[45] Kozlov, I.S., Interpretation and applications of the problem of four fixed centres, Astron zh, 52, 3, 649-656, (1975) · Zbl 0304.70017
[46] Kochiyev, A.A., The solution of the problem of the motion of a point in a field of conservative forces and its applications it in celestial mechanics, Astron zh, 54, 1, 228-232, (1974)
[47] Kochiyev, A.A., The solution of the problem of the motion of a point in the gravitational field of a rigid body, Astron zh, 64, 5, 1124-1128, (1987)
[48] Borisov, A.V.; Mamaev, I.S., Generalized problem of two and four Newtonian centers, Celest mech and dyn astron, 92, 4, 371-380, (2005) · Zbl 1129.70010
[49] Beletskii, V.V., Trajectories of space flights with a constant reactive acceleration vector, Kosmich issled, 2, 3, 408-413, (1964)
[50] Demin, V.G., The motion of an artificial satellite in a non-central gravitational field, (1968), Nauka Moscow
[51] Demin, V.G., The approximate solution of the problem of the motion of an artificial Earth satellite, Soobshch GoS astron in-ta im shternberg, 125, 3-11, (1962)
[52] Demin, V.G., A method for investigating the motion of a spacecraft in the field of action of a planet. trudy univ druzhby nar im P lumumba, Ser teor mekh, 17, 4, 13-17, (1966)
[53] Kunitsyn, A.L., A qualitative study of motions in a limit version of the problem of two fixed centres. trudy univ druzhby nar im P lumumby, Ser teor mekh, 17, 4, 32-52, (1966)
[54] Born, M., Vorlesungen über atommechanik, Göttingen 1923-1924, (1925), Springer Berlin
[55] Born, M., Vorlesungen über atommechanik, (1925), Springer Berlin · JFM 51.0727.01
[56] Moulton FR. The straight line solutions of the problem of N bodies. Ann Math Ser 2 1910; 12:1-17.(see also Bull Amer Math Soc 1900; 7:249).
[57] Arazov, G.T., The problem of three fixed centres, Pis’ma v astron zh, 1, 6, 42-45, (1975) · Zbl 0336.70013
[58] Arazov, G.T., The problem of three fixed centres, Astron zh, 53, 3, 639-646, (1976) · Zbl 0336.70013
[59] Lukashevich, YeL., An integrable case of the motion of a satellite in the Earth’s non-central gravitational field, Kosmich issled, 17, 3, 457-459, (1979)
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