Bifurcations of Liouville tori of a two fixed center problem

Original Article
  • 32 Downloads

Abstract

A complete description of the real phase topology of a two fixed center problem is introduced and studied. Moreover, all generic bifurcations of Liouville tori are determined theoretically and the periodic solution is presented. At the end of this paper, the phase portrait is studied.

Keywords

Hamilton–Jacobi’s equations Bifurcations of Liouville tori Topology of the level sets Momentum maps Periodic solution Elliptic functions Phase portrait 

Notes

Acknowledgements

The authors would like to express their sincere thanks to the editor and to anonymous reviewers for their valuable comments and suggestions which have helped immensely in improving the quality of the paper.

References

  1. Aksenov, E.P., Grebenikov, E.A., Demin, V.G.: General solution to the problem of satellite motion in the normal field of Earth’s attraction. Planet. Space Sci. 9, 491–498 (1962) ADSCrossRefGoogle Scholar
  2. Albouy, A.: The underlying geometry of the fixed centers problems. In: Brezis, H., Chang, K.C., Li, S.J., Rabinowitz, P. (eds.) Topological Methods, Variational Methods and Their Applications, Taiyuan, 2002, pp. 11–21. World Scientific, River Edge (2003) CrossRefGoogle Scholar
  3. Albouy, A., Stuchi, T.J.: Generalizing the classical fixed-centres problem in a non-Hamiltonian way. J. Phys. A 37(39), 9109–9123 (2004) ADSMathSciNetCrossRefMATHGoogle Scholar
  4. Arazov, G.T.: Investigation of motion of a satellite of spheroidal planet. Astron. Zh. 52, 891–894 (1975) ADSGoogle Scholar
  5. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60. Springer, Berlin (1989) Google Scholar
  6. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, Dynamical System III. Encyclopaedia of Mathematical Sciences, vol. 3. Springer, Berlin (2006) MATHGoogle Scholar
  7. Avdreyanov, P.P., Dushin, K.E.: Bifurcation sets in the Kovalevskaya–Yehia problem. Sb. Math. 203(4), 459–499 (2012) MathSciNetCrossRefMATHGoogle Scholar
  8. Bolsinov, A.V., Fomenko, A.T.: Integrable Hamiltonian Systems: Geometry, Topology, Classification, vols. 1 and 2. Udmurt State University Publishing House, Izhevsk (1999). English translation: Chapman and Hall/CRC, Boca Raton, vols. 1 and 2 (2004) MATHGoogle Scholar
  9. Bolsinov, A.V., Richter, P., Fomenko, A.T.: Method of circular molecules and the topology of the Kovalevskaya top. Mat. Sb. 191, 1–42 (2000) MathSciNetCrossRefMATHGoogle Scholar
  10. Borisov, A.V., Mamaev, I.S.: Generalized problem of two and four Newtonian centers. Celest. Mech. Dyn. Astron. 92(4), 371–380 (2005) ADSMathSciNetCrossRefMATHGoogle Scholar
  11. Borisov, A.V., Mamaev, I.S.: Relations between integrable systems in plane and curved spaces. Celest. Mech. Dyn. Astron. 99(4), 253–260 (2007) ADSMathSciNetCrossRefMATHGoogle Scholar
  12. Charlier, C.L.: Die Mechanik des Himmels. de Gruyter, Berlin (1927) MATHGoogle Scholar
  13. Darboux, G.: Sur un probléme de méchanique. Arch. Neerl. Sci. Exactes Nat. 6(2), 371 (1901) MATHGoogle Scholar
  14. Deprit, A.: Le probléme des deux centers fixes. Bull. Soc. Math. Belg. 14, 12–45 (1962) MathSciNetMATHGoogle Scholar
  15. Duboshin, G.N.: Celestial Mechanics: Analytical and Qualitative Methods. Nauka, Moscow (1968) MATHGoogle Scholar
  16. Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry-Methods and Applications, Part II: The Geometry and Topology of Manifolds. Springer, New York (1985) CrossRefMATHGoogle Scholar
  17. El-Sabaa, F.M.: Bifurcation of Kovalevskaya polynomial. Int. J. Theor. Phys. 34, 2071–2083 (1995) MathSciNetCrossRefMATHGoogle Scholar
  18. Euler, L.: Un corps étant attiré en raison réciproque quarreé des distances vers deux points fixes donnés. Mem. Berlin, p. 228 (1760) Google Scholar
  19. Fomenko, A.T.: Integrability and Nonintegrability in Geometry and Mechanics. Kluwer Academic, Dordrecht (1988) CrossRefMATHGoogle Scholar
  20. Fomenko, A.T.: Visual Geometry and Topology. Springer, Berlin (1994) CrossRefMATHGoogle Scholar
  21. Gavrilov, L.: Bifurcations of invariant manifolds in the generalized Hénon-Heiles system. Physica D 34, 223–239 (1989) ADSMathSciNetCrossRefMATHGoogle Scholar
  22. Gonzalez Leon, M.A., Guilarte, J.M., de la Torre Mayado, M.: Orbits in the problem of two fixed centers on the sphere. Regul. Chaotic Dyn. 22(5), 520–542 (2017) ADSMathSciNetCrossRefMATHGoogle Scholar
  23. Kaisin, V.K.: Spacecraft motion in the normal gravity field of the Earth under the action of additional forces. Kosm. Issled. 7, 686–693 (1969) ADSGoogle Scholar
  24. Kaisin, V.K.: One case of generalization of the problem of two immobile centers. Bûll. Inst. Teor. Astron. 12, 163 (1970) ADSGoogle Scholar
  25. Kharlamov, M.P.: Bifurcations of common levels of first integrals in the Kovalevskaya case. Prikl. Mat. Meh. 47, 922–930 (1994) Google Scholar
  26. Koman, G.G.: Intermediate orbits of artificial lunar satellites. Soobshch. Gos. Astron. Inst. Shternberga 186, 3–45 (1973) MathSciNetGoogle Scholar
  27. Koman, G.G.: One form of differential equations of motion of artificial lunar satellites. Astron. Zh. 52, 207–209 (1975) ADSGoogle Scholar
  28. Kozlov, V.V.: Methods of Qualitative Analysis in the Dynamics of Rigid Bodies. MGU, Moscow (1980) MATHGoogle Scholar
  29. Lukyanov, L.G., Emeljanov, N.V., Shirmin, G.I.: Generalized problem of two fixed centers or the Darboux–Gredeaks problem. Cosm. Res. 43(3), 186–191 (2005) ADSCrossRefGoogle Scholar
  30. Mamaev, I.S.: Two integrable systems on a two-dimensional sphere. Dokl. Phys. 48(3), 156–158 (2003). See also: Dokl. Akad. Nauk 389(3), 338–340 (2003) ADSMathSciNetCrossRefGoogle Scholar
  31. Moulton, F.R., Duboshin, G.N.: An Introduction to Celestial Mechanics, Suppl. 1. ONTI, Leningrad/Moscow (1935), 445 pp. (Russian translation) Google Scholar
  32. Pauli, W.: Uber das Modell des Wasserstoffmolekulions. Ann. Phys. 68, 177–240 (1922) CrossRefGoogle Scholar
  33. Seri, M.: The problem of two fixed centers: bifurcation diagram for positive energies. J. Math. Phys. 56, 1–18 (2015) MathSciNetCrossRefMATHGoogle Scholar
  34. Vozmischeva, T.G.: The two center and Lagrange problem in the Lobachevsky space. In: Proc. Int. Conf. Geometry, Integrability, and Quantization, Bulgaria, pp. 283–298 (1999) Google Scholar
  35. Vozmischeva, T.G.: Classification of motions for generalization of the two center problem on a sphere. Celest. Mech. Dyn. Astron. 77, 37–48 (2000) ADSCrossRefMATHGoogle Scholar
  36. Vozmischeva, T.G.: The Lagrange and two-center problems in the Lobachevsky space. Celest. Mech. Dyn. Astron. 84(1), 65–85 (2002) ADSMathSciNetCrossRefMATHGoogle Scholar
  37. Vozmischeva, T.G.: Integrable Problems of Celestial Mechanics in Spaces of Constant Curvature. Astrophysics and Space Science Library, vol. 180. Springer, Berlin (2003) CrossRefMATHGoogle Scholar
  38. Vozmischeva, T.G., Oshemkov, A.A.: The topological analysis of the two-center problem on the two-dimensional sphere. Sb. Math. 193(8), 3–38 (2002) MathSciNetCrossRefGoogle Scholar
  39. Vozmishcheva, T.G.: Integrable problems of celestial mechanics in spaces of constant curvature. J. Math. Sci. 125(4), 419–532 (2005) MathSciNetCrossRefMATHGoogle Scholar
  40. Waalkens, H., Dullin, H.R., Richter, P.H.: The problem of two fixed centers: bifurcation, actions, monodromy. Physica D 196, 265–310 (2004) ADSMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of EducationAin Shams UniversityCairoEgypt

Personalised recommendations