McGehee Blow-Up of the Kepler Problem on Surfaces of Constant Curvature

• Andrade, Jaime ^[1] ; Crespo, Francisco ^[1] ; Paulina, Martínez Y ^[1] ; Vidal, Claudio ^[1]
  1. [1] Universidad del Bío-Bío

     Universidad del Bío-Bío

     Comuna de Concepción, Chile

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00349-6
• Enlaces
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• Resumen

  • In this work we address the local behaviour at collision of the Kepler problem on surfaces of constant curvature. A full description of dynamics in the extended energy surface is given relying in the energy and mass values. Our analysis reveals remarkable differences with the classical Newtonian case. In particular, the collision surface does not coincides with the null energy surface.

• Referencias bibliográficas
  • 1. Andrade, J., Dávila, N., Pérez-Chavela, E., Vidal, C.: Dynamics and regularization of the Kepler problem on surfaces of constant curvature....
  • 2. Bolyai, J.: Geometrische untersuchungen, Hrsg. P. Stacke. Teubner, Leipzig (1913)
  • 3. Cariñena, J.F., Rañada, M.F., Santander, M.: Central potentials on spaces of constant curvature: the Kepler problem on the two-dimensional...
  • 4. Castelli, R., Paparella, F., Portaluri, A.: Singular dynamics under a weak potential on a sphere. NoDEA Nonlinear Differ. Equ. Appl. 20(3),...
  • 5. Delgado, J., Vidal, C.: Blow up of total collision in the tetrahedral non-rotating four body problem. In: Delgado, J., Lacomba, E.A., Llibre,...
  • 6. Delgado-Fernández, J., Pérez-Chavela, E.: The rhomboidal four body problem. Global flow on the total collision manifold. In: Ratiu, T....
  • 7. Devaney, R.L.: Singularities in classical mechanical systems. In: Katok, A. (ed.) Ergodic Theory and Dynamical Systems, I (College Park,...
  • 8. Devaney, R.L.: Triple collision in the planar isosceles three-body problem. Invent. Math. 60(3), 249– 267 (1980)
  • 9. Diacu, F.: The classical n-body problem in the context of curved space. ArXiv e-prints (2014)
  • 10. Diacu, F., Pérez-Chavela, E., Santoprete, M.: The n-body problem in spaces of constant curvature. Part I: relative equilibria. J. Nonlinear...
  • 11. Diacu, F., Pérez-Chavela, E., Santoprete, M.: The n-body problem in spaces of constant curvature. Part II: singularities. J. Nonlinear...
  • 12. do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Inc., Englewood Cliffs (1976). Translated from the Portuguese
  • 13. Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry—Methods and Applications. Part I, volume 93 of Graduate Texts in Mathematics,...
  • 14. Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds. Lecture Notes in Mathematics, vol. 583. Springer, Berlin (1977)
  • 15. Kozlov, V.V., Harin, A.O.: Kepler’s problem in constant curvature spaces. Celest. Mech. Dyn. Astron. 54(4), 393–399 (1992)
  • 16. Liebmann, H.: Die kegelschnitte un die planetenbewegung in nitchteuklidischen raum. Math. Phys. Klasse 54, 393–423 (1902)
  • 17. Liebmann, H.: Nichteuklidische Geometrie. Cornell University Library Historical Math Monographs. G. J. Göschen, Leipzig (1905)
  • 18. Lobachevsky, N.I.: Nascent non-Euclidean geometry: revisiting geometric research on the theory of parallels. Quantum 9(5), 20–25 (1999)....
  • 19. Martínez, R., Simó, C.: Relative equilibria of the restricted three-body problem in curved spaces. Celest. Mech. Dyn. Astron. 128, 221–259...
  • 20. McGehee, R.: Triple collision in the collinear three-body problem. Invent. Math. 27, 191–227 (1974)
  • 21. Meyer, K.R., Hall, G.R.: Introduction to Hamiltonian Dynamical Systems and the N-body Problem. Applied Mathematical Sciences, vol. 90....
  • 22. Santoprete, M.: Block regularization of the Kepler problem on surfaces of revolution with positive constant curvature. J. Differ. Equ....
  • 23. Tibboel, P.: Polygonal homographic orbits in spaces of constant curvature. Proc. Am. Math. Soc. 141(4), 1465–1471 (2013)
  • 24. Zhu, S.: Eulerian relative equilibria of the curved 3-body problems in S2. Proc. Am. Math. Soc. 142(8), 2837–2848 (2014)