Advertisement

Homoclinic Solutions to Infinity and Oscillatory Motions in the Restricted Planar Circular Three Body Problem

  • Marcel GuardiaEmail author
  • Pau Martín
  • Tere M. Seara
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 54)

Abstract

The circular restricted three body problem models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies, the primaries, which describe circular planar Keplerian orbits. The system has a first integral, the Jacobi constant. The existence of oscillatory motions for the restricted planar circular three body problem, that is, of orbits which leave every bounded region but which return infinitely often to some fixed bounded region, was proved by Llibre and Simó [18] in 1980. However, their proof only provides such orbits for values of the ratio between the masses of the two primaries exponentially small with respect to the Jacobi constant. In the present work, we extend their result proving the existence of oscillatory motions for any value of the mass ratio. The existence of these motions is a consequence of the transversal intersection between the stable and unstable manifolds of infinity, which guarantee the existence of a symbolic dynamics that creates the oscillatory orbits. We show that this intersection does happen for any value of the mass ratio and for big values of the Jacobi constant. We remark that, since in our setting the mass ratio is no longer small, this transversality cannot be checked by means of classical perturbation theory respect to the mass ratio. Furthermore, since our method is valid for all values of mass ratio, we are able to detect a curve in the parameter space, formed by the mass ratio and the Jacobi constant, where cubic homoclinic tangencies between the invariant manifolds of infinity appear.

Keywords

Invariant Manifold Unstable Manifold Hausdorff Dimension Homoclinic Orbit Body Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The authors have been partially supported by the Spanish MINECO-FEDER Grants MTM2009-06973, MTM2012-31714 and the Catalan Grant 2009SGR859. M. G. and P. M warmly thank the Institute for Advanced Study for their hospitality, stimulating atmosphere and support. During his stay in the Institute for Advanced Study, M. G. was also partially supported by the NSF grant DMS-0635607.

References

  1. 1.
    Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Dynamical Systems III. Volume 3 of Encyclopaedia of Mathematical Sciences. Springer, Berlin (1988)Google Scholar
  2. 2.
    Baldomá, I., Fontich, E.: Exponentially small splitting of invariant manifolds of parabolic points. Mem. Am. Math. Soc. 167(792), x–83 (2004)Google Scholar
  3. 3.
    Baldomá, I., Fontich, E., Guàrdia, M., Seara, T.M.: Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results. J. Differ. Equ. 253(12), 3304–3439 (2012)CrossRefzbMATHGoogle Scholar
  4. 4.
    Delshams, A., Seara, T.M.: An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum. Commun. Math. Phys. 150(3), 433–463 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Delshams, A., Seara, T.M.: Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom. Math. Phys. Electron. J. 3, Paper 4, 40 pp. (electronic) (1997)Google Scholar
  6. 6.
    Delshams, A., Kaloshin, V., de la Rosa, A., Seara, T.: Parabolic orbits in the restricted three body problem (2012, preprint)Google Scholar
  7. 7.
    Gelfreich, V.G.: Melnikov method and exponentially small splitting of separatrices. Phys. D 101(3–4), 227–248 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gelfreich, V.G.: Separatrix splitting for a high-frequency perturbation of the pendulum. Russ. J. Math. Phys. 7(1), 48–71 (2000)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Galante, J., Kaloshin, V.: Destruction of invariant curves using the ordering condition. Preprint, available at http://www.terpconnect.umd.edu/~vkaloshi (2010)
  10. 10.
    Galante, J., Kaloshin, V.: The method of spreading cumulative twist and its application to the restricted circular planar three body problem. Preprint, available at http://www.terpconnect.umd.edu/~vkaloshi (2010)
  11. 11.
    Galante, J., Kaloshin, V.: Destruction of invariant curves in the restricted circular planar three-body problem by using comparison of action. Duke Math. J. 159(2), 275–327 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gorodetski, A., Kaloshin, V.: Hausdorff dimension of oscillatory motions for restricted three body problems. Preprint, available at http://www.terpconnect.umd.edu/~vkaloshi (2012)
  13. 13.
    Guardia, M.: Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discret. Contin. Dyn. Syst. 33(7), 2829–2859 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guardia, M., Olivé, C., Seara, T.M.: Exponentially small splitting for the pendulum: a classical problem revisited. J. Nonlinear Sci. 20(5), 595–685 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Guardia, M., Martín, P., Seara, T.M.: Oscillatory motions for the restricted planar circular three body problem. Preprint available at http://http://arxiv.org/abs/1207.6531 (2012)
  16. 16.
    Holmes, P., Marsden, J., Scheurle, J.: Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations. In: Meyer, K.R., Saari, D.G. (eds.) Hamiltonian Dynamical Systems. Volume 81 of Contemporary Mathematics. American Mathematical Society, Providence (1988)Google Scholar
  17. 17.
    Lochak, P., Marco, J.-P., Sauzin, D.: On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems. Mem. Am. Math. Soc. 163(775), viii + 145 (2003)Google Scholar
  18. 18.
    Llibre, J., Simó, C.: Oscillatory solutions in the planar restricted three-body problem. Math. Ann. 248(2), 153–184 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Llibre, J., Simó, C.: Some homoclinic phenomena in the three-body problem. J. Differ. Equ. 37(3), 444–465 (1980)CrossRefzbMATHGoogle Scholar
  20. 20.
    McGehee, R.: A stable manifold theorem for degenerate fixed points with applications to celestial mechanics. J. Differ. Equ. 14, 70–88 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Melnikov, V.K.: On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc. 12, 1–57 (1963)Google Scholar
  22. 22.
    Moser, J.: Stable and Random Motions in Dynamical Systems. Princeton University Press, Princeton (1973). With special emphasis on celestial mechanics, Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, Annals of Mathematics Studies, No. 77Google Scholar
  23. 23.
    Martínez, R., Pinyol, C.: Parabolic orbits in the elliptic restricted three body problem. J. Differ. Equ. 111(2), 299–339 (1994)CrossRefzbMATHGoogle Scholar
  24. 24.
    Neĭshtadt, A.I.: The separation of motions in systems with rapidly rotating phase. Prikl. Mat. Mekh. 48(2), 197–204 (1984)MathSciNetGoogle Scholar
  25. 25.
    Poincaré, H.: Sur le problème des trois corps et les équations de la dynamique. Acta Math. 13, 1–270 (1890)zbMATHGoogle Scholar
  26. 26.
    Sauzin, D.: A new method for measuring the splitting of invariant manifolds. Ann. Sci. École Norm. Sup. 34(4), 159–221 (2001)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Sitnikov, K.: The existence of oscillatory motions in the three-body problems. Sov. Phys. Dokl. 5, 647–650 (1960)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Treschev, D.: Separatrix splitting for a pendulum with rapidly oscillating suspension point. Russ. J. Math. Phys. 5(1), 63–98 (1997)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Xia, Z.: Mel’ nikov method and transversal homoclinic points in the restricted three-body problem. J. Differ. Equ. 96(1), 170–184 (1992)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Mathematics BuildingUniversity of MarylandCollege ParkUSA
  2. 2.Departament de Matemàtica Aplicada IVUniversitat Politècnica de CatalunyaBarcelonaSpain
  3. 3.Departament de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations