Inventiones mathematicae

, Volume 203, Issue 2, pp 417–492 | Cite as

Oscillatory motions for the restricted planar circular three body problem

  • Marcel Guardia
  • Pau Martín
  • Tere M. SearaEmail author


The restricted three body problem models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies called the primaries. When they move along circular Keplerian orbits and the third body moves in the same plane, one has the restricted planar circular three body problem (RPC3BP). In suitable coordinates, it is a Hamiltonian system of two degrees of freedom. The conserved energy is usually called the Jacobi constant. Llibre and Simó [Math Ann 248(2):153–184, 1980] proved the existence of oscillatory motions for this system. That is, orbits which leave every bounded region but which return infinitely often to some fixed bounded region. To prove their existence they had to assume the ratio between the masses of the primaries to be small enough. In this paper we prove the existence of such motions for any value of the mass ratio \(\mu \) closing the problem of existence of oscillatory motions in the RPC3BP. To obtain such motions, we restrict ourselves to the level sets of the Jacobi constant. We show that, for any value of the mass ratio and for large values of the Jacobi constant, there exist transversal intersections between the stable and unstable manifolds of infinity in these level sets. These transversal intersections guarantee the existence of a symbolic dynamics that creates the oscillatory orbits. The main achievement is to prove the existence of these orbits without assuming the mass ratio \(\mu \) small. When \(\mu \) is not small, this transversality can not be checked by means of classical perturbation theory. Since our method is valid for all values of \(\mu \), we are able to detect a curve in the parameter space, formed by \(\mu \) and the Jacobi constant, where cubic homoclinic tangencies between the invariant manifolds of infinity appear.



The authors acknowledge useful discussions with V. Kaloshin and A. Gorodetski. They want to thank the referees for their help to improve the final version of the manuscript and Mike Jeffrey for his careful reading of the paper and helpful comments. They have been partially supported by the Spanish MINECO-FEDER Grant MTM2012-31714 and the Catalan Grant 2014SGR504. 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. T. S. has been partially supported by the Russian Scientific Foundation grant 14-41-00044 and Marie Curie Action FP7-PEOPLE-2012-IRSES, BREUDS.


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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Departament de Matemàtica Aplicada IVUniversitat Politècnica de CatalunyaBarcelonaSpain

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