Generalized periodic orbits in some restricted three-body problems


We treat the circular and elliptic restricted three-body problems in inertial frames as periodically forced Kepler problems with additional singularities and explain that in this setting the main result of Boscaggin et al. (Trans Am Math Soc 372: 677–703, 2019) is applicable. This guarantees the existence of an arbitrary large number of generalized periodic orbits (periodic orbits with possible double collisions regularized) provided the mass ratio of the primaries is small enough.

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  1. 1.

    Antoniadou, K.I., Libert, A.-S.: Origin and continuation of 3/2, 5/2, 3/1, 4/1 and 5/1 resonant periodic orbits in the circular and elliptic restricted three-body problem. Celest. Mech. Dyn. Astron. 130, 41 (2018)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Barutello, V., Ortega, R., Verzini, G.: Regularized variational principles for the perturbed Kepler problem. arXiv:2003.09383

  3. 3.

    Boscaggin, A., Dambrosio, W., Papini, D.: Periodic solutions to a forced Kepler problem in the plane. Proc. Am. Math. Soc. 148, 301–314 (2020)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Boscaggin, A., Ortega, R., Zhao, L.: Periodic solutions and regularization of a Kepler problem with time-dependent perturbation. Trans. Am. Math. Soc. 372, 677–703 (2019)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)

    MATH  Google Scholar 

  6. 6.

    Cors, J.M., Pinyol, C., Soler, J.: Analytic continuation in the case of non-regular dependency on a small parameter with an application to celestial mechanics. J. Differ. Eq. 219, 1–19 (2005)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cushman, R.H., Bates, L.M.: Global Aspects of Classical Integrable Systems, 2nd edn. Birkhäuser/Springer, Basel (2015)

    Book  Google Scholar 

  8. 8.

    Moser, J., Zehnder, E.: Notes on Dynamical Systems. American Mathemathcal Society, Philadelphia (2005)

    Book  Google Scholar 

  9. 9.

    Ortega, R.: Linear motions in a periodically forced Kepler problem. Port. Math. 68, 149–176 (2011)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Poincaré, H.: Les méthodes nouvelles de la mécanique céleste, T. 1. Gauthier-Villars, Paris (1892)

    Google Scholar 

  11. 11.

    Palacián, J.F., Yanguas, P., Fernández, S., Nicotra, M.A.: Searching for periodic orbits of the spatial elliptic restricted three-body problem by double averaging. Physica D 213, 15–24 (2006)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Sperling, H.J.: The collision singularity in a perturbed two-body problem. Celest. Mech. 1, 213–221 (1969/1970)

  13. 13.

    Weinstein, A.: Normal modes for nonlinear Hamiltonian systems. Invent. Math. 20, 47–57 (1973)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Zhao, L.: Some collision solutions of the rectilinear periodically forced Kepler problem. Adv. Nonlinear Stud. 16, 45–49 (2016)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Zhao, L.: Kustaanheimo–Stiefel regularization and the quadrupolar conjugacy. Regul. Chaotic Dyn. 20, 19–36 (2015)

    MathSciNet  Article  Google Scholar 

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R. O. is supported by MTM2017-82348-C2-1-P. L. Z. is supported by DFG ZH 605/1-1.

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Correspondence to Lei Zhao.

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Ortega, R., Zhao, L. Generalized periodic orbits in some restricted three-body problems. Z. Angew. Math. Phys. 72, 40 (2021).

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  • Forced Kepler problem
  • Restricted three-body problem
  • Generalized periodic orbits
  • Kustaanheimo–Stiefel Regularization

Mathematics Subject Classification

  • 70F15
  • 70F16