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Regular and Singular Reductions in the Spatial Three-Body Problem

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Abstract

The spatial three-body problem is a Hamiltonian system of nine degrees of freedom. We apply successive reductions in order to get the simplest reduced Hamiltonian, the one where all the continuous symmetries have been reduced out. After writing the Hamiltonian in Jacobi coordinates we use Deprit’s variables in order to perform the Jacobi elimination of the nodes. Then, we normalise with respect to the mean anomalies of the inner and outer ellipses in a region outside the resonance regime, and truncate the higher-order terms. The resulting system is expressed in the corresponding invariants that define the reduced space, which is a regular manifold of dimension eight. After that we use that the modulus of the total angular momentum and its projection onto the vertical axis of the inertial frame are integrals of the system. We obtain the invariants related to the symmetries generated by the two integrals and the corresponding reduced phase space, expressing the Hamiltonian in terms of these invariants. The reduced space has dimension six and is a singular space for some values of the parameters. Next, as the normalised Hamiltonian is independent of the argument of the pericentre of the outer ellipse up to first order in the small parameter, we reduce the system with respect to the symmetry related with the modulus of the angular momentum of the outer ellipse. We get the three invariants that generate the reduced two-dimensional space that may have up to three singular points. In this space we study the reduced system written in terms of these invariants analysing the relative equilibria, their stabilities and bifurcations.

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References

  1. Abraham R., Marsden J.E.: Foundations of Mechanics. Addison-Wesley, Reading (1985)

    Google Scholar 

  2. Arms, J.M., Cushman, R.H., Gotay, M.J.: A universal reduction procedure for Hamiltonian group actions. In: Ratiu, T. (ed.) The Geometry of Hamiltonian Systems (Berkeley, 1989), pp. 33–51. Springer, Berlin (1991)

  3. Biasco, L., Chierchia, L., Valdinoci, E.: Elliptic two-dimensional invariant tori for the planetary three-body problem. Arch. Rational Mech. Anal. 170, 91–135 (2003). Corrigendum: Arch. Rational Mech. Anal. 180, 507–509 (2006)

    Google Scholar 

  4. Celletti A., Chierchia L.: KAM tori for N-body problems: a brief history. Celest. Mech. Dyn. Astron. 95, 117–139 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chierchia L., Pinzari G.: Deprit’s reduction of the nodes revisited. Celest. Mech. Dyn. Astron. 109, 285–301 (2011)

    Article  MathSciNet  Google Scholar 

  6. Chierchia L., Pinzari G.: Planetary Birkhoff normal forms. J. Mod. Dyn. 5, 623–664 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Chierchia L., Pinzari G.: The planetary N-body problem: symplectic foliation, reductions and invariant tori. Invent. Math. 186, 1–77 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cordani B.: Global study of the 2D secular 3-body problem. Regul. Chaotic Dyn. 9, 113–128 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cushman, R.H.: Reduction, Brouwer’s Hamiltonian, and the critical inclination. Celest. Mech. 31, 401–429 (1983). Errata: Celest. Mech. 33, 297 (1984)

    Google Scholar 

  10. Cushman, R.H.: A survey of normalization techniques applied to perturbed Keplerian systems. In: Jones, K. (ed.) Dynamics Reported, vol. 1, New Series, pp. 54–112. Springer, New York (1992)

  11. Cushman R.H., Bates L.M.: Global Aspects of Classical Integrable Systems. Birkhäuser, Basel (1997)

    Book  MATH  Google Scholar 

  12. Deprit A.: Canonical transformations depending on a small parameter. Celest. Mech. 1, 12–30 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  13. Deprit A.: Delaunay normalisations. Celest. Mech. 26, 9–21 (1982)

    Article  MathSciNet  Google Scholar 

  14. Deprit A.: Elimination of the nodes in problems of N bodies. Celest. Mech. 30, 181–195 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  15. Farago F., Laskar J.: High-inclination orbits in the secular quadrupolar three-body problem. Mon. Not. R. Astron. Soc. 401, 1189–1198 (2010)

    Article  Google Scholar 

  16. Féjoz J.: Global secular dynamics in the planar three-body problem. Celest. Mech. Dyn. Astron. 84, 159–195 (2002)

    Article  MATH  Google Scholar 

  17. Féjoz J.: Quasiperiodic motions in the planar three-body problem. J. Differ. Equ. 183, 303–341 (2002)

    Article  MATH  Google Scholar 

  18. Ferrer S., Hanßmann H., Palacián J.F., Yanguas P.: On perturbed oscillators in 1-1-1 resonance: the case of axially symmetric cubic potentials. J. Geom. Phys. 40, 320–369 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ferrer, S., Osácar, C.: Harrington’s Hamiltonian in the stellar problem of three bodies: reductions, relative equilibria and bifurcations. Celest. Mech. Dyn. Astron. 58, 245–275 (1994). Erratum: Celest. Mech. Dyn. Astron. 60, 187 (1994)

    Google Scholar 

  20. Hanßmann H.: The reversible umbilic bifurcation. Physica D 112, 81–94 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Harrington R.S.: The stellar three-body problem. Celest. Mech. 1, 200–209 (1969)

    Article  Google Scholar 

  22. Henrard J.: Virtual singularities in the artificial satellite theory. Celest. Mech. 10, 437–449 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  23. Iñarrea M., Lanchares V., Palacián J.F., Pascual A.I., Salas J.P., Yanguas P.: The Keplerian regime of charged particles in planetary magnetospheres. Physica D 197, 242–268 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Iñarrea M., Lanchares V., Palacián J.F., Pascual A.I., Salas J.P., Yanguas P.: Symplectic coordinates on S 2 × S 2 for perturbed Keplerian problems: application to the dynamics of a generalised Størmer problem. J. Differ. Equ. 250, 1386–1407 (2011)

    Article  MATH  Google Scholar 

  25. Jacobi M.: Sur l’Élimination des Noeuds dans le Problème des Trois Corps. Astron. Nachr. 20, 81–98 (1843)

    Article  Google Scholar 

  26. Jefferys W.H., Moser J.: Quasi-periodic solutions for the three-body problem. Astron. J. 71, 568–578 (1966)

    Article  MathSciNet  Google Scholar 

  27. Lidov M.L., Ziglin S.L.: Non-restricted double-averaged three body problem in Hill’s case. Celest. Mech. 13, 471–489 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lieberman B.B.: Existence of quasi-periodic solutions to the three-body problem. Celest. Mech. 3, 408–426 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  29. Marsden J.E., Weinstein A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5, 121–130 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  30. McCord C.K., Meyer K.R., Wang Q.: The integral manifolds of the three body problem. Mem. Am. Math. Soc. 132, 1–91 (1998)

    MathSciNet  Google Scholar 

  31. Meyer K.R.: Symmetries and integrals in mechanics. In: Peixoto, M.M. (eds) Dynamical Systems., pp. 259–272. Academic Press, New York (1973)

    Google Scholar 

  32. Meyer K.R., Hall G.R., Offin D.: Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 2nd edn. Springer, New York (2009)

    Google Scholar 

  33. Moser J.: Regularization of Kepler’s problem and the averaging method on a manifold. Commun. Pure Appl. Math. 23, 609–636 (1970)

    Article  MATH  Google Scholar 

  34. Palacián J.F.: Normal forms for perturbed Keplerian systems. J. Differ. Equ. 180, 471–519 (2002)

    Article  MATH  Google Scholar 

  35. Palacián, J.F., Sayas, F., Yanguas, P.: Invariant tori of the spatial three-body problem reconstructed from relative equilibria. In preparation

  36. Radau R.: Sur une Transformation des Équations Différentielles de la Dynamique. Ann. Sci. École Norm. Sup. 5, 311–375 (1868)

    MathSciNet  Google Scholar 

  37. Sturmfels B.: Algorithms in Invariant Theory. Springer, New York (1993)

    MATH  Google Scholar 

  38. Yanguas P., Palacián J.F., Meyer K.R., Dumas H.S.: Periodic solutions in Hamiltonian systems, averaging, and the lunar problem. SIAM J. Appl. Dyn. Syst. 7, 311–340 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática e Informática, Universidad Pública de Navarra, 31006, Pamplona, Spain

    Jesús F. Palacián, Flora Sayas & Patricia Yanguas

Authors
  1. Jesús F. Palacián
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  2. Flora Sayas
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  3. Patricia Yanguas
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Correspondence to Jesús F. Palacián.

Additional information

To Ken Meyer with great admiration. The authors have received partial support from Project MTM 2008-03818 of the Ministry of Science and Innovation of Spain.

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Palacián, J.F., Sayas, F. & Yanguas, P. Regular and Singular Reductions in the Spatial Three-Body Problem. Qual. Theory Dyn. Syst. 12, 143–182 (2013). https://doi.org/10.1007/s12346-012-0083-z

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  • DOI: https://doi.org/10.1007/s12346-012-0083-z

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