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Structure of the centre manifold of the \(L_1,L_2\) collinear libration points in the restricted three-body problem

  • Giuseppe PucaccoEmail author
Original Article
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Part of the following topical collections:
  1. 50 years of Celestial Mechanics and Dynamical Astronomy

Abstract

We present a global analysis of the centre manifold of the collinear points in the circular restricted three-body problem. The phase-space structure is provided by a family of resonant 2-DOF Hamiltonian normal forms. The near 1:1 commensurability leads to the construction of a detuned Birkhoff–Gustavson normal form. The bifurcation sequences of the main orbit families are investigated by a geometric theory based on the reduction of the symmetries of the normal form, invariant under spatial mirror symmetries and time reversion. This global picture applies to any values of the mass parameter.

Keywords

Collinear points Lyapunov and halo orbits Normal forms 

Notes

Acknowledgements

We acknowledge useful discussions with A. Celletti, C. Efthymiopoulos, H. Hanßmann, A. Giorgilli, M. Guzzo, A. Marchesiello and D. Wilkzak. The work is partially supported by INFN, Sezione di Roma Tor Vergata and GNFM-INdAM. The author acknowledges the grant Stardust-R Marie Curie Initial Training Network.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.

References

  1. Arnol’d, V.I.: Mathematical Methods of Classical Mechanics. Springer, Berlin (1989)zbMATHGoogle Scholar
  2. Baoyin, H., McInnes, C.R.: Solar Sail halo orbits at the Sun–Earth artificial \(L_1\) point. Celest. Mech. Dyn. Astron. 94, 155–171 (2006)ADSzbMATHGoogle Scholar
  3. Bucciarelli, S., Ceccaroni, M., Celletti, A., Pucacco, G.: Qualitative and analytical results of the bifurcation thresholds to halo orbits. Ann. Mat. Pura Appl. 195, 489–512 (2016)MathSciNetzbMATHGoogle Scholar
  4. Ceccaroni, M., Celletti, A., Pucacco, G.: Halo orbits around the collinear points of the restricted three-body problem. Phys. D 317, 28–42 (2016)MathSciNetzbMATHGoogle Scholar
  5. Celletti, A., Pucacco, G., Stella, D.: Lissajous and Halo orbits in the restricted three-body problem. J. Nonlinear Sci. 25(2), 343–370 (2015)ADSMathSciNetzbMATHGoogle Scholar
  6. Cicogna, G., Gaeta, G.: Symmetry and Perturbation Theory in Nonlinear Dynamics. Springer, Berlin (1999)zbMATHGoogle Scholar
  7. Cushman, R.H., Bates, L.M.: Global Aspects of Classical Integrable Systems. Birkhauser, Basel (1997)zbMATHGoogle Scholar
  8. Cushman, R.H., Rod, D.L.: Reduction of the semi-simple 1:1 resonance. Phys. D 6, 105–112 (1982)MathSciNetzbMATHGoogle Scholar
  9. Cushman, R.H., Dullin, H.R., Hanßmann, H., Schmidt, S.: The 1:\(\pm \)2 resonance. Regul. Chaotic Dyn. 12, 642–663 (2007)ADSMathSciNetzbMATHGoogle Scholar
  10. Delshams, A., Gidea, M., Roldan, P.: Arnol’d mechanism of diffusion in the spatial circular restricted three-body problem: a semi-analytical argument. Phys. D 334, 29–48 (2016)MathSciNetzbMATHGoogle Scholar
  11. Deprit, A.: The Lissajous transformation I: basics. Celest. Mech. Dyn. Astron. 51, 201–225 (1991)ADSMathSciNetzbMATHGoogle Scholar
  12. Deprit, A., Elipe, A.: The Lissajous transformation II: normalization. Celest. Mech. Dyn. Astron. 51, 227–250 (1991)ADSMathSciNetzbMATHGoogle Scholar
  13. Efstathiou, K.: Metamorphoses of Hamiltonian systems with symmetries. In: Lecture Notes in Mathematics, vol. 1864. Springer, Berlin (2005)zbMATHGoogle Scholar
  14. Farquhar, R.W., Kamel, A.A.: Three-dimensional, periodic, ‘halo’ orbits. Celest. Mech. 7, 458–473 (1973)ADSzbMATHGoogle Scholar
  15. Farrés, A., Jorba, À., Mondelo, J.M.: Numerical study of the geometry of the phase space of the Augmented Hill Three-Body problem. Celest. Mech. Dyn. Astron. 129, 25–55 (2017)ADSMathSciNetzbMATHGoogle Scholar
  16. Gelfreich, V., Simó, C.: High-precision computations of divergent asymptotic series and homoclinic phenomena. Discrete Contin. Dyn. Syst. Ser. B 10, 511–536 (2008)MathSciNetzbMATHGoogle Scholar
  17. Giorgilli, A., Galgani, L.: Formal integrals for an autonomous Hamiltonian system near an equilibrium point. Celest. Mech. 17, 267–280 (1978)ADSMathSciNetzbMATHGoogle Scholar
  18. Gómez, G., Mondelo, J.M.: The dynamics around the collinear equilibrium points of the RTBP. Phys. D 157, 283–321 (2001)MathSciNetzbMATHGoogle Scholar
  19. Gómez, G. Jorba, À. Masdemont, J., Simó, C.: Dynamics and mission design near libration points. In: Advanced Methods for Collinear Points, vol. III. World Scientific, Singapore, ISBN: 981-02-4211-5 (2001)Google Scholar
  20. Guzzo, M.: Personal communication (2018)Google Scholar
  21. Guzzo, M., Lega, E.: Geometric chaos indicators and computations of the spherical hypertube manifolds of the spatial circular restricted three-body problem. Phys. D 373, 38–58 (2018)MathSciNetzbMATHGoogle Scholar
  22. Hanßmann, H.: Local and semi-local bifurcations in Hamiltonian dynamical systems—results and examples. In: Lecture Notes in Mathematics, vol. 1893. Springer, Berlin (2007)Google Scholar
  23. Hanßmann, H., Hoveijn, I.: The 1:1 resonance in Hamiltonian systems. J. Differ. Equ. 266(11), 6963–6984 (2018)ADSMathSciNetzbMATHGoogle Scholar
  24. Hanßmann, H., Sommer, B.: A degenerate bifurcation in the Hénon–Heiles family. Celest. Mech. Dyn. Astron. 81, 249–261 (2001)ADSzbMATHGoogle Scholar
  25. Hénon, M.: Vertical stability of periodic orbits in the restricted problem. I. Equal masses. Astron. Astrophys. 28, 415–426 (1973)ADSzbMATHGoogle Scholar
  26. Henrard, J.: Periodic orbits emanating from a resonant equilibrium. Celest. Mech. 1, 437–466 (1970)ADSMathSciNetzbMATHGoogle Scholar
  27. Hou, X.Y., Liu, L.: On motions around the collinear libration points in the elliptic restricted three-body problem. Mon. Not. R. Astron. Soc. 415, 3552–3560 (2011)ADSGoogle Scholar
  28. Howell, K.C.: Three-dimensional, periodic, ‘halo’ orbits. Celest. Mech. 32, 53–71 (1984)ADSMathSciNetzbMATHGoogle Scholar
  29. Jorba, À., Masdemont, J.: Dynamics in the center manifold of the collinear points of the restricted three body problem. Phys. D 132, 189–213 (1999)MathSciNetzbMATHGoogle Scholar
  30. Lara, M.: A Hopf variables view of the libration points dynamics. Celest. Mech. Dyn. Astron. 129, 285–306 (2017)ADSMathSciNetzbMATHGoogle Scholar
  31. Lei, H., Xu, B., Circi, C.: Polynomial expansions of single-mode motions around equilibrium points in the circular restricted three-body problem. Celest. Mech. Dyn. Astron. 130, 38 (2018)ADSMathSciNetzbMATHGoogle Scholar
  32. Marchesiello, A., Pucacco, G.: Universal unfolding of symmetric resonances. Celest. Mech. Dyn. Astron. 119, 357–368 (2014)ADSzbMATHGoogle Scholar
  33. Marchesiello, A., Pucacco, G.: Bifurcation sequences in the 1:1 Hamiltonian resonance. Int. J. Bifur. Chaos 26, 1630011 (2016)MathSciNetzbMATHGoogle Scholar
  34. McInnes, C.R.: Solar Sailing: Technology, Dynamics and Mission Applications. Springer Praxis Books/Astronomy and Planetary Sciences, Chichester (2004)Google Scholar
  35. Pucacco, G., Marchesiello, A.: An energy-momentum map for the time-reversal symmetric 1:1 resonance with \({\mathbb{Z}}_2\times {\mathbb{Z}}_2\) symmetry. Phys. D 271, 10–18 (2014)MathSciNetzbMATHGoogle Scholar
  36. Pucacco, G., Boccaletti, D., Belmonte, C.: Quantitative predictions with detuned normal forms. Celest. Mech. Dyn. Astron. 102, 163–176 (2008)ADSMathSciNetzbMATHGoogle Scholar
  37. Richardson, D.L.: Analytic construction of periodic orbits about the collinear points. Celest. Mech. 22, 241–253 (1980)ADSMathSciNetzbMATHGoogle Scholar
  38. Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems. Springer, Berlin (2007)zbMATHGoogle Scholar
  39. Scuflaire, R.: Stability of axial orbits in analytical galactic potentials. Celest. Mech. Dyn. Astron. 61, 261–285 (1995)ADSMathSciNetzbMATHGoogle Scholar
  40. Scuflaire, R.: Periodic orbits in analytical planar galactic potentials. Celest. Mech. Dyn. Astron. 71, 203–228 (1998)ADSMathSciNetzbMATHGoogle Scholar
  41. Simó, C.: Effective computations in celestial mechanics and astrodynamics. In: Rumyantsev, V.V., Karapetyan, A.V. (eds.) Modern Methods of Analytical Mechanics and their Applications. CISM Courses and Lectures, vol. 387, pp. 55–102. Springer, Vienna (1998)Google Scholar
  42. Tuwankotta, J.M., Verhulst, F.: Symmetry and resonance in Hamiltonian systems. SIAM J. Appl. Math. 61, 1369–1385 (2000)MathSciNetzbMATHGoogle Scholar
  43. Verhulst, F.: Discrete symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies. Philos. Trans. R. (Lond.) Soc. Ser. A 290, 435–465 (1979)ADSzbMATHGoogle Scholar
  44. Walawska, I., Wilczak, D.: Validated numerics for period-tupling and touch-and-go bifurcations of symmetric periodic orbits in reversible systems. Commun. Nonlinear Sci. Numer. Simul. 74, 30–54 (2019)ADSMathSciNetGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Dipartimento di Fisica and INFN – Sezione di Roma IIUniversità di Roma “Tor Vergata”RomeItaly

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