On Families of Periodic Orbits in the Restricted Three-Body Problem

• Kim, Seongchan ^[1]
  1. [1] University of Augsburg

     University of Augsburg

     Kreisfreie Stadt Augsburg, Alemania

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 18, Nº 1, 2019, págs. 201-232
• Idioma: inglés
• DOI: 10.1007/s12346-018-0288-x
• Enlaces
  • Texto completo
• Resumen

  • Since Poincaré, periodic orbits have been one of the most important objects in dynamical systems. However, searching them is in general quite difficult. A common way to find them is to construct families of periodic orbits which start at obvious periodic orbits. On the other hand, given two periodic orbits one might ask if they are connected by an orbit cylinder, i.e., by a one-parameter family of periodic orbits. In this article we study this question for a certain class of periodic orbits in the planar circular restricted three-body problem. Our strategy is to compare the Cieliebak–Frauenfelder–van Koert invariants which are obstructions to the existence of an orbit cylinder.

• Referencias bibliográficas
  • 1. Albers, P., Fish, J., Frauenfelder, U., van Koert, O.: The Conley–Zehnder indices of the rotating Kepler problem. Math. Proc. Camb. Philos....
  • 2. Arenstorf, R.F.: Periodic solutions of the restricted three body problem representing analytic continuation of Keplerian elliptic motions....
  • 3. Arnold, V.I.: Topological Invariant of Plane Curves and Caustics. AMS University Lecture Series, vol. 5. American Mathematical Society,...
  • 4. Barrar, R.: Existence of periodic orbits of the second kind in the restricted problem of three bodies. Astron. J. 70, 3–4 (1965)
  • 5. Bruno, A.D.: The Restricted 3-Body Problem: Plane Periodic Orbits. De Gruyter Expositions in Mathematics, vol. 17. Walter de Gruyter &...
  • 6. Cieliebak, K., Frauenfelder, U., van Koert, O.: Periodic orbits in the restricted three-body problem and Arnold’s J+-invariant. Regul....
  • 7. Dullin, H.R., Montgomery, R.: Syzygies in the two center problem. Nonlinearity 29, 1212–1237 (2016)
  • 8. Kim, J., Kim, S.: J+-like invariants of periodic orbits of the second kind in the restricted three-body problem. J. Topol. Anal. (to...
  • 9. Kim, S.: Dynamical convexity of the Euler problem of two fixed centers. Math. Proc. Camb. Philos. Soc. 165(2), 359–384 (2018)
  • 10. Kim, S.: Homoclinic orbits in the Euler problem of two fixed centers. J. Geom. Phys. 132, 55–63 (2018)
  • 11. Kim, S.: J+-like invariants and periodic orbits in the restricted three-body problem. Ph.D. thesis, Universität Augsburg (2018)
  • 12. Pauli, W.: Über das Modell des Wasserstoffmolekülions. Ann. Phys. 68, 177–240 (1922)
  • 13. Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste III. Gauthiers-Villars, Paris (1899)
  • 14. Strand, M.P., Reinhardt, W.P.: Semiclassical quantization of the low lying electronic states of H+ 2 . J. Chem. Phys. 70, 3812–3827...
  • 15. Verhaar, E.: On the theory of collisional orbits in the two center problem, thesis (bachelor). The University of Groningen (2014)
  • 16. Waalkens, H., Dullin, H.R., Richter, P.H.: The problem of two fixed centers: bifurcations, actions, monodromy. Physica D 196(3–4), 265–310...
  • 17. Whitney, H.: On regular closed curves on the plane. Compos. Math. 4, 276–284 (1937)