Abstract
A retrograde orbit of the planar three-body problem is a relative periodic solution with two adjacent masses revolving around each other in one direction while their mass center revolves around the third mass in the other direction. The orbit is said to be prograde or direct if both revolutions follow the same direction. Let T > 0 and \({\phi\in[0,2\pi)}\) be fixed, and consider the rotating frame which rotates the inertia frame about the origin with angular velocity \({\frac{\phi}{T}}\) . In a recent work of K.-C.Chen [5], the existence of action-minimizing retrograde orbits which are T-periodic on this rotation frame were proved to exist for a large class of masses and a continuum of \({\phi}\) . In this paper we generalize the main result in [5], provide some quantitative estimates for admissible masses and mutual distances, and show miscellaneous examples of action-minimizing retrograde orbits. We also show the existence of some prograde and retrograde solutions with additional symmetries.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barutello V., Ferrario D., Terracini S.: Symmetry groups of the planar three-body problem and action-minimizing trajectories. Arch. Ration. Mech. Anal. 190, 189–226 (2008)
Broucke R.: On relative periodic solutions of the planar general three-body problem. Celes. Mech. 12, 439–462 (1975)
Chen K.-C.: Binary decompositions for the planar N-body problem and symmetric periodic solutions. Arch. Ration. Mech. Anal. 170, 247–276 (2003)
Chen K.-C.: Removing collision singularities from action minimizers for the N-body problem with free boundaries. Arch. Ration. Mech. Anal. 181, 311–331 (2006)
Chen K.-C.: Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses. Annals of Math. 167, 325–348 (2008)
Chen, K.-C.: Variational constructions for some satellite orbits in periodic gravitational force fields. Amer. J. Math. (2009, to appear)
Chenciner, A.: Symmetries and “Simple” Solutions of the Classical N-body Problem. Proceeding of ICMP, Lisbone, 2003, River Edge, NJ, World Scientific, 2006
Chenciner A., Féjoz J., Montgomery R.: Rotating eights I. Nonlinearity 18, 1407–1424 (2005)
Chenciner A., Montgomery R.: A remarkable periodic solution of the three-body problem in the case of equal masses. Ann. Math. 152, 881–901 (2000)
Ferrario D., Terracini S.: On the existence of collisionless equivariant minimizers for the classical n-body problem. Invent. Math. 155, 305–362 (2004)
Gordon W.: A minimizing property of Keplerian orbits. Amer. J. Math. 99, 961–971 (1977)
Hénon M.: A family of periodic solutions of the planar three-body problem, and their stability. Celes. Mech. 13, 267–285 (1976)
Meyer, K.R.: Periodic Solutions of the N-body Problem. Lecture Notes in Mathematics, Vol. 1719, Berlin: Springer-Verlag, 1999
Marchal, C.: The Three-Body Problem. Studies in Astronautics, 4. Amsterdam: Elsevier Science Publishers, B.V., 1990
Moeckel R.: A topological existence proof for the Schubart orbits in the collinear three-body problem. Discrete Contin. Dyn. Syst. Ser. B 10, 609–620 (2008)
Montgomery R.: The N-body problem, the braid group, and action-minimizing periodic solutions. Nonlinearity 11, 363–376 (1998)
Moore C.: Braids in classical dynamics. Phy. Rev. Lett. 70, 3675–3679 (1993)
Palais R.: The principle of symmetric criticality. Commun. Math. Phys. 69, 19–30 (1979)
Schubart J.: Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem. Astr. Nachr. 283, 17–22 (1956)
Simó, C.: Choreographic solutions of the planar three body problem. http://www.maia.ub.es/dsg/3body.html
Szebehely V.: Theory of Orbits – The Restricted Problem of Three Bodies. Academic Press, New York (1967)
Venturelli A.: A variational proof of the existence of von Schubart’s orbit. Discrete Contin. Dyn. Syst. Ser. B 10, 699–717 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Gallavotti
Rights and permissions
About this article
Cite this article
Chen, KC., Lin, YC. On Action-Minimizing Retrograde and Prograde Orbits of the Three-Body Problem. Commun. Math. Phys. 291, 403–441 (2009). https://doi.org/10.1007/s00220-009-0769-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-009-0769-5