A simple existence proof of Schubart periodic orbit with arbitrary masses

Abstract

This paper gives an analytic existence proof of the Schubart periodic orbit with arbitrary masses, a periodic orbit with singularities in the collinear three-body problem. A “turning point” technique is introduced to exclude the possibility of extra collisions and the existence of this orbit follows by a continuity argument on differential equations generated by the regularized Hamiltonian.

Keywords

Celestial mechanics / Schubart periodic orbit / three-body problem / binary collision / periodic solution with singularity / regularization

References

[1] Aarseth S. J., Zare K. A regularization of the three-body problem. Celest Mech, 1974, 10: 185-205

[2] Bakker L., Ouyang T., Roberts G., Yan D., Simmons S. Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem. Celestial Mech Dynam Astronom, 2010, 108: 147-164

[3] Bakker L., Ouyang T., Yan D., Simmons S. Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric fourbody problem. Celestial Mech Dynam Astronom, 2011, 110: 271-290

[4] Chenciner A., Montgomery R. A remarkable periodic solution of the three-body problem in the case of equal masses. Ann of Math, 2000, 152: 881-901

[5] Conley C. The retrograde circular solutions of the restricted three-body problem via a submanifold convex to the flow. SIAM J Appl Math, 1968, 16: 620-625

[6] Hénon M. Stability and interplay motions. Celestial Mech Dynam Astronom, 1997, 15: 243-261

[7] Hietarinta J., Mikkola S. Chaos in the one-dimensional gravitational three-body problem. Chaos, 1993, 3: 183-203

[8] Hu X., Sun S. Index and stability of symmetric periodic orbits in Hamiltonian system with application to figure-eight orbit. Commun Math Phys, 2009, 290: 737-777

[9] Hu X., Sun S. Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem. Adv Math, 2010, 223: 98-119

[10] Long Y. Index Theory for Symplectic Paths with Applications, 2002, Basel-Boston-Berlin: Birkhäuser Verlag

[11] Moeckel R. A topological existence proof for the Schubart orbits in the collinear threebody problem. Discrete Contin Dyn Syst Ser B, 2008, 10: 609-620

[12] Moore C. Braids in classical dynamics. Phys Rev Lett, 1993, 70: 3675-3679

[13] Ouyang T, Simmons S, Yan D. Periodic solutions with singularities in two dimensions in the n-body problem. Rocky Mountain J of Math (to appear)

[14] Ouyang T., Yan D. Periodic solutions with alternating singularities in the collinear fourbody problem. Celestial Mech Dynam Astronom, 2011, 109: 229-239

[15] Roberts G. Linear stability of the elliptic Lagrangian triangle solutions in the threebody problem. J Differential Equations, 2002, 182: 191-218

[16] Roberts G. Linear stability analysis of the figure-eight orbit in the three-body problem. Ergodic Theory Dynam Systems, 2007, 27: 1947-1963

[17] Schubart J. Numerische aufsuchung periodischer lösungen im dreikörperproblem. Astron Narchr, 1956, 283: 17-22

[18] Shibayama M. Minimizing periodic orbits with regularizable collisions in the n-body problem. Arch Ration Mech Anal, 2011, 199: 821-841

[19] Venturelli A. A variational proof for the existence of Von Schubart’s orbit. Discrete Contin Dyn Syst Ser B, 10: 699–717