×

A simple existence proof of Schubart periodic orbit with arbitrary masses. (English) Zbl 1252.70035

Summary: 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.

MSC:

70F15 Celestial mechanics
70F10 \(n\)-body problems
70H12 Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics
70H14 Stability problems for problems in Hamiltonian and Lagrangian mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aarseth S J, Zare K. A regularization of the three-body problem. Celest Mech, 1974, 10: 185–205 · Zbl 0303.70014 · doi:10.1007/BF01227619
[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 · Zbl 1223.70029 · doi:10.1007/s10569-010-9298-y
[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 · Zbl 1270.70034 · doi:10.1007/s10569-011-9358-y
[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 · Zbl 0987.70009 · doi:10.2307/2661357
[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 · Zbl 0159.55104 · doi:10.1137/0116050
[6] Hénon M. Stability and interplay motions. Celestial Mech Dynam Astronom, 1997, 15: 243–261 · Zbl 0359.70018 · doi:10.1007/BF01228465
[7] Hietarinta J, Mikkola S. Chaos in the one-dimensional gravitational three-body problem. Chaos, 1993, 3: 183–203 · Zbl 1055.70504 · doi:10.1063/1.165984
[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 · Zbl 1231.37031 · doi:10.1007/s00220-009-0860-y
[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 · Zbl 1354.70026 · doi:10.1016/j.aim.2009.07.017
[10] Long Y. Index Theory for Symplectic Paths with Applications. Basel-Boston-Berlin: Birkhäuser Verlag, 2002 · Zbl 1012.37012
[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 · Zbl 1147.70007 · doi:10.3934/dcdsb.2008.10.609
[12] Moore C. Braids in classical dynamics. Phys Rev Lett, 1993, 70: 3675–3679 · Zbl 1050.37522 · doi:10.1103/PhysRevLett.70.3675
[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) · Zbl 1263.70015
[14] Ouyang T, Yan D. Periodic solutions with alternating singularities in the collinear fourbody problem. Celestial Mech Dynam Astronom, 2011, 109: 229–239 · Zbl 1270.70040 · doi:10.1007/s10569-010-9325-z
[15] Roberts G. Linear stability of the elliptic Lagrangian triangle solutions in the threebody problem. J Differential Equations, 2002, 182: 191–218 · Zbl 1181.70015 · doi:10.1006/jdeq.2001.4089
[16] Roberts G. Linear stability analysis of the figure-eight orbit in the three-body problem. Ergodic Theory Dynam Systems, 2007, 27: 1947–1963 · Zbl 1128.70006
[17] Schubart J. Numerische aufsuchung periodischer lösungen im dreikörperproblem. Astron Narchr, 1956, 283: 17–22 · Zbl 0070.40801 · doi:10.1002/asna.19562830105
[18] Shibayama M. Minimizing periodic orbits with regularizable collisions in the n-body problem. Arch Ration Mech Anal, 2011, 199: 821–841 · Zbl 1291.70049 · doi:10.1007/s00205-010-0334-6
[19] Venturelli A. A variational proof for the existence of Von Schubart’s orbit. Discrete Contin Dyn Syst Ser B, 10: 699–717 · Zbl 1166.70009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.