Multiple geometrically distinct closed noncollision orbits of fixed energy for N-body type problems with strong force potentials

Shiqing, Zhang

doi:10.1090/S0002-9939-96-03751-3

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ISSN 1088-6826(online) ISSN 0002-9939(print)

Multiple geometrically distinct closed
noncollision orbits of fixed energy
for N-body type problems
with strong force potentials

Author: Zhang Shiqing
Journal: Proc. Amer. Math. Soc. 124 (1996), 3039-3046
MSC (1991): Primary 34C25, 34C15, 58F05
MathSciNet review: 1371142
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Abstract: Using the equivariant Ljusternik-Schnirelmann theory, we prove that there are at least $2(N-1)2^{N-2}$ geometrically distinct noncollision orbits with prescribed energy for a class of planar N-body type problems with strong force potentials.

[1] A. Ambrosetti and V. Coti Zelati, Closed orbits of fixed energy for a class of 𝑁-body problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), no. 2, 187–200 (English, with French summary). MR 1160848 (93e:58156a)
[2] A. Bahri and P. H. Rabinowitz, Periodic solutions of Hamiltonian systems of 3-body type, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 6, 561–649. MR 1145561 (92k:58223)
[3] Ugo Bessi and Vittorio Coti Zelati, Symmetries and noncollision closed orbits for planar 𝑁-body-type problems, Nonlinear Anal. 16 (1991), no. 6, 587–598. MR 1094320 (92a:70006), http://dx.doi.org/10.1016/0362-546X(91)90030-5
[4] Vittorio Coti Zelati, A class of periodic solutions of the 𝑁-body problem, Celestial Mech. Dynam. Astronom. 46 (1989), no. 2, 177–186. MR 1044425 (91c:58116), http://dx.doi.org/10.1007/BF00053047
[5] Vittorio Coti Zelati, Periodic solutions for 𝑁-body type problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 5, 477–492 (English, with French summary). MR 1138534 (93a:70009)
[6] E. Fadell and S. Husseini, Infinite cup length in free loop spaces with an application to a problem of the 𝑁-body type, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), no. 3, 305–319 (English, with French summary). MR 1168305 (94a:58034)
[7] William B. Gordon, Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc. 204 (1975), 113–135. MR 0377983 (51 #14152), http://dx.doi.org/10.1090/S0002-9947-1975-0377983-1
[8] E. W. C. van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy, J. Math. Anal. Appl. 132 (1988), no. 1, 1–12. MR 942350 (89m:58062), http://dx.doi.org/10.1016/0022-247X(88)90039-X
[9] P. Majer and S. Terracini, Periodic solutions to some N-body type problems, preprint, 1992.
[10] Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1986. MR 845785 (87j:58024)
[11] Enrico Serra and Susanna Terracini, Collisionless periodic solutions to some three-body problems, Arch. Rational Mech. Anal. 120 (1992), no. 4, 305–325. MR 1185563 (93i:70013), http://dx.doi.org/10.1007/BF00380317
[12] Enrico Serra and Susanna Terracini, Noncollision solutions to some singular minimization problems with Keplerian-like potentials, Nonlinear Anal. 22 (1994), no. 1, 45–62. MR 1256169 (94m:58091), http://dx.doi.org/10.1016/0362-546X(94)90004-3
[13] Carl Ludwig Siegel and Jürgen K. Moser, Lectures on celestial mechanics, Springer-Verlag, New York, 1971. Translation by Charles I. Kalme; Die Grundlehren der mathematischen Wissenschaften, Band 187. MR 0502448 (58 #19464)
[14] S. Q. Zhang, Multiple closed orbits of fixed energy for N-body-type problems with gravitational potentials, preprint.

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