Some counterexamples to a generalized Saari’s conjecture

Roberts, Gareth

doi:10.1090/S0002-9947-05-03697-4

Some counterexamples to a generalized Saari’s conjecture
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by Gareth E. Roberts PDF

Trans. Amer. Math. Soc. 358 (2006), 251-265 Request permission

Abstract:

For the Newtonian $n$-body problem, Saari’s conjecture states that the only solutions with a constant moment of inertia are relative equilibria, solutions rigidly rotating about their center of mass. We consider the same conjecture applied to Hamiltonian systems with power-law potential functions. A family of counterexamples is given in the five-body problem (including the Newtonian case) where one of the masses is taken to be negative. The conjecture is also shown to be false in the case of the inverse square potential and two kinds of counterexamples are presented. One type includes solutions with collisions, derived analytically, while the other consists of periodic solutions shown to exist using standard variational methods.

References

Alain Albouy, The symmetric central configurations of four equal masses, Hamiltonian dynamics and celestial mechanics (Seattle, WA, 1995) Contemp. Math., vol. 198, Amer. Math. Soc., Providence, RI, 1996, pp. 131–135. MR 1409157, DOI 10.1090/conm/198/02494
Alain Chenciner, Action minimizing periodic orbits in the Newtonian $n$-body problem, Celestial mechanics (Evanston, IL, 1999) Contemp. Math., vol. 292, Amer. Math. Soc., Providence, RI, 2002, pp. 71–90. MR 1884893, DOI 10.1090/conm/292/04917
Alain Chenciner, Some facts and more questions about the Eight, Topological methods, variational methods and their applications (Taiyuan, 2002) World Sci. Publ., River Edge, NJ, 2003, pp. 77–88. MR 2010643
William B. Gordon, Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc. 204 (1975), 113–135. MR 377983, DOI 10.1090/S0002-9947-1975-0377983-1
C. Jacobi, Vorlesungen über dynamik, Reimer Publisher, Berlin, 1866.
C. McCord, Saari’s conjecture for the planar three-body problem with equal masses, preprint, October 2002.
R. Moeckel, A computer-assisted proof of Saari’s conjecture for the planar three-body problem, preprint, July 2003.
Richard Moeckel, On central configurations, Math. Z. 205 (1990), no. 4, 499–517. MR 1082871, DOI 10.1007/BF02571259
Richard Moeckel, Relative equilibria of the four-body problem, Ergodic Theory Dynam. Systems 5 (1985), no. 3, 417–435. MR 805839, DOI 10.1017/S0143385700003047
Richard Montgomery, Action spectrum and collisions in the planar three-body problem, Celestial mechanics (Evanston, IL, 1999) Contemp. Math., vol. 292, Amer. Math. Soc., Providence, RI, 2002, pp. 173–184. MR 1884899, DOI 10.1090/conm/292/04923
Richard Montgomery, The $N$-body problem, the braid group, and action-minimizing periodic solutions, Nonlinearity 11 (1998), no. 2, 363–376. MR 1610784, DOI 10.1088/0951-7715/11/2/011
H. Poincaré, Sur les solutions périodiques et le principe de moindre action, Comptes Rendus 123 (1896), 915-918.
Gareth E. Roberts, A continuum of relative equilibria in the five-body problem, Phys. D 127 (1999), no. 3-4, 141–145. MR 1669486, DOI 10.1016/S0167-2789(98)00315-7
D. Saari, On bounded solutions of the $n$-body problem, Periodic Orbits, Stability and Resonances, G. Giacaglia ed., D. Riedel, Dordrecht (1970), 76-81.
Donald G. Saari, On the role and the properties of $n$-body central configurations, Celestial Mech. 21 (1980), no. 1, 9–20. MR 564603, DOI 10.1007/BF01230241
Steve Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998), no. 2, 7–15. MR 1631413, DOI 10.1007/BF03025291
Aurel Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Mathematical Series, vol. 5, Princeton University Press, Princeton, N. J., 1941. MR 0005824