Some counterexamples to a generalized Saari's conjecture

Author:
Gareth E. Roberts

Journal:
Trans. Amer. Math. Soc. **358** (2006), 251-265

MSC (2000):
Primary 70F10, 70F15; Secondary 37J45

Published electronically:
January 21, 2005

MathSciNet review:
2171232

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For the Newtonian -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.

**1.**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**, 10.1090/conm/198/02494**2.**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**, 10.1090/conm/292/04917**3.**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****4.**William B. Gordon,*Conservative dynamical systems involving strong forces*, Trans. Amer. Math. Soc.**204**(1975), 113–135. MR**0377983**, 10.1090/S0002-9947-1975-0377983-1**5.**C. Jacobi,*Vorlesungen über dynamik*, Reimer Publisher, Berlin, 1866.**6.**C. McCord,*Saari's conjecture for the planar three-body problem with equal masses*, preprint, October 2002.**7.**R. Moeckel,*A computer-assisted proof of Saari's conjecture for the planar three-body problem*, preprint, July 2003.**8.**Richard Moeckel,*On central configurations*, Math. Z.**205**(1990), no. 4, 499–517. MR**1082871**, 10.1007/BF02571259**9.**Richard Moeckel,*Relative equilibria of the four-body problem*, Ergodic Theory Dynam. Systems**5**(1985), no. 3, 417–435. MR**805839**, 10.1017/S0143385700003047**10.**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**, 10.1090/conm/292/04923**11.**Richard Montgomery,*The 𝑁-body problem, the braid group, and action-minimizing periodic solutions*, Nonlinearity**11**(1998), no. 2, 363–376. MR**1610784**, 10.1088/0951-7715/11/2/011**12.**H. Poincaré,*Sur les solutions périodiques et le principe de moindre action*, Comptes Rendus**123**(1896), 915-918.**13.**Gareth E. Roberts,*A continuum of relative equilibria in the five-body problem*, Phys. D**127**(1999), no. 3-4, 141–145. MR**1669486**, 10.1016/S0167-2789(98)00315-7**14.**D. Saari,*On bounded solutions of the -body problem*, Periodic Orbits, Stability and Resonances, G. Giacaglia ed., D. Riedel, Dordrecht (1970), 76-81.**15.**Donald G. Saari,*On the role and the properties of 𝑛-body central configurations*, Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics (Math. Forschungsinst., Oberwolfach, 1978), Part I, 1980, pp. 9–20. MR**564603**, 10.1007/BF01230241**16.**Steve Smale,*Mathematical problems for the next century*, Math. Intelligencer**20**(1998), no. 2, 7–15. MR**1631413**, 10.1007/BF03025291**17.**Aurel Wintner,*The Analytical Foundations of Celestial Mechanics*, Princeton Mathematical Series, v. 5, Princeton University Press, Princeton, N. J., 1941. MR**0005824**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
70F10,
70F15,
37J45

Retrieve articles in all journals with MSC (2000): 70F10, 70F15, 37J45

Additional Information

**Gareth E. Roberts**

Affiliation:
Department of Mathematics and Computer Science, 1 College Street, College of the Holy Cross, Worcester, Massachusetts 01610

Email:
groberts@radius.holycross.edu

DOI:
https://doi.org/10.1090/S0002-9947-05-03697-4

Keywords:
Saari's conjecture,
$n$-body problems,
relative equilibria,
Hamiltonian systems

Received by editor(s):
September 12, 2003

Received by editor(s) in revised form:
February 9, 2004

Published electronically:
January 21, 2005

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.