December  2008, 1(4): 631-646. doi: 10.3934/dcdss.2008.1.631

A proof of Saari's conjecture for the three-body problem in $\R^d$

1. 

School of Mathematics, University of Minnesota, Minneapolis MN 55455

Received  January 2006 Revised  August 2008 Published  September 2008

The well-known central configurations of the three-body problem give rise to periodic solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that such rigid motions, called relative equilibrium solutions, are the only solutions with constant moment of inertia. This result will be proved here for the Newtonian three-body problem in $\R^d$ with three positive masses. The proof makes use of some computational algebra and geometry. When $d\le 3$, the rigid motions are the planar, periodic solutions arising from the five central configurations, but for $d\ge 4$ there are other possibilities.
Citation: Richard Moeckel. A proof of Saari's conjecture for the three-body problem in $\R^d$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 631-646. doi: 10.3934/dcdss.2008.1.631
[1]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[2]

Edward Belbruno. Random walk in the three-body problem and applications. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 519-540. doi: 10.3934/dcdss.2008.1.519

[3]

Richard Moeckel. A topological existence proof for the Schubart orbits in the collinear three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 609-620. doi: 10.3934/dcdsb.2008.10.609

[4]

Mitsuru Shibayama. Non-integrability of the collinear three-body problem. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 299-312. doi: 10.3934/dcds.2011.30.299

[5]

Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229

[6]

Hiroshi Ozaki, Hiroshi Fukuda, Toshiaki Fujiwara. Determination of motion from orbit in the three-body problem. Conference Publications, 2011, 2011 (Special) : 1158-1166. doi: 10.3934/proc.2011.2011.1158

[7]

Kuo-Chang Chen. On Chenciner-Montgomery's orbit in the three-body problem. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 85-90. doi: 10.3934/dcds.2001.7.85

[8]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090

[9]

Regina Martínez, Carles Simó. On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1157-1175. doi: 10.3934/dcds.2013.33.1157

[10]

Xiaojun Chang, Tiancheng Ouyang, Duokui Yan. Linear stability of the criss-cross orbit in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5971-5991. doi: 10.3934/dcds.2016062

[11]

Abimael Bengochea, Manuel Falconi, Ernesto Pérez-Chavela. Horseshoe periodic orbits with one symmetry in the general planar three-body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 987-1008. doi: 10.3934/dcds.2013.33.987

[12]

Qinglong Zhou, Yongchao Zhang. Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1763-1787. doi: 10.3934/dcds.2017074

[13]

Samuel R. Kaplan, Mark Levi, Richard Montgomery. Making the moon reverse its orbit, or, stuttering in the planar three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 569-595. doi: 10.3934/dcdsb.2008.10.569

[14]

Tiancheng Ouyang, Duokui Yan. Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3989-4018. doi: 10.3934/dcds.2017169

[15]

Niraj Pathak, V. O. Thomas, Elbaz I. Abouelmagd. The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 849-875. doi: 10.3934/dcdss.2019057

[16]

Hadia H. Selim, Juan L. G. Guirao, Elbaz I. Abouelmagd. Libration points in the restricted three-body problem: Euler angles, existence and stability. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 703-710. doi: 10.3934/dcdss.2019044

[17]

Alessandra Celletti. Some KAM applications to Celestial Mechanics. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 533-544. doi: 10.3934/dcdss.2010.3.533

[18]

Jean-Baptiste Caillau, Bilel Daoud, Joseph Gergaud. Discrete and differential homotopy in circular restricted three-body control. Conference Publications, 2011, 2011 (Special) : 229-239. doi: 10.3934/proc.2011.2011.229

[19]

Frederic Gabern, Àngel Jorba, Philippe Robutel. On the accuracy of restricted three-body models for the Trojan motion. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 843-854. doi: 10.3934/dcds.2004.11.843

[20]

Luca Biasco, Luigi Chierchia. Exponential stability for the resonant D'Alembert model of celestial mechanics. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 569-594. doi: 10.3934/dcds.2005.12.569

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]