Abstract
We show that the number of relative equilibria of the Newtonian four-body problem is finite, up to symmetry. In fact, we show that this number is always between 32 and 8472. The proof is based on symbolic and exact integer computations which are carried out by computer.
References
Albouy, A.: Integral Manifolds of the N-body problem. Invent. Math. 114, 463–488 (1993)
Albouy, A.: The symmetric central configurations of four equal masses. In: Hamiltonian Dynamics and Celestial Mechanics, Contemp. Math. 198 (1996)
Albouy, A., Chenciner, A.: Le problème des n corps et les distances mutuelles. Invent. Math. 131, 151–184 (1998)
Avis, D.: lrs – Version 4.1. http://cgm.cs.mcgill.ca/∼avis/C/lrs.html
Avis, D., Fukuda, K.: A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra. Discrete Comput. Geom. 8, 295–313 (1992)
Bernstein, D.N.: The number of roots of a system of equations. Funct. Anal. Appl. 9, 183–185 (1975)
Cabral, H.: On the integral manifolds of the n-body problem. Invent. Math. 20, 59–72 (1973)
Chazy, J.: Sur certaines trajectoires du problème des n corps. Bull. Astron. 35, 321–389 (1918)
Christof, T., Loebel, A.: PORTA: POlyhedron Representation Transformation Algorithm, Version 1.3.2. http://www.iwr.uni-heidelberg.de/∼iwr/comopt/soft/PORTA/readme.html
Cox, D., Little, J., O’Shea, D.: Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. New York: Springer 1997
Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. New York: Springer 1998
Dziobek, O.: Über einen merkwürdigen Fall des Vielkörperproblems. Astron. Nachr. 152, 33–46 (1900)
Emiris, I.: Mixvol. http://www.inria.fr/saga/emiris
Emiris, I., Canny, J.: Efficient incremental algorithm for the sparse resultant and the mixed volume. J. Symb. Comput. 20, 117–149 (1995)
Euler, L.: De motu rectilineo trium corporum se mutuo attrahentium. Novi Comm. Acad. Sci. Imp. Petrop. 11, 144–151 (1767)
Hampton, M.: Concave Central Configurations in the Four Body Problem. Thesis, University of Washington 2002
Grayson, D.R., Sullivan, M.E.: Macaulay 2, a software system for research in algebraic geometry and commutative algebra. http://www.math.uic.edu/Macaulay2/
http://www.math.umn.edu/∼rick
Huber, B., Sturmfels, B.: A polyhedral method for solving sparse polynomial systems. Math. Comput. 64, 1541–1555 (1995)
Khovansky, A.G.: Newton polyhedra and toric varieties. Funct. Anal. Appl. 11, 289–296 (1977)
Kushnirenko, A.G.: Newton polytopes and the Bézout theorem. Funct. Anal. Appl. 10, 233–235 (1976)
Kuz’mina, R.P.: On an upper bound for the number of central configurations in the planar n-body problem. Sov. Math. Dokl. 18, 818–821 (1977)
Lagrange, J.L.: Essai sur le problème des trois corps. OEuvres, vol. 6 (1772)
Lefschetz, S.: Algebraic Geometry, Princeton: Princeton University Press 1953
Lehmann-Filhés, R.: Über zwei Fälle des Vielkörpersproblems. Astron. Nachr. 127, 137–143 (1891)
Llibre, J.: On the number of central configurations in the N-body problem. Celest. Mech. Dyn. Astron. 50, 89–96 (1991)
MacDuffee, C.C.: Theory of Matrices. New York: Chelsea Publishing Co. 1946
MacMillan, W.D., Bartky, W.: Permanent configurations in the problem of four bodies. Trans. Am. Math. Soc. 34, 838–875 (1932)
McCord, C.K.: Planar central configuration estimates in the n-body problem. Ergodic Theory Dyn. Syst. 16, 1059–1070 (1996)
McCord, C.K., Meyer, K.R., Wang, Q.: The integral manifolds of the three body problem. Providence, RI: Am. Math. Soc. 1998
Moeckel, R.: Relative equilibria of the four-body problem. Ergodic Theory Dyn. Syst. 5, 417–435 (1985)
Moeckel, R.: On central configurations. Math. Z. 205, 499–517 (1990)
Moeckel, R.: Generic Finiteness for Dziobek Configurations. Trans. Am. Math. Soc. 353, 4673–4686 (2001)
Moeckel, R.: A Computer Assisted Proof of Saari’s Conjecture for the Planar Three-Body Problem. To appear in Trans. Am. Math. Soc.
Moulton, F.R.: The Straight Line Solutions of the Problem of n Bodies. Ann. Math. 12, 1–17 (1910)
Motzkin, T.S., Raiffa, H., Thompson, G.L., Thrall, R.M.: The double description method. Ann. Math. Stud. 28, 51–73 (1953)
Newton, I.: Philosophi naturalis principia mathematica. London: Royal Society 1687
Roberts, G.: A continuum of relative equilibria in the five-body problem. Phys. D 127, 141–145 (1999)
Saari, D.: On the Role and Properties of n-body Central Configurations. Celest. Mech. 21, 9–20 (1980)
Shafarevich, I.R.: Basic Algebraic Geometry 1, Varieties in Projective Space. Berlin, Heidelberg, New York: Springer 1994
Simó, C.: Relative equilibria in the four-body problem. Celest. Mech. 18, 165–184 (1978)
Smale, S.: Topology and Mechanics, II, The planar n-body problem. Invent. Math. 11, 45–64 (1970)
Smale, S.: Mathematical problems for the next century. Math. Intell. 20, 7–15 (1998)
Tien, F.: Recursion Formulas of Central Configurations. Thesis, University of Minnesota 1993
Walker, R.: Algebraic Curves. New York: Dover Publications, Inc. 1962
Wintner, A.: The Analytical Foundations of Celestial Mechanics. Princeton Math. Series 5. Princeton, NJ: Princeton University Press 1941
Wolfram, S.: Mathematica, version 5.0.1.0. Wolfram Research, Inc.
Xia, Z.: Central configurations with many small masses. J. Differ. Equations 91, 168–179 (1991)
Xia, Z.: Central configurations for the four-body and five-body problems. Preprint
Zeigler, G.: Lectures on Polytopes. Grad. Texts Math. 152. New York: Springer 1995
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Mathematics Subject Classification (2000)
70F10, 70F15, 37N05, 76Bxx
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Hampton, M., Moeckel, R. Finiteness of relative equilibria of the four-body problem. Invent. math. 163, 289–312 (2006). https://doi.org/10.1007/s00222-005-0461-0
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DOI: https://doi.org/10.1007/s00222-005-0461-0