Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

The number of planar central configurations for the $4$-body problem is finite when $3$ mass positions are fixed


Authors: Martha Alvarez, Montserrat Corbera, Joaquin Delgado and Jaume Llibre
Journal: Proc. Amer. Math. Soc. 133 (2005), 529-536
MSC (2000): Primary 70F07; Secondary 70F15
DOI: https://doi.org/10.1090/S0002-9939-04-07590-2
Published electronically: August 26, 2004
MathSciNet review: 2093077
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In the $n$-body problem a central configuration is formed when the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration vector. Lindstrom showed for $n=3$ and for $n>4$ that if $n-1$masses are located at fixed points in the plane, then there are only a finite number of ways to position the remaining $n$th mass in such a way that they define a central configuration. Lindstrom leaves open the case $n=4$. In this paper we prove the case $n=4$using as variables the mutual distances between the particles.


References [Enhancements On Off] (What's this?)

  • 1. D. COX, J. LITTLE AND D. O'SHEA, Using Algebraic Geometry, Springer-Verlag, 1998. MR 99h:13033
  • 2. O. DZIOBEK, Über einen merkwürdigen fall vielkörperproblems, Astron. Nach., 152 (1890), 32-46.
  • 3. Y. HAGIHARA, Celestial Mechanics, Vol. 1, MIT press, Cambridge, 1970. MR 57:18306
  • 4. S. LANG, Algebra, 3rd. Edition, Addison-Wesley, 1993. MR 33:5416
  • 5. P. W. LINDSTROM, The number of planar central configurations is finite when $n-1$ mass positions are fixed, Trans. Amer. Math. Soc. 353 (2001), 291-311. MR 2001b:70018
  • 6. J. LLIBRE, On the number of central configurations in the $N$-body problem. Celestial Mech. Dynam. Astronom. 50 (1991), 89-96. MR 92g:70011
  • 7. R. MCGEHEE, Triple collision in the collinear three-body problem. Invent. Math. 27 (1974), 191-227. MR 50:11912
  • 8. F.R. MOULTON, The straight line solutions of $n$ bodies, Ann. of Math. 12 (1910), 1-17.
  • 9. P. OLVER, Classical invariant theory, London Math. Soc. Student Texts, Vol. 44, Cambridge Univ. Press, New York, 1999. MR 2001g:13009
  • 10. D. 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. Celestial Mech. 21 (1980), no. 1, 9-20. MR 81a:70016
  • 11. S. SMALE, Topology and mechanics. II. The planar $n$-body problem, Invent. Math. 11 (1970), 45-64. MR 47:9671
  • 12. S. SMALE, Mathematical problems for the next century, Math. Intelligencer 20 (1998), 7-15. MR 99h:01033
  • 13. A. WINTNER, The analytical foundations of celestial mechanics, Princeton Univ. Press, 1941. MR 3:215b

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 70F07, 70F15

Retrieve articles in all journals with MSC (2000): 70F07, 70F15


Additional Information

Martha Alvarez
Affiliation: Departamento de Matemáticas, UAM–Iztapalapa, A.P. 55–534, 09340 Iztapalapa, Mexico, D.F., Mexico
Email: mar@xanum.uam.mx

Montserrat Corbera
Affiliation: Departament d’Informàtica i Matemàtiques, Universitat de Vic, 08500 Vic, Barcelona, Spain
Email: montserrat.corbera@uvic.es

Joaquin Delgado
Affiliation: Departamento de Matemáticas, UAM–Iztapalapa, A.P. 55–534, 09340 Iztapalapa, Mexico, D.F., Mexico
Email: jdf@xanum.uam.mx

Jaume Llibre
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Email: jllibre@mat.uab.es

DOI: https://doi.org/10.1090/S0002-9939-04-07590-2
Keywords: $4$--body problem, central configurations
Received by editor(s): November 12, 2002
Received by editor(s) in revised form: February 6, 2003, and July 16, 2003
Published electronically: August 26, 2004
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society