Abstract
We consider some questions on central configurations of five bodies in space. In the first one, we get a general result of symmetry for the restricted problem of n+1 bodies in dimension n-1. After that, we made the calculation of all c.c. for n=4. In our second result, we extend a theorem of symmetry due to [Albouy, A. and Libre, I.: 2002, Contemporary Math. 292, 1-16] on non-convex central configurations with 4 unit masses and an infinite central mass. We obtain similar results in the case of a big, but finite central mass. Finally, we continue the study by [Schmidt, D.S.: 1988, Contemporary Math. 81 ] of the bifurcations of the configuration with four unit masses located at the vertices of a equilateral tetrahedron and a variable mass at the barycenter. Using Liapunov-Schmidt reduction and a result on bifurcation equations, which appear in [Golubitsley, M., Stewart, L. and Schaeffer, D.: 1988, Singularties and Groups in Bifurcation Theory, Vol. II, Springer-Verlag, New York], we show that there exist indeed seven families of central configurations close to a regular tetrahedron parameterized by the value of central mass.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Albouy, A.: 1997, ‘Recherches sur le problème des N corps’, Notes Scientifiques et Techniques du Bureau des Longitudes, pp. 69–94.
A. Albouy (1995) ArticleTitle‘Symétrie des configurations centrales de quatre corps’, C.R Acad. Sci. Paris 320 217–220 Occurrence Handle0832.70011 Occurrence Handle1320359
A. Albouy (1996) ArticleTitle‘The symmetric central configurations of four equal masses’ Contemporary Math 198 131–135 Occurrence Handle1409157
A. Albouy A. Chenciner (1998) ArticleTitle‘Le problème des N corps et les distances mutuelles’ Inventiones Math 131 151–184 Occurrence Handle10.1007/s002220050200 Occurrence Handle0919.70005 Occurrence Handle1489897 Occurrence Handle1997InMat.131..151A
A. Albouy J. Libre (2002) ArticleTitle‘Spatial central configurations for the 1+4 body problem’ Contemporary Math 292 1–16
Albouy, A.: 2004, ‘On a paper of Moeckel on Central Configurations’, Regular and Chaotic Dynamics, in press
D. Saari (1980) ArticleTitle‘On the role and properties of n body central configurations’ Celestial Mech 21 9–20 Occurrence Handle10.1007/BF01230241 Occurrence Handle0422.70014 Occurrence Handle564603 Occurrence Handle1980CeMec..21....9S
Y. Long S. Sun (2002) ArticleTitle‘Four-Body central configurations with some equal masses’ Arch. Rational Mech. Anal 162 25–44 Occurrence Handle10.1007/s002050100183 Occurrence Handle1033.70004 Occurrence Handle1892230 Occurrence Handle2002ArRMA.162...25L
P. Pedersen (1944) ArticleTitle‘Librationspunke im restringierten Vierköperproblem’ Dan. Mat. Fys Medd. 21 6
Meyer R. Kenneth Schmidt S. Dieter (1988) ArticleTitle‘Bifurcations of relative equilibria in the 4- and 5-body problem’ Ergodic Theory Dyn. Syst 8* 215–225 Occurrence Handle10.1017/S0143385700009433
R.F. Arenstorf (1982) ArticleTitle‘Central configurations of four bodies with one inferior mass’ Celestial Mech 28 9–15 Occurrence Handle10.1007/BF01230655 Occurrence Handle0507.70008 Occurrence Handle682832 Occurrence Handle1982CeMec..28....9A
C. Simó (1978) ArticleTitle‘Relative equilibrium solutions in the four body problem’ Celestial Mech 18 165–184 Occurrence Handle10.1007/BF01228714 Occurrence Handle0394.70009 Occurrence Handle510556 Occurrence Handle1978CeMec..18..165S
D.S. Schmidt (1988) ArticleTitle‘Central configurations in R2and R 3‘ Contemporary Math 81 59–76
S.-N. Chow J. K. Hale (1982) ‘Methods of Bifurcation Theory’ Springer-Verlag New York
M. Golubitsky D. G> Schaeffer (1985) ‘Singularities and Groups in Bifurcation Theory’, Vol. I Springer-Verlag New York
M. Golubitsky I. Stewart D. Schaeffer (1988) ‘Singularities and Groups in Bifurcation Theory’, Vol. II Springer-Verlag New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Almeida santos, A. Dziobek’s configurations in restricted problems and bifurcation. Celestial Mech Dyn Astr 90, 213–238 (2004). https://doi.org/10.1007/s10569-004-0415-7
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10569-004-0415-7