Contemporary Mathematics Volume 292, 2002 Spatial central configurations for the 1 + 4 body problem Alain Albouy and Jaume Llibre ABSTRACT. We deal with the central configurations of the 1 + 4 body problem in space, i.e. we study the configurations without collision that are limit of central configurations of the 5 body problem in space, when the mass of one of the bodies goes to infinity. In every such limit the "small" masses are on a sphere whose center is the "big" mass. We suppose moreover that the four small masses are equal. We prove that all these central configurations have at least one plane of symmetry. We first find four "very symmetric" central configurations: the configuration where the four small masses form a regular tetrahedron, two configurations with a three-fold symmetry (we mean invariant by a rotation of angle 2Tr/3 around an axis), and a pyramidal configuration having for base a square formed by the small masses. We then obtain more information about the possible other central configurations. We prove that they are all of the "third kind", i.e. such that the four small masses are in the same hemisphere, and one of the small masses is located inside the spherical triangle formed by the other three. Finally, we conjecture that there are exactly five central configurations in our problem (we mean of course "classes of central configurations", as one can always exchange some bodies and apply some isometry or change of scale.) The fifth central configuration, the unique one having just a plane of symmetry, appears alone in a very clear two-dimensional computer picture mapping all the configurations of the third kind. 1. Introduction Central configurations are the configurations such that the total Newtonian acceleration on every body is equal to a constant . multiplied by the position vector of this body, the center of mass of the configuration being taken as the origin. Rescaling a central configuration still gives a central configuration, but the multiplicative factor . changes. The simplest property which characterizes a central configuration is that it defines a homothetic motion if the initial velocities are chosen conveniently, for example if they are taken to be all zero. We say that a motion is homothetic if during the motion only the size of the configuration changes. If we accept also 1991 Mathematics Subject Classification. Primary 70F15 Secondary 52B55. Key words and phrases. Spatial central configurations, 1 + 4 body problem. The second author is partially supported by DGES grants number PB96-1153 and by a CICYT grant number 1999SGR 00349. © 2002 American Mathematical Society
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