Central configurations with a quasihomogeneous potential function

Abstract

For the classical N-body problem, if the masses are paced at rest in the configuration that is central, then the masses will tend to the origin and result in a simultaneous collision [A. Winter, The Analytical Foundations of Celestial Mechanics, 1st ed. (Princeton University Press, Princeton, 1941)]. However, if the potential is quasihomogeneous in nature, that is, the potential is of the form U=V+W, where W=∑i≠jamimjrij−α and V=∑i≠jbmimjrij−β, then we will dramatically get different results. This paper determines the orbits that will result when the potential function is quasihomogeneous. It also shows when there existences a relative equilibrium solution for the regular N-gon configuration. This is one-half of Perko’s theorem for a homogeneous potential. Finally, we show the existence of a central configuration which is not central for either W or V. This configuration is the placement of two mutual equal pairs of masses placed at the vertices of a square. This shows that the reverse implication of Perko’s theorem for the regular N-gon does not hold for quasihomogeneous potentials.