Central configurations and homographic solutions for the quasihomogeneous N-body problem

Abstract

For the N-body problem given by quasihomogeneous force functions of the form C1/ra + C2/rb (C1, C2 real, and 0 < a ⩽ b), we prove that the simultaneous and extraneous central configurations are the only types of central configurations. Then we focus on attractive-attractive force functions (C1 and C2 positive) and analyze the connection between central configurations and homographic solutions. Namely, we extend the classical Lagrange's theorem (a solution is homographicif and only if the bodies form equivalent central configurations for all times) to attractive-attractive quasihomogeneous problems, and give sufficient conditions for the construction of homographic solutions.