Investigating a Special Class of Central Configurations for the Three-Dimensional Spatial Six Body Problem

The N-body problem has been studied since Newton's Principia Mathematica.  Given the mutual gravitational attraction between N masses and their initial positions and velocities at the current time, can one predict the position of the masses for all future times?  In the early 1900s, Poincaré discovered what we nowadays call “chaos” in the 3-body problem, and as a consequence established that the N-body problem is non-integrable: one cannot find explicit closed form formulas for all solutions.  We investigate a particular family of solutions that admit closed form solutions called a central configuration.  For these solutions, the acceleration of each mass is proportional to its position vector so that the entire system evolves by expanding or shrinking.  We investigate a class of three-dimensional central configurations for six bodies. In these configurations, three of the bodies have equal masses and lie at the vertices of an equilateral triangle.  The other three lie on a line perpendicular to this triangle.  We seek to understand when such solutions exist.  Applying the Laura-Andoyer-Dziobek form of the central configuration equations transforms the question of existence of such central configurations into an analytical problem involving a system of equations whose solutions are the values of the masses.  These new equations provide insight and may lead to an eventual solution.  We then hope to generalize our results to an N+3-body problem by making similar assumptions as in the six body case, and by considering a similar configuration where we have N masses lying on the line instead of three masses.