Analysis of Central Configurations and Bifurcations in the Planar Unequal Mass 4-Body Problem

The Newtonian N-body problem is fundamental to understanding the motion and stability of celestial systems, from planetary orbits to star clusters and galaxies. Studying central configurations provides crucial insight into these motions, as their classification governs the existence of relative equilibria and bifurcations of the integral manifolds. We study the planar 4-body problem with three equal masses and one unequal mass. This mass distribution is a natural next step beyond previously studied equal-mass cases, and the presence of an equal mass makes the problem enormously more complex. 

We systematically identify and categorize central configurations—balanced geometric arrangements of bodies, as a function of the variable mass parameter 𝜇 and potential energy U, combining numerical and computational methods, including GPU parallelization to address the problem's high computational cost. We map out the bifurcation structure in (𝜇,𝑈)-space to provide insight into these classifications and the behavior at critical mass ratios. Our results describe the transition between symmetric and asymmetric families, examine the problem’s bifurcation structure by identifying pitchfork and saddle-node bifurcations, and reveal a unique single asymmetric family that bifurcates from symmetry.