Central configurations revisited

The classical subject of equilibrium, or central, configurations of N point masses has its role in our restless universe: they describe the patterns in which N bodies engage in simultaneous collisions. In this chapter, the authors look at the classical conditions defining central configurations: they are easily seen to be expressible as systems of algebraic equations in one or several unknowns. Central configurations are important in the dynamics of the N-body problem since the equilibrium solutions involving synchronised circular motions of all bodies can be generalised. Central configurations in celestial mechanics, as well as many other geometric configurations in engineering, are determined by systems of polynomial equations. The methods of algebraic geometry provide insight into the existence of solutions and into their maximum number. For the Eulerian configurations of 3 bodies and for the symmetric collinear 4-body configurations algebraic geometry also provides competitive numerical algorithms.