Bifurcations and stability of some symmetrical classes of central configurations

In our PhD dissertation we present some new results on the subject of Central Configurations (cc's) of the Newtonian N-Body Problem. We consider the particular case of symmetrical configurations in which n₁ bodies of equal masses form a given figure circumscribed by the unit sphere S of a hyperplane H of dimension d − 1 and n₂ bodies lie on a line L perpendicular to H and intersecting it at the center of the sphere. When the bodies on H form a regular (d − 1)-simplex, we prove that there are only finitely many central cc's and that this finiteness result is independent of the dimension d of the environment space. Moreover, we study in details the bifurcation set for the case n ₂ = 2 and d = 2, 3, and conjecture the possible numbers of solutions in the case of positive masses. This is accomplished through the application of the Method of Rational Parametrization (MRP), which we developed during our research. In addition, we consider the cc's of the Planar Restricted Four Body Problem (PR4BP). Through a computation involving Gröbner Basis, we show that there are finitely many cc's in the PR4BP when two of the masses are equal. Then, by applying the MRP, we give a complete description of the symmetrical cc's of the PR4BP, including their possible numbers, bifurcation points and stability. In particular, the case of three equal masses is fully solved. Finally, we study the stability of two important classical examples of symmetrical cc's: the pseudo-centered isosceles configuration of four equal masses and the centered equilateral configuration with three equal masses and an arbitrary one at the center of the triangle. We prove that they are both spectrally unstable.