Índice de Morse de configurações centrais simétricas

Abstract

This work investigates the stability, in the sense of the Morse index, of symmetric central configurations in the N-body problem with interaction given by a general homogeneous potential. By exploiting the symmetric structure of the configurations and using tools such as circulant matrices, it is possible to compute the eigenvalues of the associated hessian matrix and determine the Morse index in various cases. We also analyze extensions involving central masses and stacked configurations, identifying instability patterns and proposing two conjectures regarding the general behavior of the index.