Resumo:
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.
