On the stability of symmetric periodic solutions of the generalized elliptic Sitnikov (N + 1)-body problem

Abstract

In this paper, we study the stability of symmetric periodic solutions of the generalized elliptic Sitnikov (N+1)-body problem. First, based on the relationship between the potential and the period as a function of the energy, we deduce the properties of the period of the solution of the corresponding autonomous equation (eccentricity e=0) in the prescribed energy range. Then, according to these properties and the stability criteria of symmetric periodic solutions of the time-periodic Newtonian equation, we analytically prove the linear stability/instability of the symmetric (m,p)-periodic solutions which emanated from nonconstant periodic solutions of the corresponding autonomous equation when the eccentricity is small, which indicates that the former stability criteria can be extended to the generalized Sitnikov problem with N≥3.