Stability Of The Elliptic Equilibria With (1+n)-gon Central Configurations

In this paper,we consider the homographic solutions generated by the(1+n)-gon central configuration.Central configuration is an important concept in celestial mechanics,by which we can construct the homographic solutions of the planar N-body problem.And up to now it is the only known way to get exact solutions of the planar N-body problem.In this kind of periodic solutions,every particle moves along a specific Keplerian orbit while the totality of the particles move on a homothety motion.If the Keplerian orbit is elliptic,we call the solution an elliptic relative equilibrium,and a relative equilibrium in the case e=0.The stability of the relative equilibrium has been widely concerned for a long time.The(1+n)-gon configurations were first proposed by Maxwell when he studied the Saturn ring system.A(1+n)-gon central configuration consists of n equal mass particles distributed on the vertices of the regular n-gon and an extra particle at the center.For they can be used as approximate models of Saturn and the Saturn ring system,the study of their stability has a strong physical background.Moeckel guess that in the planar n-body problem,a linear stable relative equilibrium always has a particle whose mass dominates and whose configuration is always a non-degenerate minimum of the function U restricted to the unit configuration sphere.However,this conjecture is far from resolved in the case of relative equilibrium.Especially,we have less results in the case of elliptical one now.In recent years,X.Hu and his coworkers studied the linear stability of periodic solutions of N-body problems by introducing Maslov-type indices,and their results show that Moeckel’s guess is also correct for elliptic relative equilibrium.Based on the above works,we calculate the Maslov-type indices of the(1+n)-gon relative equilibria when 2<n<7 and the Morse indices of the corresponding central configurations.Indicate that the Maslov-type 1-indices of the relative equilibria are greater than or equal to the Morse indices of the corresponding central configurations.In the process of calculating Maslovtype indicators,we transform the Sturm-Liouville operators into Hermitian quasi-diagonal matrices by Fourier transform.And the degenerated points and the indices are Obtained by further analyzing their eigenvalues through an image method.At the end of the paper,the ±1 degenerated curves are drawn numerically and a part of the stability regions of the(1+7)-gon elliptic relative equilibrium is obtained by analyzing its Maslov-type index.