Linear stability of the elliptic relative equilibrium with (1 + n)-gon central configurations in planar n-body problem

Central configurations and relative equilibria in the n–body problem

Stability of Relative Equilibria in the Planar n-Vortex Problem

We study the linear and nonlinear stability of relative equilibria in the planar n-vortex problem, adapting the approach of Moeckel from the corresponding problem in celestial mechanics. After establishing some general theory, a topological approach is taken to show that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum...

Planar Central Configuration Estimates in the N-body Problem

For all masses, there are at least n ?2 O 2-orbits of non-collinear planar central conngurations. In particular, this estimate is valid even if the potential function is not a Morse function. If the potential function is a Morse function, then an improved lower bound, on the order of n! ln ? n+1 3 =2, can be given.

Central Configurations for Newtonian N + 2 p + 1 - Body Problems

and Applied Analysis 3 −λ (0, 0, rp) = − N 󵄨󵄨󵄨󵄨󵄨1 + r2 p󵄨󵄨󵄨󵄨󵄨3/2 (0, 0, rp) − (0, 0, 1 r2 p)M0 − ∑ 1≤i<p ( 1 󵄨󵄨󵄨󵄨󵄨ri − rp󵄨󵄨󵄨󵄨󵄨2 + 1 󵄨󵄨󵄨󵄨󵄨ri + rp󵄨󵄨󵄨󵄨󵄨2)Mi (0, 0, 1) − 1 󵄨󵄨󵄨󵄨󵄨2rp󵄨󵄨󵄨󵄨󵄨2Mp (0, 0, 1) . (11) In order to simplify the equations, we defined a0,j = −2/|1+r2 j |3/2, aj,j = −1/4r3 j , where j = 1, . . . , p; ai,j = (1/|ri − rj|2ri − 1/|ri + rj|2ri), when i < j; ai,j = −(1/|ri − rj|2ri + 1/...

New Classes of Spatial Central Configurations for N + N + 2 - Body Problem

and Applied Analysis 3 Recently, Hampton and Santoprete 17 provided new examples of stacked spatial central configurations—central configurations for theN-body problemwhere a proper subset of the N bodies are already on a central configuration—for the 7-body problem where the bodies are arranged as concentric threeand two-dimensional simplex. New classes of stacked spatial central configuration...