\documentstyle{article}
    \title{Nonlinear instability of some relative equilibrium configurations
                  in the n-body problem}
                \author{Elmabsout Badaoui\\
Laboratoire des Syst\`eme Dynamiques, Unit\'e de Formation et\\
de la Recherche de M\'ecanique, Universit\'e Paris 6, Tour 66,\\
4 Place Jussieu, 75252 Paris cedex 05, France.}
\date{}


\begin{document}
\maketitle

   We study by an analytical method, the second order non-linear
stability of a degenerate configuration of a relative equilibrium where
n bodies having the same mass m, are at the vertices of a regular
polygon with n sides, while a body having a non-zero mass $m_0$ is at the
center of the polygon. IT is shown that, at this configuration of the
relative equilibrium, the matrix of the second derivatives of the
energy-momentum functional has always negative and positive
eigenvalues, for any choice of n and any choice of the ratio $\mu = m_0/m$
Consequently the above configuration is nonlinearly unstable for any
choice of   and n, even if for n > 6 it is linearly stable for  larger
than some third degree polynomial of n, as we had shown in a previous
paper.

\end{document}
