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Linear instability of relative equilibria for $n$-body problems in the plane

Following Smale, we study simple symmetric mechanical systems of $n$ point particles in the plane. In particular, we address the question of the linear and spectral stability properties of relative equilibria, which are special solutions of the equations of motion. Our main result is a sufficient condition to detect spectral (hence linear) instability. Namely, we prove that if the Morse index of an equilibrium point with even nullity is odd, then the associated relative equilibrium is spectrally unstable. The proof is based on some refined formulæ for computing the spectral flow. As a notable application of our theorem, we examine two important classes of singular potentials: the $α$-homogeneous one, with $α\in (0, 2)$, which includes the gravitational case, and the logarithmic one. We also establish, for the $α$-homogeneous potential, an inequality which is useful to test the spectral instability of the associated relative equilibrium.