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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:257

Article

Barutello, Vivina L.¹  Jadanza, Riccardo D.²  Portaluri, Alessandro³ 

[1] Univ Turin, Dipartimento Matemat G Peano, I-10123 Turin, Italy

[2] Politecn Torino, Dipartimento Sci Matemat JL Lagrange DISMA, I-10129 Turin, Italy

[3] Univ Turin, Dipartimento Sci Agr Forest & Alimentari DISAFA, I-10095 Grugliasco, TO, Italy

关键词: Linear instability;    Relative equilibria;    Spectral flow;    Partial signatures;    n-Body problem;    alpha-Homogeneous potential;    Logarithmic potential;   

DOI  :  10.1016/j.jde.2014.05.017

来源: Elsevier

【 摘 要 】

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 formul ae for computing the spectral flow. As a notable application of our theorem, we examine two important classes of singular potentials: the alpha-homogeneous one, with alpha is an element of(0, 2), which includes the gravitational case, and the logarithmic one. We also establish, for the alpha-homogeneous potential, an inequality which is useful to test the spectral instability of the associated relative equilibrium. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free