Spectral stability, spectral flow and circular relative equilibria for the Newtonian $n$-body problem

For the Newtonian (gravitational) $n$-body problem in the Euclidean $d$-dimensional space, $dge 2$, the simplest possible periodic solutions are provided by circular relative equilibria, (RE) for short, namely solutions in which each body rigidly rotates about the center of mass and the configuration of the whole system is constant in time and central (or, more generally, balanced) configuration. For $dle 3$, the only possible (RE) are planar, but in dimension four it is possible to get truly four dimensional (RE). A classical problem in celestial mechanics aims at relating the (in-)stability properties of a (RE) to the index properties of the central (or, more generally, balanced) configuration generating it. In this paper, we provide sufficient conditions that imply the spectral instability of planar and non-planar (RE) in $mathbb R^4$ generated by a central configuration, thus answering some of the questions raised in cite[Page 63]{Moe14}. As a corollary, we retrieve a classical result of Hu and Sun cite{HS09} on the linear instability of planar (RE) whose generating central configuration is non-degenerate and has odd Morse index, and fix a gap in the statement of cite[Theorem 1]{BJP14} about the spectral instability of planar (RE) whose (possibly degenerate) generating central configuration has odd Morse index. The key ingredients are a new formula of independent interest that allows to compute the spectral flow of a path of symmetric matrices having degenerate starting point, and a symplectic decomposition of the phase space of the linearized Hamiltonian system along a given (RE) which is inspired by Meyer and Schmidts planar decomposition cite{MS05} and which allows us to rule out the uninteresting part of the dynamics corresponding to the translational and (partially) to the rotational symmetry of the problem.