Linear stability of the regular <i>N</i>-gon periodic solutions for the planar <i>N</i>-body problem with quasi-homogeneous potential

Abstract This paper considers the planar N -body problem with a quasi-homogeneous potential given by $$ \begin{align*}W = \sum_{1\leq k&lt;j\leq N} \left[ \frac{ m_k m_j}{\|\boldsymbol{r}_k-\boldsymbol{r}_j\|} +\frac{m_k m_j C_{jk}}{\|\boldsymbol{r}_k-\boldsymbol{r}_j\|^p} \right], \end{align*} $$ where $ m_k&gt;0 $ are the masses and $C_{jk}= C_{kj}$ are nonzero real constants, and the exponent g being $ p&gt; $ 1. Generalizing techniques of the classical N -body problem, we first characterize the periodic solutions that form a regular polygon (relative equilibria) with equal masses ( $m_k= m$ , $k=1, \ldots , N$ ) and equal constants $C_{jk}= C$ , for all $j, k=1, \ldots , N$ (for short, N -gon solutions). Indeed, for $C&gt;0$ we prove that there exists a unique regular N -gon solution for each fixed positive mass m . In contrast, for the case $C &lt;0$ , we demonstrate that there can be a maximum of two distinct regular N -gon solutions for a fixed positive mass m . More precisely, there is a range of values for the mass parameter m for which no solutions of the form of an N -gon exist. Furthermore, we examine the linear stability of these solutions, with a particular focus on the special case $ N=3 $ , which is fully characterized.