  This paper addresses the classical and discrete Euler-Lagrange equations for
systems of $n$ particles interacting quadratically in $\mathbb{R}^d$. By
highlighting the role played by the center of mass of the particles, we solve
the previous systems via the classical quadratic eigenvalue problem (QEP) and
its discrete transcendental generalization. The roots of classical and discrete
QEP being given, we state some conditional convergence results. Next, we focus
especially on periodic and choreographic solutions and we provide some
numerical experiments which confirm the convergence.
