Quadratic choreographies

This paper addresses the classical and discrete Euler–Lagrange equations for systems of n   particles interacting quadratically in RdRd. 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. Next, we state a conditional convergence result, in the Hausdorff sense, for the roots of the discrete QEP to the roots of the classical one. At last, we focus especially on periodic and choreographic solutions and we provide some numerical experiments which confirm the convergence.