Quadratic choreographies
This is an incremental theoretical contribution for physicists and mathematicians studying particle dynamics.
The paper solves classical and discrete Euler-Lagrange equations for quadratically interacting particle systems using quadratic eigenvalue problems, and provides numerical experiments confirming convergence for periodic and choreographic solutions.
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.