Analytic dynamics of a one-dimensional system of particles with strong interaction

We consider the dynamics of a system of N particles on the circle with interaction of nearest neighbors, a Coulomb potential, and an analytic external force. The trajectories are real analytic functions of time. However, the series for them converge only for sufficiently small times. For zero initial velocities and a uniform initial location of particles, we prove N-dependent estimates on the coefficients of this series.