I have the Hamiltonian for system of $N$ particles moving in one dimension: $$H = \sum_{ij} \sqrt{ (p_i - p_j)^2 + \omega_0^2 (q_i - q_j)^2 }$$ where $(q_i,p_i)$ are phase space variables, $i\in\{ 1,\dots, N\}$, and $\omega_0$ is a constant. The equilibrium configuration of this system is $q_i=0$ and $p_i=0$ for all $i$. How does the system evolve after perturbed infinitesimally from the equilibrium state?