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

This is the Hamiltonian of vector resonant relaxation of a thin stellar disk.

EDIT: The equations of motions are $$\dot{q}_i = \frac{\partial H}{\partial p_i} = \sum_j M_{ij} \frac{p_i - p_j}{\sqrt{ (p_i - p_j)^2 + (q_i - q_j)^2 }}$$ $$\dot{p}_i = -\frac{\partial H}{\partial q_i} = \sum_j M_{ij} \frac{q_j - q_i}{\sqrt{ (p_i - p_j)^2 + (q_i - q_j)^2 }}$$ These equations may be written in a more compact form using 3D vectors ${\bf L}_i \equiv (q_i, p_i, 1)$ where $q_i\ll 1$ and $p_i\ll 1$: $$\dot{\bf L}_i = \sum_j M_{ij} \frac{{\bf L}_i \times {\bf L}_j}{ |{\bf L}_i - {\bf L}_j |}$$ As long as $q_i\ll 1$ and $p_i\ll 1$, the third component of this equation (let's call it $\dot{L}_{iz}$) vanishes, so $L_{iz}=1$ is conserved. Indeed, taking the scalar product with ${\bf L}_i$ gives zero, which means that $|{\bf L}_{i}|$ is conserved for all $i$, ${\bf L}_{i}$ remain unit vectors.

The equation of motion shows that the equilibrium state has all ${\bf L}_i$ parallel, and since they are unit vectors, they are equal or opposite to each other. For $N=2$, ${\bf L}_1$ and ${\bf L}_2$ rotate around ${\bf L}_1+{\bf L}_2$ with uniform angular velocity. However for infinitesimal perturbations the angular velocity approaches infinity. Is there a way to understand the normal mode oscillations of this system for "small" (maybe not infinitely small if that is ill-defined) perturbations?