Consider the following Hamiltonian which is absolute relativistic literally: only sensitive to absolute pairwise relative phase space variables of objects for a system of $N$ objects 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 $M_{ij}$ is a constant symmetric matrix. 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?

Coincidentally, this is the Hamiltonian of Newtonian vector resonant relaxation of a thin stellar disk.

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 are remarkably simple in the complex plane. Define $z_i = q_i + i p_i$, then $$H = \sum_{nm} M_{nm} |z_n - z_m| \\ \dot{z}_n = \sum_m i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}\,.$$ Notice that the "effective force" is independent of the phase space distance, it depends on only the argument of the relative coordinates. This is similar to the interaction of charged infinite insulating plates.

Is there a way to derive the normal mode oscillations of this nonlinear system for "small" perturbations around $z_i=0$?

(Note: The Hamiltonian is non-relativistic in the usual sense of the word. However this is the general relativistic Hamiltonian of 1 object moving if the metric of spacetime is $g_{\mu \nu}=x^2(-1,1,1,1)$. The Lagrangian is then $L = -\sqrt{-g_{\mu \nu}\frac{d x^{\mu}}{d\tau}\frac{d x^{\nu}}{d\tau}} =-\sqrt{x^2( 1 - \dot{x}^2)} $ which yields $H=\sqrt{x^2+ p^2}$.)

Consider the following Hamiltonian which is absolutely relativistic literally: only sensitive to absolute pairwise relative phase space variables of objects for a system of $N$ objects 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 $M_{ij}$ is a constant symmetric matrix. 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?

Coincidentally, this is the Hamiltonian of Newtonian vector resonant relaxation of a thin stellar disk.

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 are remarkably simple in the complex plane. Define $z_i = q_i + i p_i$, then $$H = \sum_{nm} M_{nm} |z_n - z_m| \\ \dot{z}_n = \sum_m i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}\,.$$ Notice that the "effective force" is independent of the phase space distance, it depends on only the argument of the relative coordinates. This is similar to the interaction of charged infinite insulating plates.

Is there a way to derive the normal mode oscillations of this nonlinear system for "small" perturbations around $z_i=0$?

(Note: The Hamiltonian is non-relativistic in the usual sense of the word. However this is the general relativistic Hamiltonian of 1 object moving if the metric of spacetime is $g_{\mu \nu}=x^2(-1,1,1,1)$. The Lagrangian is then $L = -\sqrt{-g_{\mu \nu}\frac{d x^{\mu}}{d\tau}\frac{d x^{\nu}}{d\tau}} =-\sqrt{x^2( 1 - \dot{x}^2)} $ which yields $H=\sqrt{x^2+ p^2}$.)

Consider the following Hamiltonian which is absolute relativistic literally: only sensitive to absolute pairwise relative phase space variables of objects for a system of $N$ objects 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 $M_{ij}$ is a constant symmetric matrix. 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?

Coincidentally, this is the Hamiltonian of Newtonian vector resonant relaxation of a thin stellar disk.

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 are remarkably simple in the complex plane. Define $z_i = q_i + i p_i$, then $$H = \sum_{nm} M_{nm} |z_n - z_m| \\ \dot{z}_n = \sum_m i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}\,.$$ Notice that the "effective force" is independent of the phase space distance, it depends on only the argument of the relative coordinates. This is similar to the interaction of charged infinite insulating plates.

Is there a way to derive the normal mode oscillations of this nonlinear system for "small" perturbations around $z_i=0$?

Consider the following Hamiltonian which is absolute relativistic literally: only sensitive to absolute pairwise relative phase space variables of objects for a system of $N$ objects 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 $M_{ij}$ is a constant symmetric matrix. 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?

Coincidentally, this is the Hamiltonian of Newtonian vector resonant relaxation of a thin stellar disk.

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 are remarkably simple in the complex plane. Define $z_i = q_i + i p_i$, then $$H = \sum_{nm} M_{nm} |z_n - z_m| \\ \dot{z}_n = \sum_m i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}\,.$$ Notice that the "effective force" is independent of the phase space distance, it depends on only the argument of the relative coordinates. This is similar to the interaction of charged infinite insulating plates.

Is there a way to derive the normal mode oscillations of this nonlinear system for "small" perturbations around $z_i=0$?

(Note: The Hamiltonian is non-relativistic in the usual sense of the word. However this is the general relativistic Hamiltonian of 1 object moving if the metric of spacetime is $g_{\mu \nu}=x^2(-1,1,1,1)$. The Lagrangian is then $L = -\sqrt{-g_{\mu \nu}\frac{d x^{\mu}}{d\tau}\frac{d x^{\nu}}{d\tau}} =-\sqrt{x^2( 1 - \dot{x}^2)} $ which yields $H=\sqrt{x^2+ p^2}$.)