# Name of the matrix that appears in matrix form of Hamilton's equations of motion

Consider a harmonic oscillator described by the second order differential equation $$\ddot{\phi} + \omega_0^2 \phi = 0 \, .$$ Defining $$v \equiv \dot \phi$$ we get two simultaneous equations \begin{align} \dot \phi &= v \\ \dot v &= - \omega_0^2 \phi \, . \end{align} Rescaling the variables to $$X \equiv \phi$$ and $$Y \equiv v / \omega_0$$, we get \begin{align} \dot X &= \omega_0 Y \\ \dot Y &= - \omega_0 X \end{align} which can be seen as coming from the Hamiltonian $$H = \omega_0 \left( \frac{X^2}{2} + \frac{Y^2}{2} \right)\, .$$ Writing the equations of motion in matrix form gives $$\frac{d}{dt} \left( \begin{array}{c} X \\ Y \end{array} \right) = \omega_0 \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) \left( \begin{array}{c} X \\ Y \end{array} \right) \, .$$

Is there a standard name for that matrix? In other words, is there a general name for the matrix that appears when expressing Hamilton's equations of motion in matrix form?

Perhaps this could be thought of $$i \sigma_2$$ where $$\sigma_i$$ are the Pauli matrices.
Alternatively, this is the most simple linear complex structure on $$\mathbb{R}^2$$ and it's standard symplectic form. The study of symplectic vector spaces and symplectic manifolds grew out of the study of Hamilton's equations in classical mechanics.
R = $$\begin{pmatrix} \cos\theta & -\sin \theta \\ \sin\theta & \cos\theta \end{pmatrix}$$
rotates points in the $$xy$$-plane counterclockwise through an angle $$\theta$$ about the origin of a two-dimensional Cartesian coordinate system.
So your matrix $$\begin{pmatrix}0 & 1 \\ -1 & 0 \end{pmatrix}$$ is the special case of that for the rotating angle $$\theta = -90°$$.