0
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

From Wikipedia:Rotation matrix:

The matrix
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 $\theta = -90°$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.