Introduction to Classical Mechanics · Problem 9.8

Angular momentum, Part II (General L) — Morin

Physics — Mechanics · symmetries-and-principal-axes-for-a-pancake

Problem

A rotation of the axes in the

x

y

plane through an angle

\theta

transforms the coordinates according to (you can accept this)

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}. \tag{9.94}

Use this to show that if a pancake object in the

x

y

plane has a symmetry under a rotation through

\theta \neq \pi

, then

\int xy = 0

for any choice of axes, which implies that all axes (through the origin) in the plane are principal axes.