Consider a magnetic compass needle with moment of inertia \(I\) and magnetic dipole moment \(\boldsymbol{\mu}\), free to rotate in the \(x-y\) plane. Denote the polar angle by \(\theta\). A time dependent external magnetic field \(\mathbf{B}=B_{0} \cos \omega t \hat{\boldsymbol{x}}\) applies a torque given by \(\boldsymbol{\mu} \times \mathbf{B}\) on the needle.

(a) Write the equation of motion for \(\theta\).

(b) Solve for \(\theta(t)\) numerically and generate a Poincaré map by plotting discrete points \(\theta(t=2 \pi n / \omega)\) for integer \(n\). Verify the onset of chaos for \(2 B_{0} \mu / I>\omega^{2}\).