The problem just stated that H≠T+U. I think they were trying to show that not every form of general coordinates will work for non-inertial frames. Regardless, I was missing a term of {{\omega}^2}{sin}^{2}(\phi){\phi}^2 in the kinetic energy term, which does in fact lead to H≠T+U.
The problem...
Homework Statement
A bead of mass, m is threaded on a frictionless, straight rod, which lies in the horizontal plane and is forced to spin with constant angular velocity, \omega, about a fixed vertical axis through the midpoint of the rod. Find the Hamiltonian for the bead and show that it...