Rotating-frame Lagrangian — the Hamiltonian is conserved, but isn't the energy

The Lagrangian of a particle of mass $m$ moving on the $xy$-plane which is rotating about $z$-axis with angular velocity $\omega$ is given by:

$$L=\frac12m\bigl[(\dot x-\omega y)^2+(\dot y+\omega x)^2\bigr]-V(x,y).$$

Then: