 Solution

For a regular Lagrangian, form the canonical momenta $p_i=L_{{\dot{q}}_i}$, invert for ${\dot{q}}_i(\mathbf{q},\mathbf{p},t)$, and take the Legendre transform $H={\sum }_i{p}_i{\dot{q}}_i-L$. On the upper cone let $s>0$ be distance along a generator from the apex and $\varphi$ the azimuth. Then the horizontal radius is $s\sin \theta$ and height is $s\cos \theta$, soConsequently