So I'm really stumped with this. I have a particle in a cone, like pictured. The particle orbits the z axis on the dotted line for $r$.
So knowing that $\alpha$ and $r$ remain constant in this motion, because as long as the particle is in the cone $\alpha$ won't change and as long as the particle stays in the same $z$ coordinate $r$ won't change.
So this motion would similar to a 3D Pendulum that has a constant $z$ (the height doesn't change)
Would the kinetic energy for the particle be: $$ T=\frac12mr^2\dot\theta^2 $$ or would it be $$T=\frac12mr^2\dot\theta^2\sin^2(\alpha)?$$
I know the potential would be: $$U=-mgr\cos(\alpha)$$
So the Lagrangian would be: $$L=T-U$$
Can anyone check me? I get really confused on polar coordinates.