Hamiltonian for mass on a smooth fixed hemisphere

[Jacob Flowers]

I am trying to figure out how to get the Hamiltonian for a mass on a fixed smooth hemisphere.

Using Thorton from example 7.10 page 252

My main question is about the Potential energy= mgrcosineθ is the generalized momenta Pdotθ supposed to be equal to zero because θ is cyclic? Or is Pdotθ= -∂H/∂θ= mgr sineθ
http://kylejensenphysicsblog.blogspot.com/2016/03/hamiltons-principle-and-lagrangians.htmlSorry for not being able to upload a pictuure or putting dots over P but I don't know how to do so

 

[vanhees71]

But in the link above you don't have a particle on a sphere but on a circle, and then of course $\dot{r}=0$. So the correct Lagrangian is
$$L=\frac{m}{2} r^2 \dot{\theta}^2+mgr \cos \theta.$$
I also have more conveniently pointed the $y$ axis in direction of $\vec{g}$ such that the stable stationary state is $\theta=0=\text{const}$. Then you have
$$p_{\theta}=\frac{\partial L}{\partial \dot{\theta}}=mr^2 \dot{\theta} \; \Rightarrow \; \dot{p}_{\theta}=m r^2 \ddot{\theta}=\frac{\partial L}{\partial \theta}=-mgr \sin \theta,$$
and you get the equation of motion for a mathematical pendulum, as you should:
$$\ddot{\theta}=-\frac{g}{r} \sin \theta.$$

For the more general case you should use spherical coordinates
$$\vec{x}=\begin{pmatrix} r \cos \varphi \sin \vartheta \\ r \sin \varphi \sin \vartheta \\ r \cos \vartheta \end{pmatrix},$$
again with $r=\text{const}$. Now write down the Hamiltonian, and you'll get the equations of motion for a spherical pendulum, but this problem you should discuss in the homework forum since it's way better to get some guidance to solve the problem yourself than just a solution!