A mass point moves on the wall of a hollow circle cone under influence of the homogeneous gravitational field of earth.
Use spherical coordinates to solve this problem.
a)Set up the lagrange equations of second kind.
b)If the lagrange function L isn't dependent on a coordinate q but on the associated velocity, then q is a cyclic coordinate. Is any of the generalized coordinates in this case a cyclic coordinate?
c) Check if the momentum, the z-component of the angular momentum and the total energy are conserved.
First I tried drawing the situation as best as I could. This is what I got:
I was sick the past few weeks so I basically missed everything regarding lagrangian formalism. Since we use a text book as a script to go along to I tried reading in it, but all the copies in the school library were borrowed. So I had to make use of wikipedia.
I'm not yet familiar with the terminology. The lagrange equations of second kind are the euler-lagrange equations, right?
To be honest this topic doesn't stick with me, especially the general coordinates and velocities.
Anyways, I tried setting up $L=T-V$ since this is the first step as far as I could tell from the wiki entry.
Generally it should be $L=\frac{1}{2}m(\dot{r}^2+\dot{z}^2+r^2\dot{\phi}^2)-mgz$, right?
$z=r\cot{\theta}$? Meaning, $L=\frac{1}{2}m((1+\cot^2(\theta))\dot{r}^2+r^2\dot{\phi}^2)-mgr\cot{\theta})$, correct? I hope someone could help me out here.
Edit:
@JohnHughes: I think you were right about r. Once I tried expressing the lagrangian in spherical coordinates as was asked it seemed reasonable to assume that r is actually the diagonal line closest to it.
So I got this:
a) $\vec{x}=r\begin{bmatrix}\cos{\phi}\sin{\theta}\\\sin{\phi}\sin{\theta}\\\cos{\theta}\end{bmatrix}$
Since I need the velocity for the lagrangian, I have:
$\dot{\vec{x}}=\dot{r}\begin{bmatrix}\cos{\phi}\sin{\theta}\\\sin{\phi}\sin{\theta}\\\cos{\theta} \end{bmatrix}+r\dot{\phi}\sin{\theta}\begin{bmatrix} -\sin{\phi}\\\cos{\phi}\\0\end{bmatrix}$.
Plugging that all I have: $T=\frac{m}{2}\dot{\vec{x}}^2=\frac{m}{2}(\dot{r}^2+r^2\dot{\phi}^2\sin^2{\theta})$ and $V=mgr\cos{\theta}$. $L=\frac{m}{2}(\dot{r}^2+r^2\dot{\phi}^2\sin^2{\theta})-mgr\cos{\theta}$.
That should conclude a), right?
About b): Just from the given definition in b) I can say that $\phi$ is a cyclic coordinate since the lagrangian only has $\dot{\phi}$, correct?
About c): I'm not really sure how to approach this one. My ideas so far were:
Momentum: I can imagine that the momentum isn't conserved since the mass point is accelerated due to the gravitational field, but I have no idea how to write that down mathematically correct.
Angular mometum: To be honest I'm pretty much lost here.
Total Energy: Again I would guess that the total energy is conserved since the expressions for the kinetic and potential energy don't have a dependency on t (time)?