# Metric for a surface of a cone

1. Feb 16, 2014

### gboff21

1. The problem statement, all variables and given/known data
The metric for this surface is $ds^2 = dr^2 + r^2\omega^2d\phi^2$, where $\omega = sin(\theta_0)$.
Solve the Euler-Lagrange equation for phi to show that $\dot{\phi} = \frac{k}{\omega^2r^2}$. Then sub back in to the metric to get $\dot{r}$

2. Relevant equations
$L = 1/2 g_{ab} \dot{x}^a \dot{x}^b$

3. The attempt at a solution
I've solved it to get $\ddot{\phi} + 2\frac{\dot{r}}{r}\dot{\phi} = 0$
and
$\ddot{r} - r\omega^2\dot{\phi}^2 = 0$

So how on earth do you get that answer?

2. Feb 16, 2014

### TSny

Note how you got this equation. Back up a step where you must have had $\frac{d}{dt} (\rm { expression}) = 0$

What can you conclude about the expression?

3. Feb 16, 2014

### gboff21

Ok I get it! $d/dt (\dot{\phi} r^2 \omega^2)=0$. So $\dot{\phi} = k/(r^2\omega^2)$
Thanks!!

4. Feb 16, 2014

### TSny

That's it. Good!