How Do You Solve the Euler-Lagrange Equation for the Surface of a Cone?

[gboff21]

Homework Statement

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 into the metric to get $\dot{r}$

Homework Equations

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

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?

 

[TSny]

I've solved it to get $\ddot{\phi} + 2\frac{\dot{r}}{r}\dot{\phi} = 0$

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?

 

[gboff21]

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

 

[TSny]

That's it. Good!