I'm trying to set up a functional that outputs the length of a curve on the paraboloid surface. I then want to write the Euler-Lagrange equation for the geodesic.
The surface is given in the cylindrical coordinate system $(r, \phi, z)$ with $z = \alpha r^2$.
$$ J[y] = L = \int_1^2 dl = \int_1^2 \sqrt{r^2(d\phi)^2 + (dr)^2 + (dz)^2}$$
$z = \alpha r^2 \Rightarrow dz = 2\alpha r dr.$ Therefore,
$$ J[y] = L = \int_1^2 \sqrt{r^2(d\phi)^2 + (dr)^2 + (2\alpha r)^2(dr)^2} $$ $$ = \int_1^2 \sqrt{r^2(d\phi)^2 + (1 + (2\alpha r)^2)(dr)^2} $$ $$ = \int_0^{2\pi} d\phi \underbrace{\sqrt{r^2 + (1 + (2\alpha r)^2) (\frac{dr}{d\phi})^2}}_\text{F}$$
The Euler-Lagrange equation: $$\frac{d}{d\phi}\frac{\partial F}{\partial(\frac{dr}{d\phi})} - \frac{\partial F}{\partial r} = 0$$
$$\frac{d}{d\phi}\frac{\partial F}{\partial(\frac{dr}{d\phi})} = \frac{d}{d\phi}\left[\frac{(1 + (2\alpha r)^2)\frac{dr}{d\phi}}{\sqrt{r^2 + (1 + (2\alpha r)^2) (\frac{dr}{d\phi})^2}} \right]$$
$$ \frac{\partial F}{\partial r} = \frac{r + (4\alpha^2 r)(\frac{dr}{d\phi})^2}{\sqrt{r^2 + (1 + (2\alpha r)^2) (\frac{dr}{d\phi})^2}} $$
I have the following questions:
- Are the functional and the Euler-Lagrange equation correct?
- Which conservation law can I use to obtain a simpler form of the equation for the geodesic?
