# Geodesics on a cone

1. Nov 30, 2011

### brainslush

1. The problem statement, all variables and given/known data
We shall find the equation for the shortest path between two points on a cone, using the Euler-Lagrange equation.

2. Relevant equations

3. The attempt at a solution

x = r sin(β) cos(θ)
y = r sin(β) sin(θ)
z = r cos(β)

dx = dr sin(β) cos(θ) - r sin(β) sin(θ) dθ
dy = dr sin(β) sin(θ) + r sin(β) cos(θ) dθ
dz = dr cos(β)

$ds^{2} = dx^{2} + dy^{2} + dz^{2}$

$ds^{2} = dr^{2} + r^{2} sin^{2}(β) dθ^{2}$

Setting: r' = dr/dθ

$f= \sqrt{r'^{2} + r^{2} sin^{2}(β)}$

Is this correct so far? Now I like to apply the Euler-Lagrange equations

$\frac{\partial f}{\partial r} - \frac{d}{dθ}\frac{\partial f}{\partial r'} = 0$

But I'm getting into trouble solving this:
What are the solutions for the single differentials? I'm unsure how d/dr' effects r, d/dr effects r' and d/dθ r'.

$\frac{\partial f}{\partial r} = \frac{r sin(β)}{\sqrt{r'^{2} + r^{2} sin^{2}(β) }}$ ?

In the end I like to get:

$r\frac{d^{2}r}{dθ^{2}} - 2(\frac{dr}{dθ})^{2} - r^{2}sin^{2}(β)$

Thanks.