1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Geodesics on a cone

  1. Nov 30, 2011 #1
    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)

    [itex]ds^{2} = dx^{2} + dy^{2} + dz^{2}[/itex]

    [itex]ds^{2} = dr^{2} + r^{2} sin^{2}(β) dθ^{2}[/itex]

    Setting: r' = dr/dθ

    [itex]f= \sqrt{r'^{2} + r^{2} sin^{2}(β)}[/itex]

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

    [itex] \frac{\partial f}{\partial r} - \frac{d}{dθ}\frac{\partial f}{\partial r'} = 0[/itex]

    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'.

    [itex] \frac{\partial f}{\partial r} = \frac{r sin(β)}{\sqrt{r'^{2} + r^{2} sin^{2}(β) }}[/itex] ?

    In the end I like to get:

    [itex]r\frac{d^{2}r}{dθ^{2}} - 2(\frac{dr}{dθ})^{2} - r^{2}sin^{2}(β)[/itex]

  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted