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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]

    Thanks.
     
  2. jcsd
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