Solving Geodesics on a Cone Using Euler-Lagrange

[brainslush]

Homework Statement

We shall find the equation for the shortest path between two points on a cone, using the Euler-Lagrange equation.

Homework Equations

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.

 

[mighty2000]

Thank you for your post. Your attempt at solving the problem using the Euler-Lagrange equation is a good start. However, there are a few things that need to be clarified.

Firstly, your equations for x, y, and z are correct as they represent the parametric equations for a cone. However, your expressions for dx, dy, and dz are incorrect. They should be:

dx = (dr/dθ)sin(β)cos(θ) + rcos(β)cos(θ)dθ - rsin(β)sin(θ)dθ
dy = (dr/dθ)sin(β)sin(θ) + rcos(β)sin(θ)dθ + rsin(β)cos(θ)dθ
dz = (dr/dθ)cos(β) - rsin(β)dθ

This is because you need to use the chain rule when taking the derivatives with respect to θ.

Secondly, your expression for ds^2 is incorrect. It should be:

ds^2 = (dx)^2 + (dy)^2 + (dz)^2
= (dr)^2 + r^2(sin^2(β) + cos^2(β))dθ^2
= (dr)^2 + r^2dθ^2

This is because the terms involving sin(β) and cos(β) cancel out when squared.

Now, to apply the Euler-Lagrange equation, we need to minimize the functional:

S = ∫√(ds^2) dθ = ∫√((dr)^2 + r^2dθ^2) dθ

We can write this as:

S = ∫f(r, r') dθ

where f(r, r') = √((dr)^2 + r^2dθ^2). Now, using the Euler-Lagrange equation, we have:

∂f/∂r - d/dθ(∂f/∂r') = 0

∂f/∂r = 0 (since f does not depend on r explicitly)

d/dθ(∂f/∂r') = d/dθ(r^2r'/√((dr)^2 + r^2dθ^2))

= r^2(d/dθ(r'/√((dr)^2 + r^2