- #1

lstellyl

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## Homework Statement

Consider the cylinder S in R

^{3}defined by the equation [tex]x^2+y^2=a^2[/tex]

(a). The points [tex] A=(a,0,0) \: and \: B = (a \cos{\theta}, a \sin{\theta}, b)[/tex] both lie on S. Find the geodesics joining them.

(b). Find 2 different extremals of the length functional joining [tex] A=(a,0,0) and C = (a, 0 2 \pi, b)[/tex]. How many extremals join A and C?

## Homework Equations

Euler-Lagrange generalized equations:

[tex]\frac{\partial}{\partial x_i } \{F+\sum_k \lambda_j (t) G_j \} - \frac{d}{dt} \{ \frac{\partial}{\partial x'_i } \{ F+\sum_k \lambda_j (t) G_j \} \} [/tex]

## The Attempt at a Solution

using [tex] F= \sqrt{x'^2+y'^2+z'^2} \: and \: G= x^2+y^2[/tex] and assuming that the geodesic path [tex] \gamma [/tex] was parameterized by arc length, meaning that |u'(t)|=1, i was able to get the following equations:

[tex] \lambda (t) x(t) = x''(t)[/tex]

[tex]\lambda (t) y(t) = y''(t)[/tex]

[tex]z'(t) = C [/tex] where C is a constant.

although for some reason mathematica wouldn't solve this for me... these are all simple harmonic oscillator equations... (hopefully, i say that and actually solve them correctly...)

forming the equations with constants and solving for the boundary conditions, i found the following using Mathematica's Solve function:

[tex]

x(t) = A e^{k t} + D[/tex]

[tex]y(t) = B e^{k t} + E[/tex]

[tex]z(t) = \frac{b}{L} \:\: (trivial)[/tex]

where

[tex]A = - \frac{2 a \sin{\theta/2}^2}{e^{ k L} - 1}[/tex]

[tex]B = \frac{ a \sin{\theta}}{e^{ k L} - 1}[/tex]

[tex]D = \frac{ a ( e^{k L} - \cos{\theta})}{e^{ k L} - 1}[/tex]

[tex]E = B[/tex]

where k = Sqrt[lambda] and L is the length of the curve (since it is parameterized by arc-length)

...

This solution doesn't look right to me, and I don't want to move on to part 2 with this incorrect. I feel like i am doing something implicitly wrong and should maybe be canceling out lambdas or something... because I still have a lambda in those constant values...

any help or direction would be appreciated... it has not been a good day for me and i am having trouble focusing.