Geodesic Equation/2d Surface Embedded in 3d Euclidean Space/Cylinder
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Theorem
Let $\sigma$ be the surface of a cylinder.
Let $\sigma$ be embedded in 3-dimensional Euclidean space.
Let $\sigma$ be parameterised by $\tuple {\phi, z}$ as
- $\mathbf r = \tuple {a \cos \phi, a \sin \phi, z}$
where
- $a > 0$
and
- $z, \phi \in \R$
Then geodesics on $\sigma$ are of the following form:
- $z = C_1 \phi + C_2$
where $C_1, C_2$ are real arbitrary constants.
Proof
From the given parametrization it follows that:
\(\ds E\) | \(=\) | \(\ds \mathbf r_\phi \cdot \mathbf r_\phi\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {-a \sin \phi, a \cos \phi, 0} \cdot \tuple {-a \sin \phi, a \cos \phi, 0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a^2\) | ||||||||||||
\(\ds F\) | \(=\) | \(\ds \mathbf r_\phi \cdot \mathbf r_z\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {-a \sin \phi, a \cos \phi, 0} \cdot \tuple {0, 0, 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds G\) | \(=\) | \(\ds \mathbf r_z \cdot \mathbf r_z\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {0, 0, 1} \cdot \tuple {0, 0, 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
where $E, F, G$ are the functions of the first fundamental form.
Furthermore, all derivatives of $E, F, G$ with respect to $\phi$ and $z$ vanish.
Then geodesic equations read:
- $\dfrac \d {\d t} \dfrac {a^2 \phi'} {\sqrt {a^2 \phi'^2 + z'^2} } = 0$
- $\dfrac \d {\d t} \dfrac {z'} {\sqrt {a^2 \phi'^2 + z'^2} } = 0$
Integrate these differential equations once:
- $\dfrac {a^2 \phi'} {\sqrt {a^2 \phi'^2 + z'^2} } = b_1$
- $\dfrac {z'} {\sqrt {a^2 \phi'^2 + z'^2} } = b_2$
where $b_1, b_2$ are real arbitrary constants.
Divide the first equation by the second one:
- $\dfrac {a^2 \phi'} {z'} = \dfrac {b_1} {b_2}$
To solve this in terms of $z$ as a function of $\phi$, define:
- $C_1 = \dfrac {a^2 b_2} {b_1}$
and use the chain rule:
- $\dfrac {\d z} {\d \phi} = C_1$
Integration with respect to $\phi$ yields the desired result.
In other words, geodesics are helical lines.
$\blacksquare$
Sources
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 2.9$: The Fixed End Point Problem for n Unknown Functions