Geodesic Equation/2d Surface Embedded in 3d Euclidean Space

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Theorem

Let $\sigma:U \subset \R^2 \to V \subset \R^3$ be a smooth surface specified by a vector-valued function:

$\displaystyle \mathbf r = \map{\mathbf r} {u, v}$


Then a geodesic of $\sigma$ satisfies the following system of differential equations:

$\displaystyle \frac {E_u u'^2 + 2 F_u u' v' + G_u v'^2} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \dfrac \d {\d t}\frac {2 \paren {E u' + F v'} } {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$
$\displaystyle \frac {E_v u'^2 + 2 F_v u' v' + G_v v'^2} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \dfrac \d {\d t}\frac {2 \paren {F u' + G v'} } {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$

where $E, F, G$ are the functions of the first fundamental form:

$\displaystyle E = {\mathbf r}_u \cdot {\mathbf r}_u, F = {\mathbf r}_u \cdot {\mathbf r}_v, G = {\mathbf r}_v \cdot {\mathbf r}_v$

Proof

A curve on the surface $\mathbf r$ can be specified as $u = \map u t$, $v = \map v t$

The arc length between the points corresponding to $t_0$ and $t_1$ equals

$\displaystyle J \sqbrk {u, v} = \int_{t_0}^{t_1} \sqrt {E u'^2 + 2 F u'v' + G v'^2} \rd t$

The following derivatives will appear in Euler's Equations:

\(\displaystyle \dfrac {\partial} {\partial u'} \sqrt {E u'^2 + 2 F u'v' + G v'^2}\) \(=\) \(\displaystyle \frac 1 {2 \sqrt {E u'^2 + 2 F u'v' + G v'^2} } \dfrac {\partial} {\partial u'} \paren {E u'^2 + 2 F u'v' + G v'^2}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 E u' + 2 F_u u'} {2 \sqrt {E u'^2 + 2 F u'v' + G v'^2} }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {E u' + F u'} {\sqrt {E u'^2 + 2 F u'v' + G v'^2} }\)
\(\displaystyle \dfrac {\partial} {\partial u} \sqrt {E u'^2 + 2 F u'v' + G v'^2}\) \(=\) \(\displaystyle \frac 1 {2 \sqrt {E u'^2 + 2 F u'v' + G v'^2} } \dfrac {\partial} {\partial u} \paren{E u'^2 + 2 F u'v' + G v'^2}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {E_u u'^2 + 2 F_u u' v' + G_u v'^2} {2 \sqrt {E u'^2 + 2 F u'v' + G v'^2} }\)

Analogous relations hold for derivatives with respect to $v$ and $v'$.

Euler's Equation together with the results above yield the geodesic equations.


$\blacksquare$

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