- #1
foxjwill
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Homework Statement
I'm trying (on my own) to derive the geodesic for a sphere of radius a using the geodesic equation
\ddot{u}^i + \Gamma^i_{jk}\dot{u}^j\dot{u}^k,
where \Gamma^i_{jk} are the Christoffel symbols of the second kind, \dot{u} and \ddot{u} are the the first and second derivatives w.r.t. the parameter t, and the intrinsic coordinates u^1=\phi and u^2=\theta of the sphere are given by\left\{\begin{aligned}<br />
x &= a\cos(\theta)\sin(\phi)\\<br />
y &= a\sin(\theta)\sin(\phi)\\<br />
z &= a\cos(\phi).\end{aligned}\right.
Homework Equations
\Gamma^i_{jk} = \frac{1}{2}g^{i\ell}(g_{j\ell,k} + g_{k\ell,j} - g_{jk,\ell}),
where g_{ij,k}=\frac{\partial g_{ij}}{\partial u^k} and g_{ij} is the metric tensor of the sphere.The Attempt at a Solution
I've already shown that ds^2=a^2d\phi^2 + a^2\cos^2(\phi)d\theta^2, where s is arclength, and from this I got that the only two non-zero Christoffel symbols of the second kind are
\Gamma^1_{22} = \sin(\phi)\cos(\phi) \qquad\text{and}\qquad \Gamma^2_{21} = -\tan(\phi).
Plugging these into the geodesic equation, I got the system of ODEs
\left\{\begin{aligned}<br />
\ddot\phi + \sin(\phi)\cos(\phi)\dot\theta^2 &=0\\<br />
\ddot\theta - \tan(\phi)\dot\theta\dot\phi &=0<br />
\end{aligned}\right.
Dividing the first equation by the differential d\theta^2 and the second by d\phi^2 produces \left\{\begin{aligned}<br />
\frac{d^2\phi}{d\theta^2} + \sin(\phi)\cos(\phi) &= 0\\<br />
\frac{d^2\theta}{d\phi^2} - \tan(\phi)\frac{d\theta}{d\phi} &= 0.<br />
\end{aligned}\right.
Solving the latter, I get
\theta=c_1\ln(\sec(\phi)+\tan(\phi)) + c_2.
Differentiating and then solving for \phi', we have\begin{align*}<br />
\phi' &= \frac{\cos(\phi)}{c_1}.<br />
\end{align*}
So,\begin{align*}<br />
\phi'' &= -\frac{\sin(\phi)\phi'}{c_1} = -\frac{\sin(\phi)\cos(\phi)}{c_1^2}.<br />
\end{align*}
However, this would only satisfy the first equation in the ode system if c_1=\pm1. But then the set of geodesics (i.e. the great circles) would have only one degree of freedom which doesn't seem right to me. Did I do something wrong?