Solving the Geodesic equations for a space

[Lyalpha]

I'm in an intro course and my shaky ability to solve differential equations is apparent.

How would you go about solving

\ddot{r}-r\ddot{\theta}=0

\ddot{\theta}+\frac{1}{r}\dot{r}\dot{\theta}=0

It might be obvious. They're the geodesic equations for a 2d polar coordinate system (if I'm correct).

 

[pervect]

I'm in an intro course and my shaky ability to solve differential equations is apparent.

How would you go about solving

\ddot{r}-r\ddot{\theta}=0

\ddot{\theta}+\frac{1}{r}\dot{r}\dot{\theta}=0

It might be obvious. They're the geodesic equations for a 2d polar coordinate system (if I'm correct).

The second equation is missing a factor of 2. I'm not sure exactly how you derived the geodesic equations, I'm hoping that if I remark that \Gamma^{\theta}{}_{{r}{\theta}} = \Gamma^{\theta}{}_{{\theta}{r}} = \frac{1}{r} you'll see what you omitted. If not, you'll need to write down your derivation.

After this correction, the second equation is equivalent to saying a geodesic conserves angular momentum, i.e

\frac{d}{dt} \left( r^2 \, \dot{\theta} \right) = 0

If you expand this via the chain rule you'll get

r^2 \ddot{\theta} + 2\,r\,\dot{r}\,\dot{\theta} = 0

which is equivalent to the corrected version of eq 2 (just divide both sides by r^2). This should allow you to separate the variables and solve the equatinos without too much difficulty by solving for \dot{\theta} as a function of r and the conserved, constant angular momentum - which you can also regard as a constant of integration.