Geodesics and their linear equations

[Biest]

Hi,

So I am going over on how to find a geodesic from any metric, esp. on a 2-sphere. I have been looking at my lecture notes and am confused as to how my professor solves for the equation in terms of the variables, i.e. (\theta , \phi). If i use the 2-sphere as an example here where the geodesic is given by:

\ddot{\phi} = -2\cot \theta \dot{\phi} \dot{\theta}

\ddot{\theta} = \sin \theta \cos \theta \dot{\phi}^2

So from the metric we get a first dervative

1 = \dot{\theta}^2 + \sin^2 \theta \dot{\phi}^2

My prof then goes on and simply states that

\frac{1}{\sin^2 \theta} \frac{d}{d\tau} (\sin^2 \theta \dot{\phi}) = 0

Where does the last equation come from... in lecture he simply stated it and moved on from there with

(\sin^2 \theta \dot{\phi}) = l = constant

I really thinking i got a bit rusty on this since mechanics lies two years in the past.Thank you very much any help in advance.

Cheers,

Biest

 

[Rainbow Child]

Multiply the first equation with \sin^2\theta and you will arrive to the wanted result.

 

[Biest]

Thanks... I forgot to add one more question... What do we do when we have three geodesics? Just choose one and solve from there?

 

[Rainbow Child]

Finding the geodesics is a really hard problem. In general you can not find a closed form solution for them. But if you have a killing field, i.e. \nabla_\alpha\,\xi_\beta+\nabla_\beta\,\xi_\alpha=0 then you have a constant along the geodesic, i.e. \xi_\alpha\,u^\alpha=C where u^\alpha is tangent to the geodesic.

 

[Biest]

Finding the geodesics is a really hard problem. In general you can not find a closed form solution for them. But if you have a killing field, i.e. \nabla_\alpha\,\xi_\beta+\nabla_\beta\,\xi_\alpha=0 then you have a constant along the geodesic, i.e. \xi_\alpha\,u^\alpha=C where u^\alpha is tangent to the geodesic.

I know it is hard... i had to derive the conditions for homework and was stuck on the derivative of g_{\mu \nu} anyway. At the moment we have just done polar coordinates and the 2-sphere. and now we just moved into particle orbits, which i have to work through as well cause i am trying how the Killing vector works there. I am starting to get it, but it is taking me a while already.