Geodesics and their linear equations

In summary, the conversation discusses finding a geodesic from any metric, particularly on a 2-sphere. The geodesic equations are given, and the process of solving for the equation in terms of the variables (\theta , \phi) is discussed. The last equation is derived by multiplying the first equation with \sin^2\theta. It is also mentioned that finding geodesics is a challenging problem, but if a killing field is present, there is a constant along the geodesic. The difficulty of finding geodesics is acknowledged and the speaker is currently working on understanding the concept.
  • #1
Biest
67
0
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. [tex] (\theta , \phi) [/tex]. If i use the 2-sphere as an example here where the geodesic is given by:

[tex] \ddot{\phi} = -2\cot \theta \dot{\phi} \dot{\theta} [/tex]

[tex] \ddot{\theta} = \sin \theta \cos \theta \dot{\phi}^2 [/tex]

So from the metric we get a first dervative

[tex] 1 = \dot{\theta}^2 + \sin^2 \theta \dot{\phi}^2 [/tex]

My prof then goes on and simply states that

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

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

[tex] (\sin^2 \theta \dot{\phi}) = l = constant [/tex]

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
 
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  • #2
Multiply the first equation with [itex]\sin^2\theta[/itex] and you will arrive to the wanted result.
 
  • #3
Thanks... I forgot to add one more question... What do we do when we have three geodesics? Just choose one and solve from there?
 
  • #4
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. [itex]\nabla_\alpha\,\xi_\beta+\nabla_\beta\,\xi_\alpha=0[/itex] then you have a constant along the geodesic, i.e. [itex]\xi_\alpha\,u^\alpha=C[/itex] where [itex]u^\alpha[/itex] is tangent to the geodesic.
 
  • #5
Rainbow Child said:
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. [itex]\nabla_\alpha\,\xi_\beta+\nabla_\beta\,\xi_\alpha=0[/itex] then you have a constant along the geodesic, i.e. [itex]\xi_\alpha\,u^\alpha=C[/itex] where [itex]u^\alpha[/itex] 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 [itex] g_{\mu \nu} [/itex] 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.
 

Related to Geodesics and their linear equations

What is a geodesic?

A geodesic is the shortest path between two points on a curved surface, such as a sphere or a curved plane. It is equivalent to a straight line in geometry.

How are geodesics related to linear equations?

Geodesics can be described using linear equations, specifically through the use of geodesic coordinates. These coordinates represent the points on a curved surface as solutions to a set of linear equations.

What are the applications of geodesics and their linear equations?

Geodesics and their linear equations have various applications in fields such as mathematics, physics, and engineering. They are used in the study of curved surfaces, navigation, and optimization problems.

Can geodesics be calculated on any type of curved surface?

Yes, geodesics can be calculated on any type of curved surface, including spheres, ellipsoids, and hyperbolic surfaces. The specific equations used to calculate geodesics may differ depending on the curvature of the surface.

How do geodesics and their linear equations relate to the theory of relativity?

In the theory of relativity, geodesics play a crucial role in describing the paths of particles and light in a curved spacetime. The equations used to calculate geodesics are essential in understanding the effects of gravity on the motion of objects.

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