Solving geodesic equations on the surface of a sphere

In summary, the conversation discusses finding the geodesics on the surface of a sphere of radius a. The first step is to write the geodesic equations for the spherical coordinates, which results in two equations for T and P. The next step is to find a particular solution for these equations, which the person is struggling with and hoping for guidance.
  • #1
WannabeNewton
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Homework Statement



Find the geodesics on the surface of a sphere of radius a by:
(a) writing the geodesic equations for the spherical coordinates given by:
x = rsinTcosP
y = rsinTsinP
z = rcosT

for T and P(the r - equation can be ignored as a = constant);
(b) exhibit a particular solution of these two equations; and (c) generalize (b).

Homework Equations


Geodesic equation in general form (sorry don't know how to use LaTeX)


The Attempt at a Solution


Ok so I did part (a) and ended up with the equations for T and P as follows:

d^2T / ds^2 - (sinTcosT) * (dP / ds)^2 = 0
d^2P / ds^2 - 2cotT * (dT / ds) * (dP / ds) = 0

I am terrible at solving differential equations and basically have no idea what to do from here to find a particular solution. I was hoping someone could guide me through it.
 
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  • #2
I tried to solve the first equation for dP / ds and substitute into the second but I ended up with a mess. Any help is greatly appreciated!
 

FAQ: Solving geodesic equations on the surface of a sphere

What are geodesic equations?

Geodesic equations are mathematical formulas that describe the shortest distance between two points on a curved surface, such as a sphere.

Why is it important to solve geodesic equations on the surface of a sphere?

Solving geodesic equations on the surface of a sphere is important in various fields of study, such as geology, geography, and astronomy. It allows us to calculate the shortest path between two points on Earth's surface, determine the curvature of the Earth, and navigate accurately using GPS technology.

What is the process for solving geodesic equations on the surface of a sphere?

The process involves using differential geometry and calculus to derive the geodesic equations for a sphere. These equations can then be solved using various numerical methods, such as the Runge-Kutta method, to determine the shortest distance between two points on the sphere.

How does the radius of the sphere affect the geodesic equations?

The radius of the sphere has a direct impact on the geodesic equations. As the radius increases, the curvature of the sphere decreases, and the geodesic equations become simpler to solve. Conversely, as the radius decreases, the curvature increases, and the equations become more complex.

Can geodesic equations be applied to other curved surfaces besides a sphere?

Yes, geodesic equations can be applied to any curved surface, such as an ellipsoid or a hyperboloid. However, the equations will differ depending on the specific geometry of the surface.

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