Solving geodesic equations on the surface of a sphere

Thread starter WannabeNewton
Start date Jan 11, 2011
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Geodesic Sphere Surface

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SUMMARY

The discussion focuses on solving geodesic equations on the surface of a sphere with radius 'a' using spherical coordinates. The derived equations for the angles T and P are: d²T/ds² - (sinTcosT)(dP/ds)² = 0 and d²P/ds² - 2cotT(dT/ds)(dP/ds) = 0. The user seeks assistance in finding a particular solution to these differential equations, indicating a need for guidance in solving such mathematical problems.

PREREQUISITES

Understanding of spherical coordinates and their representation.
Familiarity with differential equations and their solutions.
Knowledge of geodesic equations in differential geometry.
Basic proficiency in calculus, particularly in second-order derivatives.

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Study methods for solving second-order differential equations.
Explore the concept of geodesics in differential geometry.
Learn about specific solutions to differential equations, such as particular and general solutions.
Investigate numerical methods for approximating solutions to complex differential equations.

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Students and researchers in mathematics, physics, and engineering who are working on problems involving geodesics and differential equations on curved surfaces.

Jan 11, 2011

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.