Geodesic curves for an ellipsoid

[tossin]

Homework Statement

The problem asks to find the shortest distance between two points on Earth, assuming different equatorial and polar radii i.e. the coordinates are represented as:

x = a*cos(theta)*sin(phi)
y = a*sin(theta)*sin(phi)
z = b*cos(phi)

Homework Equations

The Attempt at a Solution

I've already done this for a sphere and it was pretty straightforward: Find the Christoffel symbols, then plug them into the geodesic equations and solve the system of ODEs for the geodesic curves. However, now I'm stuck at the beginning because I'm unsure how to treat a & b. Previously, for spherical coordinates:

x = r*cos(theta)*sin(phi)
y = r*sin(theta)*sin(phi)
z = r*cos(phi)

I calculated the metric tensor treating r, theta, and phi as variables, but when plugging into the geodesic equations, I assumed r was constant so that all derivatives of r were zero. Currently, I'm not sure whether I need to calculate new Christoffel symbols for the ellipsoid or whether I simply need to keep the derivatives of r in the geodesic equations. If the former is true, do I treat a and b as separate variables when calculating the metric tensor? I could use a little direction here. Thanks.

 

[edmundfo]

The numbers a and b are constants, just as r was in the spherical case. As the ellipsoid is a surface, you have two independent variables that you need to differentiate with respect to, i.e. theta and phi.

The Christoffel symbols are therefore on the form \Gamma_{\theta \theta}^{\theta}, \Gamma_{\theta \theta}^{\phi} etc. Several of these will be zero due to symmetry around the z-axis.