Length of a curve on a sphere using calc of variations

[waddles]

I have an assignment question here using calculus of variations.

Question:
Show that the length ℓ of a curve on the surface of a sphere of radius a is

ℓ=int sqrt(theta_dot^2+(sin(theta))^2*phi_dot^2) dt

Hence show that the shortest distance between two points on the sphere is given by part of a
great circle, i.e., a circle with origin at the centre of the sphere.

I used spherical polar coordinates and managed to get as far as deriving the above integral, and then tried to use Lagrange's equations to determine eqns of motion for theta and phi but got some really nasty equations out. I also wasn't sure what initial values to use.

Any help would be appreciated.

 

[Winzer]

Waddles,

Take a look at: http://en.wikipedia.org/wiki/Line_element.
Now you should be able to find the correct element. Perform some simplifying.
Don't forget, you have to parameterize your line element.