TITLE: Geodesics of a Sphere in Cartesian Coordinates
QUESTION [5 upvotes]: I want to minimize $I = \int |\dot{x}|^2 dt$
subject to the constraint $|x|^2=1$ (sphere)
which gives an Euler equation of $\lambda x - \ddot{x} = 0$.
I have to show that the Euler equation is actually $|\dot{x}|^2 x - \ddot{x} = 0$.
Is it right to assume that $\lambda=|\dot{x}|^2$ simply by the fact that it minimizes $I^* = \int |\dot{x}|^2- \lambda (|x|^2-1) dt$ which is $\geq 0$, so the $\lambda$ that minimizes $I^*$ is $|\dot{x}|^2$?
If I then try to integrate the Euler equation, then I get a SHM equation:
$x1= A \cos(|\dot{x}| t - C)$ where A, C are constants
and similarly for x2, x3
But how do I combine them to give the equation of a great circle, since I don't know the $C$'s?
Thank you for any enlightenment!

REPLY [4 votes]: Let the initial data be given by:
$x(t=0) = x_0,\quad\dot{ x}(t=0) = v_0  $
The equations of motion of the constrained Lagrangian define a simple harmonic motion. After the substitution of the initial data, the solution has the form
$x = x_0 \cos(\sqrt{\lambda} t) + \frac{ v_0}{\sqrt{\lambda}} \sin(\sqrt{\lambda} t) $
The requirement that at all times $t$
$|x|^2 = 1$
Implies (by the substitution of the solution into the constraint equation)
$|x_0|^2 = 1,  v_0. x_0 = 0, | v_0|^2 = \lambda$
Therefore (again by substitution)
$|\dot{x}(t)|^2 = \lambda = |v_0|^2$
Therefore 
$\lambda = |\dot{ x}(t)|^2$
To see that the trajectory is a great circle, we notice that it lies on the plane (passing through the sphere's center): 
$ w. x(t)= 0$
Where: 
$\vec w = x_0 \times  v_0$