I just completed a Calculus IV course and taught myself basic Calculus of Variations, and wanted to extend some of the basic principles of optimization from planes to surfaces. The arc length minimization on a plane is the first problem presented in most CoV texts, and it seemed like a trivial problem to extend it to more complex surfaces. The problem hasn't proven to be so trivial, and I was hoping for some help.
The first surface that came to mind was a unit sphere, because I knew the answer would be one of the so-called great circles.
My first step was to parameterize the sphere using theta and phi (or $u$ and $v$), yielding:
$$\vec{T}(u,v)=\langle X(u,v), Y(u,v), Z(u,v) \rangle$$
$$\vec{T}(u,v)=\langle \cos(u)\sin(v),\sin(u)\sin(v),\cos(v) \rangle$$
And then to find the total derivative of the sphere with respect to t, hoping to express the arc length functional as the integral of the magnitude of the total derivative.
$$\frac{\vec{dT}(u(t),(v(t))}{dt} = \langle X_u\dot{u} + X_v\dot{v}, Y_u\dot{u} + Y_v\dot{v}, Z_u\dot{u} + Z_v\dot{v} \rangle$$
The actual computation is too long to bother typesetting, but the final expression for the Langrangian is $L(\xi,\dot{\xi}) = \sqrt{\cos^2(v)\dot{u}^2 + \dot{v}^2}$ and the integral is $\int_{t1}^{t2}{\sqrt{\cos^2(v)\dot{u}^2 + \dot{v}^2}} dt$.
And now I imagine I'd solve a system of Euler-Lagrange equations for u and v, which seems mathematically extremely difficult.
\begin{align} \frac{\partial L}{\partial u} - \frac{d}{dt}\frac{\partial L}{\partial \dot{u}} = 0 \\[10pt] \frac{\partial L}{\partial v} - \frac{d}{dt}\frac{\partial L}{\partial \dot{v}} = 0 \end{align}
Is there some way to solve the system of equations, or is there a better way to approach the problem? This seems to be a significant problem in differential geometry, which I have no experience in, so this may be something covered in detail in later courses.
Thanks