Note: Graph Geodesics Variational (+Gravity)

This is the second in a series of posts on multiple ways to compute the geodesic equation for surfaces in $\mathbb{R}^3$ described as the graph of a function $z=f(x,y)$. Ive ended up re-doing these calculations several times in the past while writing various programs, so want to record them in one spot for my future reference (and hopefully, the benefit of others!). This time we will approach the problem via the calculus of variations where we minimize the energy (or total integral of $\|\gamma^\prime\|^2$ over the curve).

We begin by fixing the same notation as before.

The Euler Lagrange Equations

Generalization: Motion under Uniform Gravity