In this problem we try to find the critical points (ideally minima) of the energy functional on curves in the hyperbolic plane. Generally, for any Riemannian manifold (U, g) the energy functional is given by $$E(\gamma) = \frac{1}{2} \int_I g_{\gamma (t)} (\gamma '(t), \gamma '(t))dt$$
for curves γ : I → U.
(i) For the upper half plane model of hyperbolic space H$^2$ = { z = x+iy $\in$ R; y>0} with riemannian metric (ds$^2$ = $\frac{dx^{2} + dy^{2}}{y^{2}}$) calculate the ODE a curve γ : I → H$^2$ has to satisfy in order to be a critical point of the energy functional under compactly supported variations.
(ii) Recall that SL(2, R) acts by isometries on H$^2$ via A · z = $\frac {az + b} {cz + d}$ for A ∈ SL(2, R). Show that the energy does not change when a curve γ is moved by an isometry to A · γ.
For...
i) Would I simply take the Euler Lagrange equations and say that the point, $\gamma$ is a solution to the E-L equations? I'm not sure how to solve this, isn't there only one equation which, when satisfied denotes a critical point? I.e., the Euler-Lagrange Equations?
ii) Here I would basically take $\gamma$ and act on it by A so that it'd have $\gamma$ where z is in the definition but idk how exactly to show the energy doesn't change. I'm not sure what form $\gamma$ would even have so I'm not sure I understand how to show this. I know this is basically saying we want to show E($\gamma$) is invariant under multiplication and addition of its solutions but can't quite get started.
First let me answer your second question. By definition isometry $A$ acts on a Riemannian manifold in the following way. Let $p\in M$ and $v,w\in T_pM$ (tangent space of $M$). Then $$g_p(v,w)=g_{A(p)}(D_pA(v),D_pA(w)).$$ Now in $\mathbb{H}^2$ this is equivalent to the fact that $d(z_1,z_2)=d(Az_1,Az_2)$ where $d$ is the hyperbolic distance (one of the way to prove this is to use the explicit formulas of hyperbolic distances between two points). Now use the above formula and chain rule to show that $E(A(\gamma))=E(\gamma)$.
Now coming back to your first question. The main computational difficulty in EL equation is to compute the integrand which is the metric in this case. And so you can choose two paths. One is through crude computation as you have suggested. There is another way from which this whole question is coming from. As a general question this could be asked for any Riemannian manifold and try to get a characterization of the critical points. It turns out that the critical points are exactly the geodesics (modulo some generic conditions on the manifold but you can ignore them in this context). This whole story has a name: The first variation formula.