How does $r$ depend on $\varphi$ in the Schwarzschild metric? I am confused about the Wikipedia derivation of the equation

for geodesic motion in the Schwarzschild spacetime. The derivation of this equation involves variation with respect to the longitude $\varphi$ only then the variation with respect to the time $t$ only.
My question is how can we vary with respect to $\varphi$ only when $r$ clearly depends on $\varphi$ since $dr/d\varphi\neq0$ so a variation of $\varphi$ leads to a variation of $r$ also?
 A: Before varying a Lagrangian or an action, all variables are considered independent. The classical equations of motion then correspond to the subset of possible trajectories that leave the action stationary. That is, before varying the action, $r$ and $\varphi$ are indeed independent coordinates. They only depend on one another after you require that the action is stationary.
To be a little more precise, the action $S$ is a functional which takes possible trajectories and returns a real number. Let $\Gamma$ (the "phase space") be the set of all possible trajectories. Within $\Gamma$, $r$ and $\varphi$ are completely independent. Now, let $\Sigma$ be the subset of $\Gamma$ that minimizes the action (we call $\Sigma$ the "shell"). $\Sigma$ simply consists of all solutions to the classical equations of motion. In general, coordinates on $\Sigma$ will have some nontrivial relation to each other.
This is exactly like how the coordinates $x,y,z$ on $\mathbb{R}^3$ are completely independent, but if I were to embedd a sphere into $\mathbb{R}^3$, then, on this subset, $x,y,z$ depend on each other in a nontrivial way.
A: $\let\lam=\lambda \let\th=\vartheta \let\phi=\varphi 
\def\cS{{\cal S}} \def\D#1#2{{d#1 \over d#2}}$
Your question can be answered from a more general viewpoint, without thinking of Schwarzschild metric and its geodesics.
You have a 4D spacetime $\cS$ with coordinates $t$, $r$, $\th$, $\phi$. A curve in $\cS$ is meant as a mapping $\Bbb R\to\cS$ represented by parametric equations
$$t = t(\lam) \qquad r = r(\lam) \qquad \th = \th(\lam) \qquad \phi = \phi(\lam) \tag1$$
$\lam$ being a real parameter you may choose at will.
If the curve is the worldline of a moving body $\lam$ is usually identified with proper time $\tau$ but this isn't mandatory. Any other parameter can be used if it's in a one-to-one correspondence with the original one.
So you may e.g. assume as a parameter $\phi$ if in the motion it's an increasing function of $\tau$. If ODE's (with independent variable $\tau$) are given for functions (1) you'll use the chain rule to transform them into ODE's with independent variable $\phi$. E.g.
$$\D r\tau = \D r\phi\,\D\phi\tau$$
wherefrom
$$\D r\phi = {dr/d\tau \over d\phi/d\tau}.$$
