I've just came up with this coordinate system (don't know if it exists or it has been previously invented): $$ \mathbf r = (r\cos{\theta}\cos{\varphi}, \; r\sin{\varphi}, \; r\sin{\theta}\cos{\varphi}) $$ This are the coordines of a vector confined to a plane with a normal that is allowed to rotate around the $y$ axes. Then the rest of the coordinates variables are found by creating a "local" system polar coordinates in that plane.
Now, all orbital-like motion would simply be described by $\varphi$. For example an orbit of radius $1$, with $\theta = \pi/2$, would be written as: $$ \varphi(t) = \omega t + \varphi_0 \\ \mathbf r = (0, \sin(\omega t + \varphi_0), \cos(\omega t + \varphi_0)) $$
This could be useful for describing circular-ish trajectories. For example, geodesics on a sphere. Is easily to see that: $$ s = \int_\gamma \sqrt{ dr^2 + r^2 d\varphi^2 + r^2 \cos^2 (\varphi) d \theta^2 } $$ By appliying the Euler-Lagrange equations I get the following (note that $r'(t) = 0$): $$ \varphi''(t) = -\cos(\varphi(t)) \sin(\varphi(t)) \theta'(t)^2 $$ But I was expecting something like: $$ \varphi''(t) = 0 $$
My question is: Is it possible to get the equation I described before, using Euler-Lagrange Equations. Is a solution like: $\varphi(t) = \omega t + \varphi_0$, to be expected from this?