What is the argument purely from Lagrangian formalism (without using forces), that the Lagrangian of a mass in a central force would exhibit motion in 2 dimensions as opposed to 3 dimensions?
Details:
The Lagrangian in central force must be:
$L = \frac{1}{2}m\dot{r}^2+\frac{1}{2}mr^2\dot{\theta}^2+\frac{1}{2}m(r\sin\theta)^2\dot{\phi}^2-U(r)$
I saw in several notes, people eliminate the $\phi$ coordinate using the fact that angular momentum $\vec{L}$ is conserved, and hence the motion must happen in 2d. But I feel that this uses elements of Newtonian Mechanics. Is there a purely lagrangian argument using the fact that the potential is that of a central force and hence there must only be two generalized coordinates instead of two?
