= Solution
In <spherical coordinates>, the <Lagrangian> is
$$
\mathcal L=\frac m2\left(\dot r^2+r^2\dot\theta^2+r^2\sin^2\theta\,\dot\phi^2\right)-V(r).
$$
A <central force> exerts zero <torque>, so the <angular momentum> vector is constant and the orbit lies in a fixed plane. Choose that plane as $\theta=\pi/2$; then $\ell=mr^2\dot\phi$ is conserved. Substituting this relation into the <energy> gives
$$
\boxed{E=\frac m2\dot r^2+V_{\rm eff}(r),\qquad V_{\rm eff}(r)=V(r)+\frac{\ell^2}{2mr^2}.}
$$
The <effective potential> reduces radial motion to a one-dimensional problem. Allowed radii have $E\geq V_{\rm eff}$, radial turning points satisfy equality, and a <circular orbit> occurs at $V_{\rm eff}'=0$. A strict local minimum gives stable radial oscillations. Eliminating $\dot\phi$ directly inside the original <Lagrangian> would require a <Routhian>; the energy derivation avoids giving the centrifugal term the wrong sign.
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