= Radial effective potential on a surface of revolution
{title2=$U_h(r)=h^2/(2r^2)+gf(r)$}
A particle constrained to a stationary smooth <surface of revolution> $z=f(r)$ under uniform <gravity> conserves $h=r^2\dot\theta$ and <mechanical energy> per unit <mass>. Substitute $\dot z=f'(r)\dot r$ and $\dot\theta=h/r^2$ into that <energy> to obtain
$$
\frac12[1+f'(r)^2]\dot r^2+U_h(r)=E.
$$
This proves the radial reduction. Simple roots of $E-U_h$ are <turning points>; a stationary point $U_h'(r_0)=0$ supports an exact <circular orbit>. At a local maximum of $U_h$, another trajectory at the same <energy> can approach the orbit asymptotically as a <separatrix>. Existence of roots elsewhere is insufficient for confinement: their dynamically accessible ordering relative to the launch radius matters.
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