= Circular orbit condition for a radial kinetic Lagrangian
{title2=$p_\theta^2=-4b(R)^2c'(R)/b'(R)$}
Let a planar <Lagrangian> have the form $L=a(r)\dot r^2+b(r)\dot\theta^2+c(r)$, with smooth coefficients and $b(r)>0$. Its <cyclic coordinate> $\theta$ has constant <conjugate momentum> $p_\theta=2b(r)\dot\theta$. For a <circular orbit> $r=R$, the radial <Euler-Lagrange equation> reduces to $b'(R)\dot\theta^2+c'(R)=0$. If $b'(R)\ne0$, eliminating $\dot\theta$ gives the displayed condition. A real orbit requires $-c'(R)/b'(R)\geq0$; the radial kinetic coefficient $a$ does not enter this constant-radius condition, although it affects radial perturbations. If $b'(R)=0$, the original condition must be used rather than divided by zero.
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