Circular orbit condition for a radial kinetic Lagrangian ($p_{\theta }^2=-4b(R{)}^2{c}^{\prime }(R)/b^{\prime }(R)$)

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 $b^{\prime }(R){\dot{\theta }}^2+c^{\prime }(R)=0$. If $b^{\prime }(R)\ne 0$, eliminating $\dot{\theta }$ gives the displayed condition. A real orbit requires $-c^{\prime }(R)/b^{\prime }(R)\ge 0$; the radial kinetic coefficient $a$ does not enter this constant-radius condition, although it affects radial perturbations. If $b^{\prime }(R)=0$, the original condition must be used rather than divided by zero.