= Solution
The first term, $m\dot r^2/2$, is radial <kinetic energy>. The second, $L^2/(2mr^2)$, is tangential <kinetic energy>, since conservation of <angular momentum> gives $L=mr^2\dot\varphi$. The final term, $-GMm/r$, is the gravitational <potential energy>, with zero at infinite separation. Thus
$$
E=\frac12m\dot r^2+U_{\rm eff}(r),\qquad
\boxed{U_{\rm eff}(r)=\frac{L^2}{2mr^2}-\frac{GMm}{r}.}
$$
The second and third terms together form the radial <effective potential> for a <Kepler orbit>. At an apsis the radial <velocity> vanishes, so $E=U_{\rm eff}$; for a bound noncircular orbit the two accessible roots give the nearest and furthest radii.
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