 Solution

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