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THE GEOMETRY OF MATHEMATICAL METHODS

Section 21.15 Effective Potential

If we compare the equation for the velocity of a one-dimensional harmonic oscillator (21.13.3)
\begin{equation} \dot{x} = \pm\sqrt{\frac{2}{m}\bigl(E-\frac12 (x-x_0)^2\bigr)}\tag{21.15.1} \end{equation}
from Section 21.13 with the equation for the radial coordinate (21.14.4)
\begin{align} \dot{r} \amp = \pm\sqrt{\frac{2}{\mu}\bigl(E-U(r)\bigr)-\frac{\ell^2}{\mu^2r^2}}\tag{21.15.2}\\ \amp = \pm\sqrt{\frac{2}{\mu}\Bigl(E -(U(r)+\frac{\ell^2}{2\mu r^2})\Bigr)}\tag{21.15.3} \end{align}
from Section 21.14, we can see an intriguing possibility. If we add together the ordinary potential \(U(r)\) and the angular part of the kinetic energy, \(-\frac{\ell^2}{2\mu r^2}\text{,}\) this sum acts the same way in (21.15.3) as the potential does in (21.13.3). Then, we can make a graph analogous to Figure 21.10 and analyze the classical turning points of the motion. The sum of the ordinary potential, \(U(r)\) and the angular part of the kinetic energy, \(-\frac{\ell^2}{2\mu r^2}\) is called the effective potential.
\begin{equation} U_\text{eff} \equiv U(r)-\frac{\ell^2}{2\mu r^2}\text{.}\tag{21.15.4} \end{equation}
The applet below shows you how the effective potential depends on the parameters: (magnitude of the) angular momentum \(\ell\text{,}\) strength of the force \(k\text{,}\) and reduced mass \(\mu\) for the case \(-\frac{k}{r}\) for spherically symmetric gravitational or electrostatic forces. (Note that the dependence of \(k\) on \(\mu\) for the case of gravitational forces has been ignored.)

Activity 21.7.

Explore how the shape of the effective potential (shown in black) depends on the parameters \(\ell\text{,}\) \(k\text{,}\) and \(\mu\text{.}\)
Figure 21.11. The effective potential, shown in black is the sum of two terms: the ordinary potential, shown in blue, and the angular part of the kinetic energy, shown in red. For a given effective potential, the total energy \(E\text{,}\) shown in green will determine the shape of the orbit.