21.2: Finding the Effective Potential Generated by the Oscillating Force

As stated above, our system is a particle of mass $m$ moving in one dimension in a time-independent potential $V(x)$ and subject to a rapidly oscillating force $f=f_{1} \cos \omega t+f_{2} \sin \omega t$.

The oscillation’s strength and frequency are such that the particle only moves a small distance in $V(x)$ during one cycle, and the oscillation is much faster than any oscillation possible in the potential alone.

The equation of motion is

$$\begin{equation} m \ddot{x}=-d V / d x+f \end{equation}$$

The particle will follow a path

$$\begin{equation} x(t)=X(t)+\xi(t) \end{equation}$$

where $\xi(t)$ describes rapid oscillations about a smooth path $X(t)$, and the average value $\overline{\xi(t)} \text { of } \xi(t)$ over a period $2 \pi / \omega$ is zero.

Expanding to first order in $\xi$,

$$\begin{equation} m \ddot{X}+m \ddot{\xi}=-\dfrac{d V}{d x}-\xi \dfrac{d^{2} V}{d x^{2}}+f(X, t)+\xi \dfrac{\partial f}{\partial X} \end{equation}$$

This equation has smooth terms and rapidly oscillating terms on both sides, and we can equate them separately. The leading oscillating terms are

$$\begin{equation} m \ddot{\xi}=f(X, t) \end{equation}$$

We’ve dropped the terms on the right of order $\xi, \text { but kept } \ddot{\xi}, \text { because } \ddot{\xi} \sim \omega^{2} \xi \gg \xi$.

So to leading order in the rapid oscillation,

$$\begin{equation} \xi=-f / m \omega^{2} \end{equation}$$

Now, averaging the full equation of motion with respect to time (smoothing out the jiggle, matching the slow-moving terms), the $m \ddot{\xi}$ on the left and the $f(X, t)$ on the right both disappear (but cancel each other anyway), the $\xi d^{2} V / d x^{2}$ term averages to zero on the assumption that the variation of $d^{2} V / d x^{2}$ over a cycle of the fast oscillation is negligible, but we cannot drop the average

$$\begin{equation} \overline{\xi \dfrac{\partial f}{\partial X}}=-\dfrac{1}{m \omega^{2}} \overline{f \dfrac{\partial f}{\partial X}}=-\dfrac{1}{m \omega^{2}} \nabla_{X} \overline{f^{2}} \end{equation}$$

Incorporating this nonzero term, we have an equation of “slow motion”

$$\begin{equation} m \ddot{X}=-d V_{\mathrm{eff}} / d X \end{equation}$$

where, using $|\dot{\xi}|=|f| / m \omega$,

$$\begin{equation} V_{\mathrm{eff}}=V+\overline{f^{2}} / 2 m \omega^{2}=V+\dfrac{1}{2} m \overline{\dot{\xi}^{2}} \end{equation}$$

The effective potential is the original plus a term proportional to the kinetic energy of the oscillation.