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}=-\frac{d V}{d x}-\xi \frac{d^{2} V}{d x^{2}}+f(X, t)+\xi \frac{\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 \frac{\partial f}{\partial X}}=-\frac{1}{m \omega^{2}} \overline{f \frac{\partial f}{\partial X}}=-\frac{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+\frac{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.