21.3: Stability of a Pendulum with a Rapidly Oscillating Vertical Driving Force

Recall now the Lagrangian for the simple (rigid)pendulum of length \(ℓ\), mass \(m\), angle from vertically down \(\phi\), constrained to move in a vertical plane, point of support driven to oscillate vertically with amplitude \(\alpha\) and frequency \(\Omega\) (from the section on parametric resonance),

\begin{equation}
L=\frac{1}{2} m \ell^{2} \dot{\phi}^{2}+m a \ell \Omega^{2} \cos \Omega t \cos \phi+m g \ell \cos \phi
\end{equation}

Our previous analysis of this system was for driving frequencies near double the natural frequency. Now we’ll investigate the behavior for driving frequencies far more rapid than the natural frequency.

The equation of motion,

\begin{equation}
\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{\phi}}\right)=\frac{\partial L}{\partial \phi}
\end{equation}

is

\begin{equation}
m \ell^{2} \ddot{\phi}=-m a \ell \Omega^{2} \cos \Omega t \sin \phi-m g \ell \sin \phi
\end{equation}

so evidently the external driving force is \(f=-m a \Omega^{2} \cos \Omega t \sin \phi\)

(Landau has a misprint—an extra \(ℓ\) in this, p 95) and, from the previous section, (except that for the pendulum we are using \(\Omega\), not \(\omega\), for the external driving frequency)

\begin{equation}
V_{\mathrm{eff}}=V+\overline{f^{2}} / 2 m \Omega^{2}=m g \ell\left[-\cos \phi+\left(a^{2} \Omega^{2} / 4 g \ell\right) \sin ^{2} \phi\right]
\end{equation}

\(\text { For } \phi=\pi+\varepsilon, \quad \varepsilon \text { small, }\)

\begin{equation}
V_{\mathrm{eff}}(\varepsilon) \cong m g \ell\left[1-\frac{1}{2} \varepsilon^{2}+\left(a^{2} \Omega^{2} / 4 g \ell\right) \varepsilon^{2}\right]
\end{equation}

and for \(a^{2} \Omega^{2}>2 g \ell\)

the upward position is stable!

At first glance, this may seem surprising: the extra term in the potential from the oscillations is like a kinetic energy term for the oscillating movement. Surely the pendulum is oscillating more in the vertically up position than when it’s to one side? So why isn’t that a maximum of the added effective potential? The point is that the relevant variable is not the pendulum’s height above some fixed point, the variable is \(\phi\) —and the rapid oscillations of \(\phi\) are minimum (zero) in the vertically up position.