2024/1/15, Hiizu Nakanishi

The Theory of Kapitza Pendulum

First, let us consider the case where the vibration is imposed vertically on the pivot; \[ f(t) = f\sin(\Omega t) . \]

Lagrangian

Let $(x,y)$ be the position of the pendulum weight, then the position and the velocity are given as \[\left\{\begin{array}{rl} x & = -\ell \sin\varphi \\ y & = \ell\cos\varphi + f(t) \end{array}\right. \hspace{5em} \left\{\begin{array}{rl} \dot x & = -\ell\dot\varphi \cos\varphi \\ \dot y & = -\ell\dot\varphi\sin\varphi + \dot f(t) \end{array}\right. \] and Lagrangian is obtained as \[ L = {1\over 2}m \Big( \ell^2\dot\varphi^2 -2\ell\dot\varphi\sin\varphi \dot f(t) +\dot f^2(t) \Big) - mg\big(\ell\cos\varphi + f(t)\big) . \]

Equation of Motion

The equation of motion is given by Lagrangian as ラグランジュの運動方程式は、 \[ {d p\over dt} = {\partial L\over\partial \varphi},\qquad p := {\partial L\over\partial\dot\varphi} = m\Big(\ell^2\dot\varphi -\ell\dot f(t)\sin\varphi\Big) \] $$ \Rightarrow\qquad \ddot\varphi = \left( \omega_0^2 + {\ddot f(t)\over\ell}\right)\sin\varphi \,;\qquad \omega_0:=\sqrt{g\over\ell} . \tag{1} $$ In the following, we consider the case with \[ f\ll\ell, \qquad \Omega\gg \omega_0. \]

Approximate Solution

We decompose the angle of the pendulum $\varphi(t)$ into two components as \[ \varphi(t) = \Phi(t) + \xi(t), \tag{2} \] where $\Phi(t)$ is the slowly moving component and $\xi(t)$ is the fast moving component comparable with the imposed vibration $\Omega$, and find the equation for $\Phi(t)$. Here, $\xi(t)$ is much smaller than $\Phi(t)$, and reduces to zero when it is averaged over the period of the vibration $2\pi/\Omega$; \[ \xi(t)\ll\Phi(t), \qquad \overline{\xi(t)}:={\Omega\over 2\pi}\int_t^{t+2\pi/\Omega} \xi(t') dt'=0. \]

By inserting Eq.(2) into Eq.(1), we may obtain the approximate equation \begin{align*} \ddot\Phi(t) + \textcolor{red}{\ddot\xi(t) } & = \left( \omega_0^2 + {\ddot f(t)\over \ell}\right) \sin\big(\Phi(t)+\xi(t)\big) \\ & \approx \left( \omega_0^2 + \textcolor{red}{{\ddot f(t)\over \ell}}\right) \Big( \sin\Phi(t) + \textcolor{red}{\xi(t)\, \cos\Phi(t)} \Big), \tag{3} \end{align*} where we have used $\Phi\ll\xi$.

Assuming that both $\xi(t)$ and $f(t)$ vibrate by the angular frequency $\Omega$, the magnitudes of the related terms in Eq.(3) may be estimated as \[ \ddot\xi \sim \Omega^2\xi,\quad {\ddot f(t)\over\ell}\sin\Phi \sim \Omega^2{f\over\ell},\quad \omega_0^2\xi\cos\Phi\sim \omega_0^2\xi,\quad {\ddot f(t)\over\ell}\,\xi\cos\Phi \sim \Omega^2{f\over\ell}\,\xi . \] If we adopt $\Omega\gg\omega_0$ and $\xi\sim f/\ell$, then we have the relations among these terms \[ \ddot\xi,\quad {\ddot f(t)\over\ell}\sin\Phi \qquad\gg\qquad \omega_0^2\xi\cos\Phi,\quad {\ddot f(t)\over\ell}\,\xi\cos\Phi. \] The two large terms should balance in Eq.(3) \[ \ddot\xi(t) = {\ddot f(t)\over\ell}\,\sin\Phi(t). \] Noting that $\Phi(r)$ is slowly varying, we can approximate as \[ \xi(t)\approx {f(t)\over\ell}\,\sin\Phi(t). \] After inserting this into Eq.(3) and averaging over the period of vibration $2\pi/\Omega$, we obtain \begin{align*} \ddot\Phi(t) & = \omega_0^2 \sin\Phi(t) + \omega_0^2\,\overline{\xi(t)}\,\cos\Phi(t) + \overline{\ddot f(t) f(t)\over \ell^2}\sin\Phi(t)\cos\Phi(t) \\ & = \omega_0^2 \sin\Phi(t) - {\Omega^2f^2\over 2 \ell^2}\sin\Phi(t)\cos\Phi(t) \\ & := -{d\over d\Phi} U_{\rm eff}(\Phi) . \tag{4} \end{align*} Here, we have used $\overline{\xi(t)}=0$ and $U_{\rm eff}(\Phi)$ is the effective potential for the slow variable $\Phi$: \[ U_{\rm eff}(\Phi):= \omega_0^2\cos\Phi + {\Omega^2 f^2\over 4\,\ell^2}\sin^2\Phi . \tag{5} \] writing down Eq.(4) in the form of harmonic oscillator, we have \[ m\,{d\big(\ell\dot\Phi\big)\over dt} = -{d\over d(\ell\Phi)}\left( mg\ell\cos\Phi + \textcolor{red}{{1\over 2}m \overline{\big(\ell\dot\xi\,\big)^2 }} \right); \qquad \dot\xi(t) := {\Omega f(t)\over\ell}\sin\Phi, \] thus the second term in the effective potential may be considered as the time average of the kinetic energy of the vibration due to external force.

Expanding Eq.(5) by $\Phi$ as \[ U_{\rm eff} (\Phi) = \omega_0^2\left[ 1 +{1\over 2}\left( -1 + {1\over 2}\left({\Omega f\over\omega_0\ell}\right)^2\right)\Phi^2 + O\big(\Phi^4\big)\right], \] thus the inverted state $\Phi=0$ is stabilized when \[ f\,\Omega > \sqrt 2\; \ell\,\omega_0 . \]

The case where the direction of the external vibration is $\theta$

Similar analysis leads us to the effective potential \[ U_{\rm eff}(\Phi) = \omega_0^2\cos\Phi + {\Omega^2 f^2\over 4\,\ell^2}\sin^2\big(\Phi-\theta\big) . \] The term due to the external vibration has the minima in the directions $\theta$ and $\theta+\pi$.