Kapitza's pendulum or Kapitza pendulum is a rigid pendulum in which the pivot point vibrates in a vertical direction, up and down. It is named after Russian Nobel Prize laureate physicist Pyotr Kapitza, who in 1951 developed a theory which successfully explains some of its unusual properties. The unique feature of the Kapitza pendulum is that the vibrating suspension can cause it to balance stably in an inverted position, with the bob above the suspension point. In the usual pendulum with a fixed suspension, the only stable equilibrium position is with the bob hanging below the suspension point; the inverted position is a point of unstable equilibrium, and the smallest perturbation moves the pendulum out of equilibrium. In nonlinear control theory the Kapitza pendulum is used as an example of a parametric oscillator that demonstrates the concept of "dynamic stabilization". The pendulum was first described by Andrew Stephenson in 1908, who found that the upper vertical position of the pendulum might be stable when the driving frequency is fast. Yet until the 1950s there was no explanation for this highly unusual and counterintuitive phenomenon. Pyotr Kapitza was the first to analyze it in 1951. He carried out a number of experimental studies and as well provided an analytical insight into the reasons of stability by splitting the motion into "fast" and "slow" variables and by introducing an effective potential. This innovative work created a new subject in physics – vibrational mechanics. Kapitza's method is used for description of periodic processes in atomic physics, plasma physics and cybernetical physics. The effective potential which describes the "slow" component of motion is described in "Mechanics" volume (§30) of Landau's Course of Theoretical Physics. Another interesting feature of the Kapitza pendulum system is that the bottom equilibrium position, with the pendulum hanging down below the pivot, is no longer stable. Any tiny deviation from the vertical increases in amplitude with time. Parametric resonance can also occur in this position, and chaotic regimes can be realized in the system when strange attractors are present in the Poincaré section.
Notation
Denote the vertical axis as y {\displaystyle y} and the horizontal axis as x {\displaystyle x} so that the motion of pendulum happens in the ( x {\displaystyle x} - y {\displaystyle y} ) plane. The following notation will be used
g {\displaystyle g} — free fall acceleration,
l {\displaystyle l} — length of rigid and light pendulum,
m {\displaystyle m} — mass of the (idealized, mathematical) pendulum,
ν {\displaystyle \nu } —frequency of the vertical oscillations of the suspension,
a {\displaystyle a} — amplitude of the oscillations of the suspension,
ω 0 = g / l {\displaystyle \omega _{0}={\sqrt {g/l}}} — proper frequency of the mathematical pendulum. Denoting the angle between pendulum and downward direction as φ {\displaystyle \varphi } the time dependence of the position of pendulum gets written as
{ x = l sin φ y = − l cos φ − a cos ν t {\displaystyle {\begin{cases}x&=l\sin \varphi \\y&=-l\cos \varphi -a\cos \nu t\end{cases}}}
Energy The potential energy of the pendulum is due to gravity and is defined by, in terms of the vertical position, as
E P O T = m g y = − m g ( l cos φ + a cos ν t ) . {\displaystyle E_{\mathrm {POT} }=mgy=-mg(l\cos \varphi +a\cos \nu t).\,}
The kinetic energy includes, in addition to the standard term E K I N = m l 2 φ ˙ 2 / 2 {\displaystyle E_{\mathrm {KIN} }=ml^{2}{\dot {\varphi }}^{2}/2} for the velocity of a mathematical pendulum, a contribution due to vibrations of the suspension and a cross-term
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