Kapitza's pendulum
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.
[1] 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".
[2] Yet until the 1950s there was no explanation for this highly unusual and counterintuitive phenomenon.
[1] 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.
[3] 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.
[4] 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.
The following notation will be used Denoting the angle between pendulum and downward direction as
the time dependence of the position of pendulum gets written as The potential energy of the pendulum is due to gravity and is defined by, in terms of the vertical position, as The kinetic energy in addition to the standard term
, describing velocity of a mathematical pendulum, there is a contribution due to vibrations of the suspension The total energy is given by the sum of the kinetic and potential energies
The total energy is conserved in a mathematical pendulum, so time
According to the virial theorem the mean kinetic and potential energies in harmonic oscillator are equal.
This means that the line of symmetry corresponds to half of the total energy.
The kinetic energy is more sensitive to vibration compared to the potential one.
reads up to irrelevant total time derivative terms.
The differential equation which describes the movement of the pendulum is nonlinear due to the
then the pendulum will oscillate close to the only stable point
Technically, we perform a perturbative expansion in the "coupling constants"
The perturbative treatment becomes exact in the double scaling limit
is defined as The equation of motion for the "slow" component
-oscillation yields to leading order The "slow" equation of motion becomes by introducing an effective potential It turns out[1] that the effective potential
as the mathematical pendulum and the other minimum is in the upper vertical position
[5] For even larger amplitude, the stable oscillation becomes unstable, and the pendulum would start rotating.
At even larger amplitudes, the rotating mode is also destroyed, and a strange attractor appears by period-doubling cascade.
[7][8] The rotating solutions of the Kapitza's pendulum occur when the pendulum rotates around the pivot point at the same frequency that the pivot point is driven.
Interesting phase portraits might be obtained in regimes which are not accessible within analytic descriptions, for example in the case of large amplitude of the suspension
[9][10] Increasing the amplitude of driving oscillations to half of the pendulum length
