Kapitza's pendulum
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 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 A. 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 formula_1 and the horizontal axis as formula_2 so that the motion of pendulum happens in the (formula_2-formula_1) plane. The following notation will be used Denoting the angle between pendulum and downward direction as formula_11 the time dependence of the position of pendulum gets written as Energy . The potential energy of the pendulum is due to gravity and is defined by of the vertical position as The kinetic energy in addition to the standard term formula_14, 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 formula_16 and the Lagrangian by their difference formula_17. The total energy is conserved in a mathematical pendulum, so time formula_18 dependence of the potential formula_19 and kinetic formula_20 energies is symmetric with respect to the horizontal line. 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. In the case of vibrating suspension the system is no longer a closed one and the total energy is no longer conserved. The kinetic energy is more sensitive to vibration compared to the potential one. The potential energy formula_21 is bound from below and above formula_22 while the kinetic energy is bound only from below formula_23. For high frequency of vibrations formula_5 the kinetic energy can be large compared to the potential energy. Equations  of motion. Motion of pendulum satisfies Euler–Lagrange equations. The dependence of the phase formula_11 of the pendulum on its position satisfies the equation: where the Lagrangian formula_27 reads up to irrelevant total time derivative terms. The differential equation which describes the movement of the pendulum is nonlinear due to the formula_30 factor. Equilibrium positions. Kapitza's pendulum model is more general than the simple pendulum. The Kapitza model reduces to the latter in the limit formula_31. In that limit, the tip of the pendulum describes a circle: formula_32. If the energy in the initial moment is larger than the  maximum of the potential energy formula_33 then the trajectory will be closed and cyclic. If the initial energy is smaller formula_34 then the pendulum will oscillate close to the only stable point formula_35. When the suspension is vibrating with a small amplitude formula_36 and with a frequency formula_37 much higher than the proper frequency formula_38, the angle formula_39 may be viewed as a superposition formula_40 of a "slow" component formula_41 and a rapid oscillation formula_42 with small amplitude due to the small but rapid vibrations of the suspension. Technically, we perform a perturbative expansion in the "coupling constants" formula_43 while  treating the ratio formula_44 as fixed. The perturbative treatment becomes exact in the double scaling limit formula_45. More precisely, the rapid oscillation formula_42 is defined as The equation of motion for the "slow" component formula_41 becomes Time-averaging over the rapid formula_5-oscillation yields to leading order The "slow" equation of motion becomes by introducing an effective potential It turns out that the effective potential formula_54 has two minima if formula_55, or equivalently, formula_56. The first minimum is in the same position formula_57 as the mathematical pendulum and the other minimum is in the upper vertical position formula_58. As a  result the upper vertical position, which is unstable in a mathematical pendulum, can become stable in Kapitza's pendulum. Rotating solutions. 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. There are two rotating solutions, one for a rotation in each direction. We shift to the rotating reference frame using formula_59 and the equation for formula_39 becomes: Again considering the limit in which formula_5 is much higher than the proper frequency formula_38, we find that the rapid-formula_5 slow-formula_65 limit leads  to the equation: The effective potential is just that of a simple pendulum equation. There is a stable equilibrium at formula_67 and an unstable equilibrium at formula_68. Phase portrait. 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 formula_69. Increasing the amplitude of driving oscillations to half of the pendulum length formula_70 leads to the phase portrait shown in the figure. Further increase of the amplitude to formula_71 leads to full filling of the internal points of the phase space: if before some  points of the phase space were not accessible, now system can reach any of the internal points. This situation holds also for larger values of formula_6.