Kapitza's pendulum or Kapitza pendulum is a rigid
pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...
in which the pivot point vibrates in a vertical direction, up and down. It is named after Russian
Nobel laureate
The Nobel Prizes ( sv, Nobelpriset, no, Nobelprisen) are awarded annually by the Royal Swedish Academy of Sciences, the Swedish Academy, the Karolinska Institutet, and the Norwegian Nobel Committee to individuals and organizations who make out ...
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
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...
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 In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form ...
, and the smallest perturbation moves the pendulum out of equilibrium. In
nonlinear control theory
Nonlinear control theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of dyn ...
the Kapitza pendulum is used as an example of a
parametric oscillator
A parametric oscillator is a harmonic oscillator#Driven harmonic oscillators, driven harmonic oscillator in which the oscillations are driven by varying some parameter of the system at some frequency, typically different from the natural frequenc ...
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
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
,
and
cybernetical physics Cybernetical physics is a scientific area on the border of cybernetics and physics which studies physical systems with cybernetical methods. Cybernetical methods are understood as methods developed within control theory, information theory, syst ...
. The effective potential which describes the "slow" component of motion is described in "Mechanics" volume (§30) of
Landau
Landau ( pfl, Landach), officially Landau in der Pfalz, is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990 ...
's ''
Course of Theoretical Physics
The ''Course of Theoretical Physics'' is a ten-volume series of books covering theoretical physics that was initiated by Lev Landau and written in collaboration with his student Evgeny Lifshitz starting in the late 1930s.
It is said that Landau ...
''.
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 attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
s are present in the
Poincaré section
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* Luc ...
.
Notation
Denote the vertical axis as
and the horizontal axis as
so that the motion of pendulum happens in the (
-
) plane. The following notation will be used
*
—frequency of the vertical oscillations of the suspension,
*
— amplitude of the oscillations of the suspension,
*
— proper frequency of the mathematical pendulum,
*
— free fall acceleration,
*
— length of rigid and light pendulum,
*
— mass.
Denoting the angle between pendulum and downward direction as
the time dependence of the position of pendulum gets written as
:
Energy
The
potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...
of the pendulum is due to gravity and is defined by, in terms of the vertical position, as
:
The
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
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
and the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
by their difference
.
The total energy is conserved in a mathematical pendulum, so time
dependence of the potential
and kinetic
energies is symmetric with respect to the horizontal line. According to the
virial theorem
In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...
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
is bound from below and above