Encyclopedia of Nonlinear Science
With a history covering many centuries, the pendulum has been of primary importance in the development of physics. It is well known that in the 17th century, Dutch experimental physicist Christiaan Huygens discovered the synchronization phenomenon while studying pendulum clocks (Huygens, 1673). Using a pendulum as an example, Huygens formulated the conservation law for mechanical energy. In the 18th century, the French scientist Jean Charles Borda suggested using a pendulum for measuring the acceleration of gravity (Borda, 1792), and Wilhelm Bessel used a pendulum for an accurate verification of the equality of inertial and gravitational masses. There are many other applications of pendula for physical measurements; for example, a pendulum with a randomly vibrated suspension axis has turned out to be a good model for control of turbulence in subsonic submerged jets (Landa, 1996).
Two models of a pendulum are well known—mathematical and physical. The mathematical pendulum is defined as a ball suspended by a weightless thread in vacuum (Figure 1a). It is described by the equation
where φ is the pendulum angle of deviation, 
is the angular frequency of small oscillations, g is the acceleration due to gravity, l is the thread length, and dots indicate derivative with respect to time. The physical pendulum can be defined as a body moving about a fixed horizontal axis O (Figure 1b).
Taking account of air friction, the motion equation for such a body is
where I is the body’s moment of inertia, h is the friction factor, m is the body’s mass, and a is the distance between the axis and the body’s center of gravity.
Equation (1) belongs to the class of nonlinear conservative (Hamiltonian) systems, whereas Equation (2) is a dissipative system. Solutions of these equations
can be conveniently analyzed with the phase plane φ, 
We illustrate this for Equation (1).
Using the energy conservation law, we obtain an equation for a trajectory in the phase plane φ, 
in the form
where E is the total pendulum energy. For E<2mgl, Equation (3) describes closed trajectories corresponding to pendulum oscillations about the stable equilibrium position (φ=0), and for E>2mgl, nonclosed trajectories corresponding to the rotation of the pendulum (Figure 2a). These two kinds of trajectories are separated by peculiar trajectories passing through singular saddle points (
φ=±π, ±3π,…). Such trajectories are said to be separatrices. Because values of φ differing from each other by 2π are physically in-distinguishable, the phase plane shown in Figure 2a can be rolled into a cylinder.
In general, solutions of Equations (1) and (3) are Jacobi elliptic functions, but more simple solutions can be obtained in the case of small oscillations 
and in the case when E=2mgl (this value of E corresponds to the motion of a representative point along a separatrix). In the latter case, Equation (3) is split into two equations:
By integrating (4) we find
The time dependencies of φ=φ+ and 
are shown in Figure 2b and c. These solutions play an important part in soliton theory.
In the case of sufficiently small oscillations, when sin φ≈φ−φ3/6, Equation (1) reduces to the Duffing equation
The general solution of Equation (6) is
where sn is a Jacobi elliptic function of modulus 
and 
Solution (7) is valid for 
when k≤1. This constraint is certainly fulfilled because Equation (6) follows from (1) only for A<1. Solution (7) describes periodic oscillations of period 4K(k)/Ω, where K(k) is the full elliptic integral of the first kind. It follows from this that the period of oscillations increases with increasing amplitude. Such a property inherent in nonlinear systems is called anisochronism. It should be noted that Galileo was the first to discover the isochronism of small pendulum oscillations.
If a pendulum consisting of an iron ball suspended by a thread of length l is placed between the opposite poles of a magnet, its behavior essentially changes (Landa, 1996, 2001). Approximating the magnetic force acting on the ball by F(φ)=ml(a1φ−b1φ3) and restricting ourselves to small oscillations of the ball, then the pendulum angular deviation φ obeys the following approximate equation:
where 
In the case that 
(the equilibrium position x=0 becomes aperiodically unstable (the corresponding singular point φ=0, 
in the phase plane φ, 
becomes of saddle type). If, in addition to this, the inequality 
holds, two stable equilibrium positions with coordinates 
appear. These equilibrium positions correspond to singular points of center type. But if 
and a>0, then the ball adheres to one of the magnet poles. Equation (8) describes a so-called two-well oscillator, which is the subject of recent widespread interest in connection with stochastic and vibrational resonances (Landa, 2001).
If a pendulum is suspended from a uniformly rotating shaft, it can execute self-oscillations. Such a pendulum was discovered by William Froude and mentioned in the famous treatise by Lord Rayleigh (Rayleigh, 1877). Rayleigh showed that oscillations of such a pendulum are approximately described by an equation which came to be known as the Rayleigh equation. A controlled Froude pendulum was suggested by Neimark to be a model of stochastic oscillations (Neimark & Landa, 1992).
POLINA LANDA
See also Damped-driven anharmonic oscillator; Duffing equation; Elliptic functions; Hamiltonian systems; Solitons
Further Reading
Borda, J. 1792. Mémoires sur la mesure du pendule (Mesure de la 
), Paris
Huygens, C. 1673. Christiani Hvgenii Zvlichemii Horologivm oscillatorivm, sive, De motv pendvlorvm ad horologia aptato demonstrationes geometricœ. Paris: Muguet; as Christiaan Huygens’ The Pendulum Clock, or, Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks, translated with notes by R.J.Blackwell, Ames: Iowa State University Press, 1986
Landa, P.S. 1996. Nonlinear Oscillations and Waves in Dynamical Systems, Dordrecht and Boston: Kluwer
Landa, P.S. 2001. Regular and Chaotic Oscillations, Berlin and New York: Springer
Neimark, Yu.I. & Landa, P.S. 1992 Stochastic and Chaotic Oscillations, Dordrecht and Boston: Kluwer (original Russian edition 1987)
Rayleigh, Lord (Strutt, J.W.) 1877–78. The Theory of Sound, London: Macmillan; reprinted New York: Dover, 1945
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at the motion of a representative point along a separatrix.
