# Pendulum equation

An ordinary differential equation of the form

$$\tag{* } \dot{x} dot = - a \sin x,$$

where $a$ is a positive constant. A pendulum equation arises in the study of free oscillations of a mathematical pendulum in a gravity field — a point mass with one degree of freedom attached to the end of a non-extendible and incompressible weightless suspender, the other end of which is fastened on a hinge which permits the pendulum to rotate in a vertical plane. The unknown function $x( t)$ is the deviation angle of the pendulum at time $t$ from the lower equilibrium position, measured in radians;

$$a = \frac{g}{l} ,$$

where $l$ is the length of the suspender and $g$ is the gravitational acceleration. The (approximate) equation describing the small oscillations of the pendulum about the lower equilibrium position has the form

$$\dot{x} dot = - ax.$$

The qualitative investigation of the pendulum equation is carried out using the law of conservation of energy, which relates the position and the velocity of the pendulum:

$$\frac{\dot{x} ^ {2} }{2} - a \cos x = E,$$

where $E = \textrm{ const }$ is the total energy of the pendulum. The time scale can be chosen so that $a= 1$. Then for an energy value $E < 1$ the pendulum performs an oscillatory movement (the velocity changes its sign periodically), whereas for $E > 1$ it rotates (the velocity has constant sign). The solution $x( t)$ of (*) with initial condition $x( 0) = 0$, $x ^ \prime ( 0) = \alpha$ for $E = - 1+ \alpha ^ {2} /2 < 1$, satisfies the relation

$$\sin x( \frac{t)}{2} = \frac \alpha {2} \mathop{\rm sn} t,$$

where the Jacobi elliptic function $\mathop{\rm sn}$ has modulus $\alpha /2$( cf. Jacobi elliptic functions).

Of great practical importance are equations close to the pendulum equation. The presence of a small friction that depends on the position and velocity of the pendulum leads to the equation

$$\dot{x} dot = - a \sin x + \epsilon f( x, \dot{x} );$$

the small oscillations of the pendulum with friction are described by the equation

$$\dot{x} dot = - ax + \epsilon f( x, \dot{x} ),$$

a particular case of which is the van der Pol equation. The oscillations of a pendulum for which the length of the suspender varies periodically (the motion of a swing) are described by the Hill equation, an important particular case of which is the Mathieu equation.

#### References

 [1] V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian) [2] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971) [3] A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian)

For $a = 1$ and $x( 0) = 0$, $x ^ \prime ( 0)= \alpha$ with $| \alpha | < 2$ the motion is periodic with amplitude

$$A = \ \left | \mathop{\rm arccos} \left ( 1 - \frac{\alpha ^ {2} }{2} \right ) \right |$$

and period

$$T = 4 \int\limits _ { 0 } ^ { \pi /2 } \frac{d \theta }{\sqrt {1- \sin ^ {2} ( \alpha / 2 ) \sin ^ {2} \theta } } ,$$

which is a complete elliptic integral of the first kind, see [a1]. The periodically-forced damped pendulum

$$\dot{x} dot + \epsilon \dot{x} + \sin x = \delta \cos \omega t ,\ \ 0 < \epsilon ,\ \delta \ll 1,$$

and the parametrically-forced damped pendulum

$$\dot{x} dot + \epsilon \dot{x} + ( 1+ \delta \cos \omega t) \sin x = 0 ,$$

give rise to chaotic solutions. These are analyzed with Melnikov's method in, respectively, [a1] and [a3]. In [a4] the class of problems

$$\dot{x} dot + x = \epsilon f ( x, \dot{x} , t ),\ \ 0 < \epsilon \ll 1,$$

is studied with perturbation methods (cf. also Perturbation theory). Special attention is given to the averaging method (cf. e.g. Krylov–Bogolyubov method of averaging).

#### References

 [a1] J.K. Hale, "Ordinary differential equations" , Wiley (Interscience) (1969) [a2] J. Guckenheimer, P. Holmes, "Nonlinear oscillations, dynamical systems, and bifurcation of vectorfields" , Springer (1983) [a3] S. Wiggins, "Global bifurcations and chaos" , Springer (1988) [a4] J.A. Sanders, F. Verhulst, "Averaging methods in nonlinear dynamical systems" , Springer (1985) [a5] V.I. Arnol'd, A. Avez, "Problèmes ergodiques de la mécanique classique" , Gauthier-Villars (1967) (Translated from Russian)
How to Cite This Entry:
Pendulum equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pendulum_equation&oldid=48152
This article was adapted from an original article by Yu.S. Il'yashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article