# 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.

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