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)

Comments

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)