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An ordinary differential equation of the form
 
An ordinary differential equation of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p0720001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\dot{x} dot  = - a  \sin  x,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p0720002.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p0720003.png" /> is the deviation angle of the pendulum at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p0720004.png" /> from the lower equilibrium position, measured in radians;
+
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;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p0720005.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{g}{l}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p0720006.png" /> is the length of the suspender and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p0720007.png" /> is the gravitational acceleration. The (approximate) equation describing the small oscillations of the pendulum about the lower equilibrium position has the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p0720008.png" /></td> </tr></table>
+
$$
 +
\dot{x} dot  = - ax.
 +
$$
  
 
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:
 
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p0720009.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\dot{x}  ^ {2} }{2}
 +
- a  \cos  x  = E,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200010.png" /> is the total energy of the pendulum. The time scale can be chosen so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200011.png" />. Then for an energy value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200012.png" /> the pendulum performs an oscillatory movement (the velocity changes its sign periodically), whereas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200013.png" /> it rotates (the velocity has constant sign). The solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200014.png" /> of (*) with initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200016.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200017.png" />, satisfies the relation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200018.png" /></td> </tr></table>
+
$$
 +
\sin  x(
 +
\frac{t)}{2}
 +
  =
 +
\frac \alpha {2}
 +
  \mathop{\rm sn}  t,
 +
$$
  
where the Jacobi elliptic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200019.png" /> has modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200020.png" /> (cf. [[Jacobi elliptic functions|Jacobi elliptic functions]]).
+
where the Jacobi elliptic function $  \mathop{\rm sn} $
 +
has modulus $  \alpha /2 $(
 +
cf. [[Jacobi elliptic functions|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
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200021.png" /></td> </tr></table>
+
$$
 +
\dot{x} dot  = - a  \sin  x + \epsilon f( x, \dot{x} );
 +
$$
  
 
the small oscillations of the pendulum with friction are described by the equation
 
the small oscillations of the pendulum with friction are described by the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200022.png" /></td> </tr></table>
+
$$
 +
\dot{x} dot  = - ax + \epsilon f( x, \dot{x} ),
 +
$$
  
 
a particular case of which is the [[Van der Pol equation|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|Hill equation]], an important particular case of which is the [[Mathieu equation|Mathieu equation]].
 
a particular case of which is the [[Van der Pol equation|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|Hill equation]], an important particular case of which is the [[Mathieu equation|Mathieu equation]].
Line 33: Line 82:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Arnol'd,  "Ordinary differential equations" , M.I.T.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Andronov,  A.A. Vitt,  A.E. Khaikin,  "Theory of oscillators" , Dover, reprint  (1987)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Arnol'd,  "Ordinary differential equations" , M.I.T.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Andronov,  A.A. Vitt,  A.E. Khaikin,  "Theory of oscillators" , Dover, reprint  (1987)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200025.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200026.png" /> the motion is periodic with amplitude
+
For $  a = 1 $
 +
and  $  x( 0) = 0 $,  
 +
$  x  ^  \prime  ( 0)= \alpha $
 +
with $  | \alpha | < 2 $
 +
the motion is periodic with amplitude
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200027.png" /></td> </tr></table>
+
$$
 +
= \
 +
\left |  \mathop{\rm arccos} \left ( 1 -  
 +
\frac{\alpha  ^ {2} }{2}
 +
\right ) \right |
 +
$$
  
 
and period
 
and period
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200028.png" /></td> </tr></table>
+
$$
 +
= 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 [[#References|[a1]]]. The periodically-forced damped pendulum
 
which is a complete elliptic integral of the first kind, see [[#References|[a1]]]. The periodically-forced damped pendulum
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200029.png" /></td> </tr></table>
+
$$
 +
\dot{x} dot + \epsilon \dot{x} + \sin  x  = \delta  \cos  \omega t ,\ \
 +
0 < \epsilon ,\  \delta \ll  1,
 +
$$
  
 
and the parametrically-forced damped pendulum
 
and the parametrically-forced damped pendulum
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200030.png" /></td> </tr></table>
+
$$
 +
\dot{x} dot + \epsilon \dot{x} + ( 1+ \delta  \cos  \omega t) \sin  x  = 0 ,
 +
$$
  
 
give rise to chaotic solutions. These are analyzed with Melnikov's method in, respectively, [[#References|[a1]]] and [[#References|[a3]]]. In [[#References|[a4]]] the class of problems
 
give rise to chaotic solutions. These are analyzed with Melnikov's method in, respectively, [[#References|[a1]]] and [[#References|[a3]]]. In [[#References|[a4]]] the class of problems
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072000/p07200031.png" /></td> </tr></table>
+
$$
 +
\dot{x} dot + x  = \epsilon f ( x, \dot{x} , t ),\ \
 +
0 < \epsilon \ll  1,
 +
$$
  
 
is studied with perturbation methods (cf. also [[Perturbation theory|Perturbation theory]]). Special attention is given to the averaging method (cf. e.g. [[Krylov–Bogolyubov method of averaging|Krylov–Bogolyubov method of averaging]]).
 
is studied with perturbation methods (cf. also [[Perturbation theory|Perturbation theory]]). Special attention is given to the averaging method (cf. e.g. [[Krylov–Bogolyubov method of averaging|Krylov–Bogolyubov method of averaging]]).

Latest revision as of 08:05, 6 June 2020


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)
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