Editing Double pendulum

{{ Accessibility dispute|date=February 2025}} 
{{Short description|Pendulum with another pendulum attached to its end}}{{more footnotes|date=June 2013}}
[[Image:Double-Pendulum.svg|upright|thumb|A double pendulum consists of two [[pendulum]]s attached end to end.]]
In [[physics]] and [[mathematics]], in the area of [[dynamical systems]], a '''double pendulum''', also known as a '''chaotic pendulum''', is a [[pendulum]] with another pendulum attached to its end, forming a simple [[physical system]] that exhibits rich [[dynamical systems|dynamic behavior]] with a [[butterfly effect|strong sensitivity to initial conditions]].<ref>{{cite journal |last=Levien |first=R. B. |last2=Tan |first2=S. M. |title=Double Pendulum: An experiment in chaos |journal=[[American Journal of Physics]] |year=1993 |volume=61 |issue=11 |page=1038 |doi=10.1119/1.17335 |bibcode=1993AmJPh..61.1038L }}</ref> The motion of a double pendulum is governed by a pair of coupled [[ordinary differential equation]]s and is [[chaos theory|chaotic]].

==Analysis and interpretation==
Several variants of the double pendulum may be considered; the two limbs may be of equal or unequal lengths and masses, they may be [[simple pendulum]]s or [[compound pendulum]]s (also called complex pendulums) and the motion may be in three dimensions or restricted to one vertical plane. In the following analysis, the limbs are taken to be identical compound pendulums of length {{mvar|ℓ}} and mass {{mvar|m}}, and the motion is restricted to two dimensions.

[[Image:Double-compound-pendulum-dimensioned.svg|right|thumb|Double compound pendulum]]
[[Image:Double pendulum simulation.gif|right|thumb|Motion of the double compound pendulum (from [[numerical integration]] of the equations of motion)]]

In a compound pendulum, the mass is distributed along its length. If the double pendulum mass is evenly distributed, then the [[center of mass]] of each limb is at its midpoint, and the limb has a [[moment of inertia]] of {{math|1=''I'' = {{sfrac|1|12}}''mℓ''<sup>2</sup>}} about that point.<!-- The moment of inertia of a rod rotating around an axis attached to one of its ends equals {{math|1=''I'' = {{sfrac|1|3}}''mℓ''<sup>2</sup>}}. -->

It is convenient to use the angles between each limb and the vertical as the [[generalized coordinates]] defining the [[Configuration space (physics)|configuration]] of the system. These angles are denoted {{math|''θ''<sub>1</sub>}} and {{math|''θ''<sub>2</sub>}}. The position of the center of mass of each rod may be written in terms of these two coordinates. If the origin of the [[Cartesian coordinate system]] is taken to be at the point of suspension of the first pendulum, then the center of mass of this pendulum is at:<math display="block">\begin{align}
x_1 &= \tfrac{1}{2} \ell \sin \theta_1 \\
y_1 &= -\tfrac{1}{2} \ell \cos \theta_1 
\end{align}</math>

and the center of mass of the second pendulum is at
<math display="block">\begin{align}
x_2 &= \ell \left ( \sin \theta_1 + \tfrac{1}{2} \sin \theta_2 \right ) \\
y_2 &= -\ell \left ( \cos \theta_1 + \tfrac{1}{2} \cos \theta_2 \right )
\end{align}</math>
This is enough information to write out the Lagrangian.

===Lagrangian===
The [[Lagrangian mechanics|Lagrangian]] is given by
<math display="block">\begin{align}
L &= \text{kinetic energy} - \text{potential energy} \\
&= \tfrac{1}{2} m \left ( v_1^2 + v_2^2 \right ) + \tfrac{1}{2} I \left ( \dot\theta_1^2 + \dot\theta_2^2 \right ) - m g \left ( y_1 + y_2 \right ) \\
&= \tfrac{1}{2} m \left ( \dot x_1^2 + \dot y_1^2 + \dot x_2^2 + \dot y_2^2 \right ) + \tfrac{1}{2} I \left ( \dot\theta_1^2 + \dot\theta_2^2 \right ) - m g \left ( y_1 + y_2 \right )
\end{align}</math>
The first term is the ''linear'' [[kinetic energy]] of the [[center of mass]] of the bodies and the second term is the ''rotational'' kinetic energy around the center of mass of each rod. The last term is the [[potential energy]] of the bodies in a uniform gravitational field. The [[Newton's notation|dot-notation]] indicates the [[time derivative]] of the variable in question.

Using the values of <math>x_1</math> and <math>y_1</math> defined above, we have
<math display="block">
\begin{align}
\dot x_1 &= \dot \theta_1 \left(\tfrac{1}{2}\ell \cos \theta_1 \right)  \\[1ex]
\dot y_1 &= \dot \theta_1 \left(\tfrac{1}{2} \ell \sin \theta_1 \right)
\end{align} 
</math>
which leads to
<math display="block">
v_1^2 = \dot x_1^2 + \dot y_1^2
= \tfrac{1}{4} \dot \theta_1^2 \ell^2 \left(\cos^2 \theta_1 + \sin^2 \theta_1 \right)
= \tfrac{1}{4} \ell^2 \dot \theta_1^2 .
</math>

Similarly, for <math>x_2</math> and <math>y_2</math> we have
<math display="block">
\begin{align}
\dot x_2 &= \ell \left(\dot \theta_1 \cos \theta_1 + \tfrac{1}{2} \dot \theta_2 \cos \theta_2 \right) \\
\dot y_2 &= \ell \left(\dot \theta_1 \sin \theta_1 + \tfrac{1}{2} \dot \theta_2 \sin \theta_2 \right)
\end{align} 
</math>

and therefore

<math display="block">
\begin{align}
v_2^2
&= \dot x_2^2 + \dot y_2^2 \\[1ex]
&= \ell^2 \left(
    \dot \theta_1^2 \cos^2 \theta_1 + \dot \theta_1^2 \sin^2 \theta_1
    + \tfrac{1}{4} \dot \theta_2^2 \cos^2 \theta_2 + \tfrac{1}{4} \dot \theta_2^2 \sin^2 \theta_2
    + \dot \theta_1 \dot \theta_2 \cos \theta_1 \cos \theta_2 + \dot \theta_1 \dot \theta_2 \sin \theta_1 \sin \theta_2
\right) \\[1ex]
&= \ell^2 \left(
\dot \theta_1^2 + \tfrac{1}{4} \dot \theta_2^2 
+ \dot \theta_1 \dot \theta_2 \cos \left(\theta_1 - \theta_2 \right)
 \right).
\end{align} 
</math>

Substituting the coordinates above into the definition of the Lagrangian, and rearranging the equation, gives
<math display="block">
\begin{align}
L &=
\tfrac{1}{2} m \ell^2 \left(
    \dot \theta_1^2
    + \tfrac{1}{4} \dot \theta_1^2 + \tfrac{1}{4} \dot \theta_2^2 
    + \dot \theta_1 \dot \theta_2 \cos \left(\theta_1 - \theta_2 \right)
\right)
+ \tfrac{1}{24} m \ell^2 \left( \dot \theta_1^2 + \dot \theta_2^2 \right)
- m g \left(y_1 + y_2 \right)
\\[1ex]
&= \tfrac{1}{6} m \ell^2 \left (
    \dot \theta_2^2 + 4 \dot \theta_1^2
    + 3 {\dot \theta_1} {\dot \theta_2} \cos (\theta_1-\theta_2)
\right)
+ \tfrac{1}{2} m g \ell \left ( 3 \cos \theta_1 + \cos \theta_2 \right ).
\end{align}
</math>

The equations of motion can now be derived using the [[Euler–Lagrange equation]]s, which are given by
<math display="block">
\frac{d}{dt} \frac{\partial L}{\partial \dot{\theta}_i} - \frac{\partial L}{\partial \theta_i} = 0, \quad i = 1,2.
</math>
We begin with the equation of motion for <math>\theta_1</math>. The derivatives of the Lagrangian are given by
<math display="block">
\frac{\partial L}{\partial \theta_1} = -\tfrac{1}{2} m \ell^2 \dot{\theta}_1 \dot{\theta}_2 \sin(\theta_1 - \theta_2) - \tfrac{3}{2} mg\ell \sin\theta_1
</math>
and
<math display="block">
\frac{\partial L}{\partial \dot{\theta}_1} = \tfrac{4}{3} m\ell^2 \dot{\theta}_1 + \tfrac{1}{2} m\ell^2 \dot{\theta}_2 \cos(\theta_1-\theta_2).
</math>
Thus
<math display="block">
\frac{d}{dt} \frac{\partial L}{\partial \dot{\theta}_1} 
= \tfrac{4}{3} m\ell^2 \ddot{\theta}_1 
+ \tfrac{1}{2} m\ell^2 \ddot{\theta}_2 \cos(\theta_1-\theta_2)
- \tfrac{1}{2} m\ell^2 \dot{\theta}_2(\dot{\theta}_1 - \dot{\theta}_2) \sin(\theta_1 - \theta_2).
</math>
Combining these results and simplifying yields the first equation of motion,
<math display="block">
\tfrac{4}{3} \ell \ddot{\theta}_1 + \tfrac{1}{2} \ell \ddot{\theta}_2 \cos(\theta_1 - \theta_2) + \tfrac{1}{2} \ell \dot{\theta}_2^2 \sin(\theta_1-\theta_2) + \tfrac{3}{2} g \sin\theta_1 = 0.
</math>

Similarly, the derivatives of the Lagrangian with respect to <math>\theta_2</math> and <math>\dot{\theta}_2</math> are given by
<math display="block">
\frac{\partial L}{\partial \theta_2} = \tfrac{1}{2} m \ell^2 \dot{\theta}_1 \dot{\theta}_2 \sin(\theta_1 - \theta_2) - \tfrac{1}{2} mg\ell \sin\theta_2
</math>
and
<math display="block">
\frac{\partial L}{\partial \dot{\theta}_2} = \tfrac{1}{3} m\ell^2 \dot{\theta}_2 + \tfrac{1}{2} m\ell^2 \dot{\theta}_1 \cos(\theta_1-\theta_2).
</math>
Thus
<math display="block">
\frac{d}{dt} \frac{\partial L}{\partial \dot{\theta}_2} 
= \tfrac{1}{3} m\ell^2 \ddot{\theta}_2 
+ \tfrac{1}{2} m\ell^2 \ddot{\theta}_1 \cos(\theta_1-\theta_2)
- \tfrac{1}{2} m\ell^2 \dot{\theta}_1(\dot{\theta}_1 - \dot{\theta}_2) \sin(\theta_1 - \theta_2).
</math>
Plugging these results into the Euler-Lagrange equation and simplifying yields the second equation of motion,
<math display="block">
\tfrac{1}{3} \ell \ddot{\theta}_2 + \tfrac{1}{2} \ell \ddot{\theta}_1 \cos(\theta_1 - \theta_2) - \tfrac{1}{2} \ell \dot{\theta}_1^2 \sin(\theta_1-\theta_2) + \tfrac{1}{2} g \sin\theta_2 = 0.
</math>

No [[closed form expression|closed form]] solutions for <math>\theta_1</math> and <math>\theta_2</math> as functions of time are known, therefore the system can only be solved [[numerical integration|numerically]], using the [[Runge–Kutta methods|Runge Kutta method]] or [[numerical methods for ordinary differential equations|similar techniques]].

[[File:Double-pendulum.png|thumb|Parametric plot for the time evolution of the angles of a double pendulum. Note that the graph resembles [[Brownian motion]].]]

==Chaotic motion==
[[File:Double pendulum flip time 2021.png|thumb|Graph of the time for the pendulum to flip over as a function of initial conditions]] 
[[Image:DPLE.jpg|right|thumb|Long exposure of double pendulum exhibiting chaotic motion (tracked with an [[LED]])]]
[[File:3-double-pendulums.gif|right|thumb|Three double pendulums with nearly identical starting conditions diverge over time, demonstrating the chaotic nature of the system.]]

The double pendulum undergoes [[chaotic motion]], and clearly shows a sensitive dependence on [[initial conditions]]. The image to the right shows the amount of elapsed time before the pendulum flips over, as a function of initial position when released at rest. Here, the initial value of {{math|''θ''<sub>1</sub>}} ranges along the {{mvar|x}}-direction from −3.14 to 3.14. The initial value {{math|''θ''<sub>2</sub>}} ranges along the {{mvar|y}}-direction, from −3.14 to 3.14. The color of each pixel indicates whether either pendulum flips within:
* <math>\sqrt{\frac{\ell}{g}}</math> (black)
* <math>10\sqrt{\frac{\ell}{g}}</math> (red)
* <math>100\sqrt{\frac{\ell}{g}}</math> (green)
* <math>1000\sqrt{\frac{\ell}{g}}</math> (blue) or
* <math>10000\sqrt{\frac{\ell}{g}}</math> (purple).

Initial conditions that do not lead to a flip within <math>10000\sqrt{\frac{\ell}{g}}</math> are plotted white.

The boundary of the central white region is defined in part by energy conservation with the following curve:
<math display="block">3 \cos \theta_1 + \cos \theta_2 = 2. </math>

Within the region defined by this curve, that is if<math display="block">3 \cos \theta_1 + \cos \theta_2 > 2, </math>then it is energetically impossible for either pendulum to flip. Outside this region, the pendulum can flip, but it is a complex question to determine when it will flip. Similar behavior is observed for a double pendulum composed of two [[point mass]]es rather than two rods with distributed mass.<ref>Alex Small, ''[https://drive.google.com/file/d/11UMDzK4_V5AeeqJcm9GsuXLH2YErMlyv/view?usp=drive_link Sample Final Project: One Signature of Chaos in the Double Pendulum]'', (2013). A report produced as an example for students. Includes a derivation of the equations of motion, and a comparison between the double pendulum with 2 point masses and the double pendulum with 2 rods.</ref>

The lack of a natural excitation frequency has led to the use of [[Tuned mass damper|double pendulum systems in seismic resistance designs]] in buildings, where the building itself is the primary inverted pendulum, and a secondary mass is connected to complete the double pendulum.{{cn|date=April 2025}}

==See also==
* [[Double inverted pendulum]]
* [[Pendulum (mechanics)]]
* [[Trebuchet]]
* [[Bolas]]
* [[Mass damper]]
* Mid-20th century physics textbooks use the term "double pendulum" to mean a single bob suspended from a string which is in turn suspended from a V-shaped string. This type of [[pendulum]], which produces [[Lissajous curves]], is now referred to as a [[Blackburn pendulum]].

==References==
{{reflist}}

== Further reading ==
{{refbegin}}
*{{cite book
 | last = Meirovitch
 | first = Leonard
 | year = 1986
 | title = Elements of Vibration Analysis
 | edition = 2nd
 | publisher = McGraw-Hill Science/Engineering/Math
 | isbn = 0-07-041342-8
}}
* Eric W. Weisstein, ''[http://scienceworld.wolfram.com/physics/DoublePendulum.html Double pendulum]'' (2005), ScienceWorld ''(contains details of the complicated equations involved)'' and "[http://demonstrations.wolfram.com/DoublePendulum/ Double Pendulum]" by Rob Morris, [[Wolfram Demonstrations Project]], 2007 (animations of those equations).
* [[Peter Lynch (meteorologist)|Peter Lynch]], ''[https://web.archive.org/web/20030608233118/http://www.maths.tcd.ie/~plynch/SwingingSpring/doublependulum.html Double Pendulum]'', (2001). ''(Java applet simulation.)''
* Northwestern University, ''[http://www.physics.northwestern.edu/vpl/mechanics/pendulum.html Double Pendulum] {{Webarchive|url=https://web.archive.org/web/20070603131902/http://www.physics.northwestern.edu/vpl/mechanics/pendulum.html |date=2007-06-03 }}'', ''(Java applet simulation.)''
* Theoretical High-Energy Astrophysics Group at UBC, ''[https://web.archive.org/web/20070310213326/http://tabitha.phas.ubc.ca/wiki/index.php/Double_pendulum Double pendulum]'', (2005).
{{refend}}
==External links==
*Animations and explanations of a [http://www.physics.usyd.edu.au/~wheat/dpend_html/ double pendulum] and a [http://www.physics.usyd.edu.au/~wheat/sdpend/ physical double pendulum (two square plates)] by Mike Wheatland (Univ. Sydney)

*Interactive Open Source Physics JavaScript simulation with detailed equations [http://iwant2study.org/ospsg/index.php/interactive-resources/physics/02-newtonian-mechanics/02-dynamics/454-e-double-pendulum-drivenwee double pendulum]
*Interactive Javascript simulation of a [http://bestofallpossibleurls.com/double-pendulum.html double pendulum]
*Double pendulum physics simulation from [https://www.myphysicslab.com/dbl_pendulum.html www.myphysicslab.com] using [https://github.com/myphysicslab/myphysicslab open source JavaScript code]
*Simulation, equations and explanation of [http://www.chris-j.co.uk/rott.php Rott's pendulum]
*{{YouTube|O2ySvbL3-yA|Comparison videos of a double pendulum with the same initial starting conditions}}
* [http://freddie.witherden.org/tools/doublependulum/ Double Pendulum Simulator] - An open source simulator written in [[C++]] using the [[Qt (toolkit)|Qt toolkit]].
* [http://www.imaginary2008.de/cinderella/english/G2.html Online Java simulator] {{Webarchive|url=https://web.archive.org/web/20220816023756/http://www.imaginary2008.de/cinderella/english/G2.html |date=2022-08-16 }} of the [[Imaginary (exhibition)|Imaginary exhibition]].

{{Chaos theory}}

[[Category:Chaotic maps]]
[[Category:Dynamical systems]]
[[Category:Mathematical physics]]
[[Category:Pendulums]]