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| {{more footnotes|date=June 2013}}
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| [[Image:Double-Pendulum.svg|upright|thumb|A double pendulum consists of two [[pendulum]]s attached end to end.]]
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| In [[physics]] and [[mathematics]], in the area of [[dynamical systems]], a '''double pendulum''' is a [[pendulum]] with another pendulum attached to its end, and is a simple [[physical system]] that exhibits rich [[dynamical systems|dynamic behavior]] with a strong sensitivity to initial conditions.<ref>Levien RB and Tan SM. Double Pendulum: An experiment in chaos.''American Journal of Physics'' 1993; 61 (11): 1038</ref> The motion of a double pendulum is governed by a set of coupled [[ordinary differential equation]]s. For certain [[energy|energies]] its motion is [[chaos theory|chaotic]].
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| ==Analysis and interpretation==
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| 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 the vertical plane. In the following analysis, the limbs are taken to be identical compound pendulums of length <math>\ell</math> and mass <math>m</math>, and the motion is restricted to two dimensions.
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| [[Image:Double-compound-pendulum-dimensioned.svg|right|thumb|Double compound pendulum]] | |
| In a compound pendulum, the mass is distributed along its length. If the 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>\textstyle I=\frac{1}{12} m \ell^2</math> about that point.<!-- The moment of inertia of a rod rotating around an axis attached to one of its ends equals <math>\textstyle I=\frac{1}{3} m \ell^2</math>. -->
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| It is convenient to use the angles between each limb and the vertical as the [[generalized coordinates]] defining the [[configuration space|configuration]] of the system. These angles are denoted θ<sub>1</sub> and 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:
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| :<math>
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| x_1 = \frac{\ell}{2} \sin \theta_1,
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| </math>
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| :<math>
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| y_1 = -\frac{\ell}{2} \cos \theta_1
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| </math>
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| and the center of mass of the second pendulum is at
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| :<math>
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| x_2 = \ell \left ( \sin \theta_1 + \frac{1}{2} \sin \theta_2 \right ),
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| </math>
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| :<math>
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| y_2 = -\ell \left ( \cos \theta_1 + \frac{1}{2} \cos \theta_2 \right ).
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| </math>
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| This is enough information to write out the Lagrangian.
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| ===Lagrangian===
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| The [[Lagrangian]] is
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| :<math>
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| \begin{align}L & = \mathrm{Kinetic~Energy} - \mathrm{Potential~Energy} \\
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| & = \frac{1}{2} m \left ( v_1^2 + v_2^2 \right ) + \frac{1}{2} I \left ( {\dot \theta_1}^2 + {\dot \theta_2}^2 \right ) - m g \left ( y_1 + y_2 \right ) \\
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| & = \frac{1}{2} m \left ( {\dot x_1}^2 + {\dot y_1}^2 + {\dot x_2}^2 + {\dot y_2}^2 \right ) + \frac{1}{2} I \left ( {\dot \theta_1}^2 + {\dot \theta_2}^2 \right ) - m g \left ( y_1 + y_2 \right ) \end{align}
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| </math>
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| 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.
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| Substituting the coordinates above and rearranging the equation gives
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| :<math> | |
| L = \frac{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 ] + \frac{1}{2} m g \ell \left ( 3 \cos \theta_1 + \cos \theta_2 \right ).
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| </math>
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| [[Image:double-compound-pendulum.gif|right|frame|Motion of the double compound pendulum (from numerical integration of the equations of motion)]]
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| [[Image:DPLE.jpg|right|thumb|Long exposure of double pendulum exhibiting chaotic motion (tracked with an [[LED]])]]
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| There is only one conserved quantity (the energy), and no conserved [[generalized momentum|momenta]]. The two momenta may be written as
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| :<math>
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| p_{\theta_1} = \frac{\partial L}{\partial {\dot \theta_1}} = \frac{1}{6} m \ell^2 \left [ 8 {\dot \theta_1} + 3 {\dot \theta_2} \cos (\theta_1-\theta_2) \right ]
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| </math>
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| and
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| :<math>
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| p_{\theta_2} = \frac{\partial L}{\partial {\dot \theta_2}} = \frac{1}{6} m \ell^2 \left [ 2 {\dot \theta_2} + 3 {\dot \theta_1} \cos (\theta_1-\theta_2) \right ].
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| </math>
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| These expressions may be inverted to get
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| :<math>
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| {\dot \theta_1} = \frac{6}{m\ell^2} \frac{ 2 p_{\theta_1} - 3 \cos(\theta_1-\theta_2) p_{\theta_2}}{16 - 9 \cos^2(\theta_1-\theta_2)}
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| </math>
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| and
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| :<math>
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| {\dot \theta_2} = \frac{6}{m\ell^2} \frac{ 8 p_{\theta_2} - 3 \cos(\theta_1-\theta_2) p_{\theta_1}}{16 - 9 \cos^2(\theta_1-\theta_2)}.
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| </math>
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| The remaining equations of motion are written as
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| :<math>
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| {\dot p_{\theta_1}} = \frac{\partial L}{\partial \theta_1} = -\frac{1}{2} m \ell^2 \left [ {\dot \theta_1} {\dot \theta_2} \sin (\theta_1-\theta_2) + 3 \frac{g}{\ell} \sin \theta_1 \right ]
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| </math>
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| and
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| :<math>
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| {\dot p_{\theta_2}} = \frac{\partial L}{\partial \theta_2}
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| = -\frac{1}{2} m \ell^2 \left [ -{\dot \theta_1} {\dot \theta_2} \sin (\theta_1-\theta_2) + \frac{g}{\ell} \sin \theta_2 \right ].
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| </math>
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| These last four equations are explicit formulae for the time evolution of the system given its current state. It is not possible to go further and integrate these equations analytically{{Citation needed|date=June 2011}}, to get formulae for θ<sub>1</sub> and θ<sub>2</sub> as functions of time. It is however possible to perform this integration numerically using the [[Runge–Kutta methods|Runge Kutta]] method or similar techniques.
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| ==Chaotic motion==
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| [[Image:Double_pendulum_flips_graph.png|thumb|Graph of the time for the pendulum to flip over as a function of initial conditions]]
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| The double pendulum undergoes [[chaotic motion]], and 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 conditions. Here, the initial value of θ<sub>1</sub> ranges along the ''x''-direction, from −3 to 3. The initial value θ<sub>2</sub> ranges along the ''y''-direction, from −3 to 3. The colour of each pixel indicates whether either pendulum flips within <math>10\sqrt{\ell/g }</math> (green), within <math>100\sqrt{\ell/g }</math> (red), <math>1000\sqrt{\ell/g }</math> (purple) or <math>10000\sqrt{\ell/g }</math> (blue). Initial conditions that don't lead to a flip within <math>10000\sqrt{\ell/g }</math> are plotted white.
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| The boundary of the central white region is defined in part by energy conservation with the following curve:
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| :<math>
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| 3 \cos \theta_1 + \cos \theta_2 = 2. \,
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| </math>
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| Within the region defined by this curve, that is if
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| :<math>
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| 3 \cos \theta_1 + \cos \theta_2 > 2, \,
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| </math>
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| 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 masses rather than two rods with distributed mass.<ref>Alex Small, ''[https://12d82b32-a-62cb3a1a-s-sites.googlegroups.com/site/physicistatlarge/Computational%20Physics%20Sample%20Project-Alex%20Small-v1.pdf 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>
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| The lack of a natural excitation frequency has led to the use of 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.
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| ==See also==
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| * [[Double inverted pendulum]]
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| * [[Pendulum (mathematics)]]
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| * 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]]. An artistic application of this can be seen here: http://paulwainwrightphotography.com/pendulum_gallery.shtml .
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite book
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| | last = Meirovitch
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| | first = Leonard
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| | year = 1986
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| | title = Elements of Vibration Analysis
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| | edition = 2nd edition
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| | publisher = McGraw-Hill Science/Engineering/Math
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| | isbn = 0-07-041342-8
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| }}
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| * 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).
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| * Peter Lynch, ''[http://www.maths.tcd.ie/~plynch/SwingingSpring/doublependulum.html Double Pendulum]'', (2001). ''(Java applet simulation.)''
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| * Northwestern University, ''[http://www.physics.northwestern.edu/vpl/mechanics/pendulum.html Double Pendulum]'', ''(Java applet simulation.)''
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| * Theoretical High-Energy Astrophysics Group at UBC, ''[http://tabitha.phas.ubc.ca/wiki/index.php/Double_pendulum Double pendulum]'', (2005).
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| ==External links==
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| *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)
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| *[http://www.youtube.com/watch?v=Uzlccwt5SKc&NR=1 Video] of a double square pendulum with three (almost) identical starting conditions.
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| *Double pendulum physics simulation from [http://www.myphysicslab.com/dbl_pendulum.html www.myphysicslab.com]
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| *Simulation, equations and explanation of [http://www.chris-j.co.uk/rott.php Rott's pendulum]
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| *Comparison videos of a double pendulum with the same initial starting conditions on [http://www.youtube.com/watch?v=O2ySvbL3-yA YouTube]
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| * [http://freddie.witherden.org/tools/doublependulum/ Double Pendulum Simulator] - An open source simulator written in [[C++]] using the [[Qt (toolkit)|Qt toolkit]].
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| * [http://www.imaginary2008.de/cinderella/english/G2.html Online Java simulator] of the [[Imaginary_(exhibition)|Imaginary exhibition]].
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| * Vadas Gintautas, [[Alfred Hubler|Alfred Hübler]] (2007). [http://pre.aps.org/abstract/PRE/v75/i5/e057201 Experimental evidence for mixed reality states in an interreality system] Phys. Rev. E 75, 057201 Presents data on an experimental, mixed reality system in which a real and virtual pendulum complexly interact.
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| {{Chaos theory}}
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| [[Category:Chaotic maps]]
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| [[Category:Pendulums]]
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