Editing Double pendulum (section)

==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}}