# Talk:Double pendulum

## What is a double pendulum?

This article states that "a double pendulum is a pendulum with another pendulum attached to its end", with link to pendulum. Looking at "pendulum", I see that it consists of two masses connected by two massless rods. Here, however, pendulum consists of two massive rods. I think, by double pendulum, people usually think about the case with two massless rods and two masses, see most of the references given at the end of the article. Understanding that this case might also be called double pendulum, not even mentioning the more usual meaning is not satisfactory.

Cumi 13:41, 24 March 2006 (UTC)

I agree with the previous poster. The usual "simple" double pendulum is masses at the ends of massless rods, and the figure suggests this. However the derivation here is for something different, i.e. a "compound" double pendulum consisting of rods with mass but with no masses at their ends. It is quite odd. —Preceding unsigned comment added by 129.78.64.102 (talk) 00:46, 3 November 2007 (UTC)

I suggest the following terms :

• "punctual pendulum" for a pendulum made of a massive point and a massless rod ;
• "solid pendulum" for a pendulum made of a massive rod.

We can make it more general by adding "double", "triple" etc to the expression. Example : "double punctual pendulum" Bete spatio-temporelle 193.54.238.42 (talk) 15:54, 7 March 2013 (UTC)

On WP we are obliged to use standard terminology, and not invent our own. However there are other ways (parenthetical explanations), to avoid forcing the reader to follow links for the definition of the terms. --catslash (talk) 15:39, 9 March 2013 (UTC)
Creating knowledge is forbidden on Wikipedia... Maybe there are good reasons for that, but it annoys me a lot :-( . Bete spatio-temporelle 152.77.24.34 (talk) 16:38, 23 March 2013 (UTC)

## No analytical solution ?

I wonder if there is really no known analytical solution to the double punctual pendulum move.

• Is there an exact analytical solution ?
• If no, is there an approximate analytical solution ?
• If no, what are the precise obstacles to overcome to find an analytical solution ?

Research advances... when we search ! Bete spatio-temporelle152.77.24.34 (talk) 16:03, 7 March 2013 (UTC)

No analytical solution is not quite what it says.
• The trivial solution θ1(t) = θ2(t) = 0 is exact though somewhat dull.
• More interestingly, the double pendulum displays periodic behaviour for certain initial conditions, and any particular instance of this could surely be represented to arbitrary accuracy with a fourier series.
• The chaotic behaviour for other initial conditions makes a general analytic solution seem unlikely (though I don't know whether this proves anything).
The article would benefit greatly from some proof of chaotic behaviour and/or non-integrability if such exist. Perhaps you could read-up on this topic and fix the article? --catslash (talk) 14:58, 9 March 2013 (UTC)

Since a double pendulum exhibits periodic motion, there exist approximate analytical solutions as demonstrated in Floquet theory and Hill differential equation.

The two problems with obtaining an analytical solution using these methods is in the first place approximating the sine and cosine in the equations of motion and on the other hand you need to truncate the Hill differential equation to a finite number of harmonics.

For the rest it can work quite well for 2 degree of freedom dynamical systems with a periodic term, as long as the steady state solution is stable. 134.157.34.132 (talk) 17:16, 10 March 2016 (UTC) Barend

## chaos

What units are used for angles < θ1, θ2; CHAOTIC MOTION, Double pendulum >

in the x-y graph where: x axis: angle θ1 is plotted for -3 to +3 (left to right) y axis: angle θ2 is plotted for -3 to +3 (top to bottom)

the graph is described as: < Graph of the time for the pendulum to flip over as a < function of initial conditions

thanks for the information!

veritus 16:15, 24 February 2007 (UTC)

## the period of a double pendulum with a small angle

Ok what is the period of a double pendulum with a small angle?

anyone?

hello? —Preceding unsigned comment added by 128.226.162.108 (talk) 18:57, 20 February 2008 (UTC)

## Human Double Pendulum?

Hello, As a dance and movement experiment, I am interested in constructing a human double pendulum - that is, with humans as bobs or masses. I want to hang the first person in a climbing harness by a rope suspended from a truss. Then I want to hang the second person from the first person. Any suggestions?

Heather @ Helium Aerial Dance www.heliummm.com 64.131.241.205 (talk) 17:06, 25 February 2008 (UTC)

## Animated Picture

Could we include an animation of a double pendulum? I am interested in how it moves and how the bottom one swings in relation to the top one. 203.134.124.36 (talk) 04:11, 6 August 2008 (UTC)

Scroll down the page a bit. This an animation of a compound double pendulum (because it's an integration of the equations given in the article). I should think the motion of a simple double pendulum would look pretty similar though. --catslash (talk) 09:50, 6 August 2008 (UTC)

## Chaotic motion: time elapsed until the double pendulum "flips over"

1) I assume that "to flip over" means to have the bar l2 vercital (θ2 = π).

2) The mentioned quantities ${\displaystyle 10{\sqrt {g/\ell \ }}}$, ${\displaystyle 100{\sqrt {g/\ell \ }}}$, etc. are not times but inverse of times. Maybe ${\displaystyle 10{\sqrt {\ell \ /g}}}$, ${\displaystyle 100{\sqrt {\ell \ /g}}}$, etc.? —Preceding unsigned comment added by 81.208.31.210 (talk) 12:07, 13 September 2008 (UTC)

Someone should verify the first assumption and clarify in the article. It is not apparent. LokiClock (talk) 17:24, 22 December 2009 (UTC)

3) I don't understand how this plot was produced. What is the source of this plot? As explained the pendulum is a chaotic dynamical system. From some simulations I did, if one perturbs the initial condition by some small delta, the trajectory diverges quickly. In non-dimensional units, at t=10, the perturbation grows to about 10 delta (to simplify). This means that for example if delta = 10^-12, around time t=120, this delta has grown to about 1 and the two trajectories become completely different. The exact numbers are not important. The point is that it is very difficult to integrate exactly the equations of motion beyond t=100 or 200. One needs to use extended precision arithmetic. Even then, here there is a mention of 10,000. I did not attempt that calculation but that seems like a very challenging problem. Therefore back to my question, how was this plot produced? It would be easy to consider a numerical integrator with some fixed time step and produce such a plot without considering the accuracy. However this plot would then completely depends on the choice of numerical integrator and time step. I suspect this is what is happening here. This plot is not showing the exact solution to this problem but rather the behavior of some approximate numerical integrator. I might be wrong but I would like to have a clarification of this point. Zirhom (talk) 18:36, 5 March 2013 (UTC)

4) see -http://nldlab.gatech.edu/w/index.php?title=Double_pendulum#References -Bender, Feinberg, Hook and Weir (2009): Chaotic systems in complex phase space, Journal of Physics, Vol. 73, No. 3, September 2009, pp.453-470 for similar plots/explanations. — Preceding unsigned comment added by 136.172.247.44 (talk) 13:55, 15 June 2016 (UTC)

5) How excatly was the equation for the ellipse derived? It seems correct but isn't proven — Preceding unsigned comment added by 136.172.247.44 (talk) 14:25, 15 June 2016 (UTC)

6) Considering the figure of the time for the pendulum to 'flip', I figured out that this plot has to show the time for a flip, only considering the second pendulum to flip. The formula derived actually describes the area of sets of initial conditions with enough potential Energy to make the second pendulum flip, while the first pendulum has angle zero. This can easily be proven by calculating the Epot_flip of the whole system for angle_1 = 0 and angle_2 = 180degree (situation of a flip of the second pendulum) Plotting the energy Epot and cutting removing values for Epot<Epot_flip, one obtains this cat-eye shaped area. EDIT: This was my edit, now signing it. I didn't have an account back then. --FridtjofNansen (talk) 08:12, 31 March 2017 (UTC)

## Why we use the inertia about the centre of mass, not about the end

Of the two bodies the first pendulum is certainly rotating about its end. The second pendulum is only rotating about its end when the when the first one is stationary. When the first pendulum is moving, then the second is rotating about some point in space which is harder to locate.

The quantity wanted to write the Lagrangian is the total kinetic energy of the two the pendulums. We could write the KE of the first pendulum as

${\displaystyle \mathrm {KE} _{1}={\frac {1}{2}}I_{end}{\dot {\theta _{1}}}^{2}}$

where (as you stated)

${\displaystyle I_{end}={\frac {1}{3}}ml^{2}}$

Alternatively we could write this energy (and have written it) as the sum of the kinetic energy due to the motion of the centre of mass of the pendulum, plus the kinetic energy due to rotation about this centre of mass (see the kinetic energy article). That is

${\displaystyle \mathrm {KE} _{1}={\frac {1}{2}}mv_{1}^{2}+{\frac {1}{2}}I_{cm}{\dot {\theta _{1}}}^{2}}$

where

${\displaystyle v_{1}={\frac {l}{2}}{\dot {\theta _{1}}}}$

(the centre of mass being halfway along the rod, and hence at a radius of ${\displaystyle l/2}$ from the pivot), and where

${\displaystyle I_{cm}={\frac {1}{12}}ml^{2}}$

Either way we get

${\displaystyle \mathrm {KE} _{1}={\frac {1}{6}}ml^{2}{\dot {\theta _{1}}}^{2}}$

Both ways are correct. The first way (using the moment of inertia about the end) is simpler, and therefore the preferred way to do it for a single pendulum.

The motion of the second pendulum is more complicated, it's both swinging and moving as whole. To use the first method we'd have to work out the centre of rotation for the combined motion, and then the moment of inertia about that point. This seems difficult, so we use the second method.

Of course we could use one method for the first pendulum and another method for the second, and write

${\displaystyle \mathrm {KE} _{\mathrm {total} }={\frac {1}{2}}I_{end}{\dot {\theta _{1}}}^{2}+{\frac {1}{2}}mv_{2}^{2}+{\frac {1}{2}}I_{cm}{\dot {\theta _{2}}}^{2}}$

but somebody has decided that it's neater to write

${\displaystyle \mathrm {KE} _{\mathrm {total} }={\frac {1}{2}}mv_{1}^{2}+{\frac {1}{2}}I_{cm}{\dot {\theta _{1}}}^{2}+{\frac {1}{2}}mv_{2}^{2}+{\frac {1}{2}}I_{cm}{\dot {\theta _{2}}}^{2}}$

which comes to the same thing.

Sorry that was a bit long - hope it made sense! --catslash (talk) 00:21, 20 September 2008 (UTC)

Thank you for this clarification, it was really helpful. User:Nillerdk (talk) 14:12, 22 January 2009 (UTC)

## removed horology definition

It seemed to be talking about a clock with two pendulums hanging down, that are swinging precisely in anti-phase. That's not what this article is about, the pendulums are hung end-to-end in this article. The maths and everything else would be completely different. Because it was unreferenced and clearly distinct, I removed it.- Wolfkeeper 17:08, 1 April 2010 (UTC)

## Error in Lagrangian

Why do you include the "rotational energy" in the Lagrangian? Kinetic energy in terms of x1,y1 and y1,y2 is enough to specify the Lagrangian => transform to theta1 and theta2. Seems to be an error, unless I'm missing something. Someone should look into that and fix it. Danski14(talk) 20:10, 12 September 2010 (UTC)

I see the description above, but that is a very confusing way to do it. I would recommend changing to the normal way of solving this problem.Danski14(talk) 20:12, 12 September 2010 (UTC)
You may have overlooked the fact that the analysis and description relate to compound pendulums of equal length, while the first diagram shows simple pendulums of unequal length. For simple pendulums there need be no rotational energy terms since they are point masses. The article history shows that the diagram was removed in April for this reason - but it was then replaced. There is probably less work in replacing the first diagram, rather than rest of the article. --catslash (talk) 21:02, 12 September 2010 (UTC)
If you are not in a hurry, I could make an appropriate SVG diagram. --catslash (talk) 21:12, 12 September 2010 (UTC)
Done: the diagram now exactly matches the text of the analysis. It should make more sense now. --catslash (talk) 01:25, 22 September 2010 (UTC)

## Article citation

Most of the content of this article appear to be copied directly from this paper: http://tabitha.phas.ubc.ca/wiki/images/archive/3/37/20080811183757!Double.pdf — Preceding unsigned comment added by 24.56.46.24 (talk) 17:16, 4 January 2014 (UTC)

The paper is dated 11 Aug 2008, so if anything the article includes material copied from this article (check the Jun 2008 version of this article). Another possibility is that the author of the paper and the relevant contributor to this article is the one and the same person. --catslash (talk) 21:59, 4 January 2014 (UTC)
In fact a contributor to this article gives on his user-page a real name and affiliation identical to that of the author of the paper (caution: read WP:OUTING before mentioning real names). --catslash (talk) 22:29, 4 January 2014 (UTC)
That makes sense then. I don't think the author would have copied from here, since the fractal image's origin isn't really explained here, but it is in the paper. — Preceding unsigned comment added by 24.56.46.24 (talk) 02:41, 5 January 2014 (UTC)

## Practical example of Double-Pendulum in real life

In a practical, real life example, the double pendulum can be seen inthe very dangerous design of the double trailer trucks in highways. Frequently, their chaotic behaviour has produced large accidents. If Highway authorities could understand some mathematics and physics, their use would be banned immediately! Their dynamic behaviour is called "Jackknifing" (there is a Wikipedia page on this). Recently, a double tank trailer went out of control near Mexico City, the tanks were carrying Liquified Petroleum Gas or LPG, causing a huge series of explosions and fire, killing several families that lived at the side of the highway. — Preceding unsigned comment added by 192.100.180.21 (talk) 17:08, 23 July 2014 (UTC)

## Chaos or not enough energy?

The introduction infers that the double pendulum is chaotic ONLY "for certain energies". However the section on Chaos states simply that it: "The double pendulum undergoes chaotic motion, and shows a sensitive dependence on initial conditions" which infers that motion of the pendulum is chaotic.

A cursory search on Google confirmed my old school knowledge, that a double pendulum behaves chaotically, so I have edited the introduction to conform to the current snese of the latter statement in the article: "The motion of a double pendulum is governed by a set of coupled ordinary differential equations and is chaotic."

However if the second defintion was intended to read: "The double pendulum shows a sensitive dependence on initial conditions and under certain of these undergoes chaotic motion" then when someone edits it accordingly would they ALSO please explain in the text what the initial conditions must be to result in chaos. Such an explanation MUST in the first instance be verbal not mathematical.

LookingGlass (talk) 12:16, 15 February 2016 (UTC)

For certain initial conditions, the double pendulum appears to undergo periodic motion without any great sensitivity to those conditions. For example, when both pendulums swing together with a small amplitude, then the system seems to behave as a single pendulum which flexes slightly at its centre. I assume that this is the reason for [this edit], which inserts the words Above a certain energy its motion is chaotic. It occurred to me that if this low energy state was not chaotic, then a very high energy state with both pendulums rapidly rotating as one though full revolutions was unlikely to be chaotic either. Consequently I weakened above a certain energy to for certain energies in [this edit].
However, it could be that double pendulum has chaotic behaviour for all initial conditions of non-zero energy, but for certain energies the divergence of initially similar trajectories is too slow to be discernible numerically. After all, the evolution of the solar system is chaotic, despite the apparently regular orbits of the planets. --catslash (talk) 00:14, 16 February 2016 (UTC)

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