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How does one linearize and analyze such a system?

Just noticed I could edit this, so from my comment below:

I am trying to get a feel for what analysis us used beyond the introduction I have had. The equations for the double pendulum were derived from the second derivative of the equations for position of each mass and the tension of the mass against the rods. So far the only understanding I have been able to get from it is using numerically generated phase planes and plots of motion made in maple.

I have seen an overview of the analysis mentioned on wolfram's site and am reading into that, but these books on mechanics are fairly broad, and I am looking to see if there is a specific field I need to look into.

Edit in response to comment: Not looking for this to be simplified to the level of a pendulum, I just need to know how to approach the problem. I do not know what analytical tools would be used to better understand the system(as I understand it, there is no solving for such a complicated system in the same form that a regular nonlinear pendulum can be).

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Jeremy- Your last question was closed for being a homework problem. Writing another one that's so vague most people won't follow it won't help matters. –  Ben Webster Dec 5 '09 at 16:36
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fpqc: that's not really true. We do have mathematical physics questions here, and studying the differential equations of systems of pendulums should be fair game, so long as the question is phrased clearly and mathematically. Now, Jeremy seems to be asking about how to take a nonlinear system of ODEs and linearize it, probably hoping for an analogy to the single pendulum where you got from $x''+sin x=0$ to $x''+x=0$ for small amplitude approximation. I don't know if there's a way to do this, because I don't know this stuff at all, but if he clarified, it should be a perfectly good question. –  Charles Siegel Dec 5 '09 at 20:42
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@fpqc: there is no need to be so rude. Also, MINUS 1 POWER1111111111111 = MINUS 1 ;) –  Alberto García-Raboso Dec 5 '09 at 22:43
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@fpqc: all these homework questions come from different people. If all of them were coming from the same person, then maybe rudeness would have some effect, even though I don't find it appropiate. I think there's a discussion to be had on tea.mathoverflow.net about this, since MO will keep growing and inevitably will me more visible on Google and other search engines, which will most probably attract more people looking to get there homework problems solved. –  Alberto García-Raboso Dec 6 '09 at 14:46
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@fpqc. Rudeness is unacceptable. I've failed at this in the past, but I'll try to reform too. :-) –  Scott Morrison Dec 9 '09 at 8:45
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2 Answers

up vote 2 down vote accepted

This is an answer to Charles' restatement of the question.

Recall that equation F(x,x',x'') = 0 (e.g. x'' + sin x = 0) can be written as a system

X' = f(X), where X = (x,x')^T (e.g. f(x',x) = (-sin x, x')^T) and that system can be linearized about an equilibrium E = (x_,x'_)^T to obtain a linear equation X' = AX where A is the 2 x 2 matrix given by the derivative of f at E.

So too a larger system F(x,x',x'',y,y',y'') = 0 can be written as a system

X' = f(X) where X = (x,x',y,y')

Given an equilibrium E = (x_,x'_,y_,y'_), the linearization of X' = f(X) about E is again X' = AX where X = (x',x,y',y) and A is the derivative of f evaluated at E. A is a 4 by 4 matrix. If all eigenvalues of A have negative real part, the system is stable. If one eigenvalue has positive real part, the system is unstable. If there are no eigenvalues with positive real part, and there are eigenvalues which lie on the imaginary axis, then the equilibrium is "spectrally stable," and further analysis is required to determine the nonlinear stability.

With regard to the energy of the system, you are looking for a function of the form V(x,x',y,y') whose time-derivative along solutions is constant. A good place to start is with the guess 1/2((x')^2 + (y')^2) + F(x,y). You should be able to figure out what F needs to be in this particular example.

For future reference, this forum is (I believe) intended primarily for questions from students and practitioners who are a little further along in their study. I decided to answer the question because I can imagine being very frustrated at not having some information that it would take an expert five minutes to explain and the question didn't strike me as the kind which would encourage others to attempt to turn this forum into a homework help site. If the moderators disagree, I apologize.

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More generally, nonlinear equations can be linearized along certain "manifolds", i.e. only linear terms are needed to describe the essential dynamics near these manifolds.

http://en.wikipedia.org/wiki/Center_manifold

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