# Stability analysis of a system of 2 second order nonlinear differential equations [closed]

I am working on a project to write a paper on the stability of a double pendulum. The only problem is that we never covered anything that could cover a system of two second order nonlinear ODEs (taylor expansion and a little of perturbation, to make phase planes). My professor suggested(Based on the wolfram site) that I analyze the energy of the system, so I have picked up some graduate textbooks on mechanics to try and learn this.

So, I ask you if there is anything specific you can tell me to look into to be able to provide some analysis of this system for my paper.

This the system, in the form of 4 first order equations:

deq1 := diff(v1(t), t) = (2*sin(a1(t))*m1*g+m2*g*sin(a1(t)-2*a2(t))+sin(a1(t))*m2*g+m2*v1(t)^2*L1*sin(-2*a2(t)+2*a1(t))+2*m2*v2(t)^2*L2*sin(a1(t)-a2(t)))/(-2*L1*m1-L1*m2+L1*m2*cos(-2*a2(t)+2*a1(t)))
deq2 := diff(a1(t), t) = v1(t)
deq3 := diff(v2(t), t) = (-2*v1(t)^2*L1*m1*sin(a1(t)-a2(t))-m2*v2(t)^2*L2*sin(-2*a2(t)+2*a1(t))-m2*g*sin(-a2(t)+2*a1(t))+sin(a2(t))*g*m1-m1*g*sin(-a2(t)+2*a1(t))+sin(a2(t))*g*m2-2*m2*v1(t)^2*L1*sin(a1(t)-a2(t)))/(-2*L2*m1-L2*m2+L2*m2*cos(-2*a2(t)+2*a1(t)))
deq4 := diff(a2(t), t) = v2(t)

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I don't agree with this being closed for being localized. It's motivated by a specific example, yes, but then so are most mathematical problems. (Moreover, the specific example in this question looks like it might be interesting in its own right.) The real question seems to be (a) generally, how to analyze stability, and (b) how to get mileage out of energy conservation in cases like this. Both parts sound like perfectly respectable questions to me. –  Darsh Ranjan Dec 5 '09 at 9:52
I agree with Darsh; this question probably needs some editing to sound more like one for MO, but it does have good pieces to it. I think if Jeremy combines his two questions then it'll end up being a nice question. –  Ben Weiss Dec 5 '09 at 16:11
Ben and Darsh- I think "too localized" really means "do your own homework." Obviously this is a bit more interesting than many of the HW problems that show up, but MO still has a "no homework" policy in the FAQ mathoverflow.net/faq#whatnot –  Ben Webster Dec 5 '09 at 16:20
Sounds good Ben, thanks for explaining. –  Ben Weiss Dec 5 '09 at 18:13
I am having trouble finding the part where I asked anyone to do my homework for me, the only thing I asked for was a survey of analysis of equations more complicated then I am familiar with. –  Jeremy Dec 5 '09 at 19:36
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## closed as too localized by Scott Morrison♦Dec 5 '09 at 8:43

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