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Determining stability about a critical point using eigenfunctions

  1. Aug 2, 2012 #1
    I admit I am a bit out of practice when it comes to DiffEq. I think I am either forgetting a simple step or getting my methods mixed up.
    1. The problem statement, all variables and given/known data
    The problem concerns a pendulum defined by

    d2θ/dt2 + (c/mL)(dθ/dt) + (g/L)sinθ = 0
    where m=1, L=1, c=0.5, and of course g=9.8

    After converting the DE to a first order system:

    set x = θ and y = x' so that
    dx/dt = y
    dy/dt = -9.8sinx - 0.5y

    And identifying critical points:

    (n∏,0) where n is an integer

    I am asked to linearize the nonlinear system and determine stability about the critical points. If i can get some help with the first point, I should be able to figure out the others.

    2. Relevant equations

    dx/dt = y
    dy/dt = -9.8sinx - 0.5y

    3. The attempt at a solution

    I have been rummaging through my notes on eigenfunctions and am more or less at a loss how an eigenvalue determines stability. I think I want to take the partial derivatives of the system and evaluate at the point (0,0)...

    fx(dy/dt) = -9.8cosx
    fy(dy/dt) = -0.5
    fx(dx/dt) = 0
    fy(dx/dt) = 1

    evaluated @ (0,0):
    fx(dy/dt) = -9.8
    fy(dy/dt) = -0.5
    fx(dx/dt) = 0
    fy(dx/dt) = 1

    But I'm not sure why I care or how to proceed to determine the stability about that point.

    Clarification: I would like some tips on how to linearize my function and how to get it into the form with which I can determine its eigenvalues ( A-lambda)v=0
     
    Last edited: Aug 2, 2012
  2. jcsd
  3. Aug 2, 2012 #2

    vela

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    Expand sin x as a first-order Taylor polynomial. That's the linear approximation of sin x.
     
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