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Phase space

  1. Feb 23, 2015 #1
    1. The problem statement, all variables and given/known data

    For a double pendulum, how do we plot the phase space for ##\theta_2## (the lower of the pendulum), i.e. the plot ##\theta_2, \ \dot{\theta}_2?##

    • ##x## = horizontal position of pendulum mass
    • ##y## = vertical position of pendulum mass
    • ##\theta## = angle of pendulum (0 = vertical downwards, counter-clockwise is positive)
    • ##L## = length of rod (constant)

    2. Relevant equations

    F = ma
    ##x_1 = L_1\sin \theta_1##
    ##y_1 = L_1\cos \theta_1##


    (the ##_1## subscript is the upper pendulum while ##_2## is the lower pendulum)

    3. The attempt at a solution

    I found the equation for ##\theta''_2## (which is pretty long to write in here, but I will write if you guys want me to) and I converted the second order equation into a first order, by substituting ##\theta''_2## to ##\omega'_2##, i.e. ##\omega'_2 = \theta''_2##, but I am wondering what exactly is ##\theta_2## suppose to be here?

    Will our ##\theta_2## just be the ##\theta_2## in ##x_2 = x_1 + L_2 \sin\theta_2##?
     
  2. jcsd
  3. Feb 23, 2015 #2

    BvU

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    Can you clarify the question ? I mean, the answer "the same way as for ##\theta_1##" is probably not what you are looking for...

    Idem for "yes" to the question "will our ##\theta_2## just be the ##\theta_2## in ##x_2 = x_1 + L_2\;\sin\theta_2## ?"

    What is it that we can do for you ? You are aware that there is no analytic solution for this seemingly simple device (see wiki) ?
     
  4. Feb 23, 2015 #3
    The link that you provided has a good picture. I want to plot a graph of the angular velocity and the ##\theta_2## that is given in the picture of the wiki link you provided. But how do I find ##\theta_2## and ##\dot{\theta_2}##?
     
  5. Feb 23, 2015 #4

    BvU

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    Same link says
    See also wolfram alpha
     
  6. Feb 23, 2015 #5
    In the link, it provided ##\dot{\theta_2}## so in order to find ##\theta_2## I have to integrate ##\dot{\theta_2}## by using Runge Kutta?
     
  7. Feb 23, 2015 #6

    BvU

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    The expression for ##\dot\theta_2## happens to contain all four time-dependent variables ( ## \theta_1,\; \theta_2,\; p_{\theta_1}, \; p_{\theta_2}\; ##), so you will have to provide initial values for all four of them and then integrate all four of them...

    It's a bit of work, but then you'll get the nicest phase space plots corresponding to the spectacular animated pictures in the link !
     
  8. Feb 23, 2015 #7
    Oh wow, that will be a lot of work. So that's the only option huh. What will be good initial conditions for them?
     
  9. Feb 23, 2015 #8

    BvU

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    Check with the animated pics in the link ! But ##\pi/2, \pi/2, 0,0## seems to be one of them and it sure rocks !

    PS plotting while stepping is much more attractive than calculating the whole thing first and then producing a still picture. It will be chaotic anyway, most of the time.
     
  10. Feb 23, 2015 #9
    Thank you very much!
     
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