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Energy of a pendulum (variable length, Lyapunov)

  1. Mar 1, 2014 #1
    Hello, question about the energy of a variable length pendulum.

    Suppose you have a pendulum in the standard sense where θ is the angle, and we let the length r be a function of time r = r(t). What is the energy of the pendulum?

    So far, I have determined that kinetic energy is = (1/2)m(r*dθ/dt)^2 + (1/2)m*(dr/dt)^2
    and the potential energy is = mgr(1-cosθ).

    In my homework problem i need to come up with a suitable Lyapunov function to study the stability of the pendulum and the typical approach approach is to set the lyapunoc function V = E (energy).

    But the problem is that this is a time varying lyapunov function, i.e. V = V(θ,t). And i have to satisfy a positive definite constraint on V that is V(0,t) = 0 for all t. the problem is that dr/dt term.

    Is there another lyapunov function i can choose here? or am i misrepresenting the energy?
    Thank you
     
  2. jcsd
  3. Mar 6, 2014 #2

    BvU

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    Hello robbed,

    I don't see an answer here, but I do have a question: what can be a possible equilibrium point if you don't know anything about r(t) ?
     
  4. Mar 6, 2014 #3
    So I already solved this question myself. Turns out that I did have the proper Lyapunov function.

    Also BvU, in this homework problem there were some bounds given for r(t), and its derivatives which I omitted because it was irrelevant to my question. To answer your question though, there aren't any equilibrium points unless r(t) is constant. homework problem was about stability anyways,
     
  5. Mar 6, 2014 #4

    BvU

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    Very good ! Thanks for the extra info.
     
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