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I Two different equations of motion from the same Lagrangian?

  1. Aug 5, 2016 #1
    The equation of motion of a pendulum with a support oscillating horizontally sinusoidally with angular frequency ##\omega## is given by (5.116). (See attached.)

    I get a different answer by considering the Euler-Lagrange equation in ##x## and then eliminating ##\ddot{x}## in (5.115):

    Referring to (5.114), we have
    ##\frac{d}{dt}\frac{\partial L}{\partial\dot{x}}=\frac{\partial L}{\partial x}##
    ##\frac{d}{dt}(m\dot{x}+ml\dot{\theta}\cos\theta)=0##
    ##m\ddot{x}+ml\ddot{\theta}\cos\theta-ml\dot{\theta}^2\sin\theta=0##
    ##\ddot{x}=l\dot{\theta}^2\sin\theta-l\ddot{\theta}\cos\theta##

    Substituting this into (5.115), we have
    ##l\ddot{\theta}+l\dot{\theta}^2\sin\theta\cos\theta-l\ddot{\theta}\cos^2\theta=-g\sin\theta##
    ##l\ddot{\theta}\sin\theta+\dot{\theta}^2\cos\theta=-g##

    The solution ##\theta (t)## would in general be different from the solution ##\theta (t)## of (5.116). Why are there two solutions? What does the former solution represent?

    Screen Shot 2016-08-06 at 2.00.02 am.png
     
    Last edited: Aug 5, 2016
  2. jcsd
  3. Aug 5, 2016 #2
    The reason you're getting a different answer is because you are not solving the same system. The problem gives you x (t), but you are trying to solve for it as though it is an unknown. By doing this you are changing the support from moving at a known oscillation to having its oscillating based on the swinging of the pendulum. Hope that helps
     
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