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Finding equation of motion from lagrangian

  1. Jan 22, 2009 #1
    Hi, I am trying to solve this problem here:

    http://img201.imageshack.us/img201/7006/springqo9.jpg [Broken]

    We're supposed to find the equation of motion from the lagrangian and not newton's equations.

    Attempted solution:

    [tex] L = T - U = \frac{I\omega^2}{2} + \frac{mv^2}{2} - \frac{kr^2}{2} [/tex]
    [tex]I = m(r^2 + l^2) [/tex]
    [tex]v = \frac{dr}{dt}[/tex]

    From the euler-lagrange equation [tex]\frac{\partial L}{\partial r} = \frac{d}{dt}\frac{\partial L}{\partial v}[/tex] I get:

    [tex]m\omega^2r - kr = m \frac{d^2r}{dt^2}[/tex]
    [tex](\omega^2-\frac{k}{m})r = \frac{d^2r}{dt^2}[/tex]

    If anyone can see any mistakes i'd appreciate it if they could let me know. Thanks
     
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Jan 22, 2009 #2
    Your equation

    [tex](\omega^2-\frac{k}{m})r = \frac{d^2r}{dt^2}[/tex]

    appears correct. Recall that

    [tex]\frac{k}{m} = \omega^2_{spring}[/tex]

    So, this shows the special value for

    [tex]\omega^2[/tex]

    Particularly,

    [tex]\omega^2=\omega^2_{spring}[/tex]
     
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