1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lagrangian of pendulum

  1. Apr 3, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider a pendulum of mass m and length b in the gravitational field whose point of attachment moves horizontally [tex]x_0=A(t)[/tex] where [tex]A(t)[/tex] is a function of time.

    a) Find the Lagrangian equation of motion.
    b) Give the equation of motion in the case of small oscillations. What happens in that case when [tex]A(t)=cos\left(\sqrt{\frac {a} {b}}t\right)[/tex]


    2. Relevant equations

    [tex] {\cal L} = T - U [/tex]

    3. The attempt at a solution

    a) The position of the pendulum would be given by:

    [tex] x = A(t) + bsin\left(\theta\right) [/tex] [tex] \dot{x} = \dot{A}(t) + b\dot{\theta}cos\left(\theta\right) [/tex]
    [tex] y = bcos\left(\theta\right) [/tex] [tex] \dot{y} = -b\dot{\theta}sin\left(\theta\right) [/tex]

    The kinetic energy [tex]T[/tex] would be equal to:

    [tex] T = \frac {m} {2} \left({\dot{A}(t)}^2 + 2\dot{A}(t) b \dot{\theta} cos\left(\theta\right) + b^2\dot{\theta}^2\right) [/tex]

    and taking the zero potential to be at [tex] x = 0 [/tex] I get that the potential is equal to :

    [tex] U = -mgy = -mgbcos\left(\theta\right) [/tex]

    And the Lagrangian would be:

    [tex] {\cal L} = T - U = \frac {m} {2} \left({\dot{A}(t)}^2 + 2\dot{A}(t) b \dot{\theta} cos\left(\theta\right) + b^2\dot{\theta}^2\right) + mgbcos\left(\theta\right) [/tex]

    I would like to know if I have represented the position of the pendulum the right way because I get a non-linear differential equation for part b and I doubt that's right. Thanks!
     
  2. jcsd
  3. Apr 4, 2012 #2
    I'm sure the system will be nonlinear. What you should assume is that the oscillations are small, θ << 1. Then you can, by construction, drop all the nonlinear terms.
     
  4. May 7, 2012 #3
    May i know what will be the phase plot for the same? I mean how should i proceed to get the phase plot for the same pendulum as above.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Lagrangian of pendulum
Loading...