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Euler-Lagrange equation for paraboloid plane

  1. Jun 5, 2006 #1
    I have a classical mechanics question I couldn't conclude. The reason seems to be mathematical. It's this:
    There's a paraboloid shaped plane of mass M, which is standing on a frictionless surface and can slide freely. It's surface is [tex]y=ax^2[/tex]. A point mass m is place on the plane. Solve the Euler-Lagrange equations of the system for little mass.

    My (unsuccessful) solution is as follows:

    I chose [tex]x_m[/tex] as the x (horizontal) coordiante of point mass, and for sliding, plane it's [tex]x_M[/tex]. I defined another coordinate for point mass: [tex]x[/tex] is the horizontal distance of point mass from the center (or bottom) of the parabol. So, coordinates of m are
    [tex]x_m = x_M + x[/tex]
    [tex]y_m = ax^2[/tex]
    so velocities are
    [tex]\dot{x_m} = \dot{x_M} + \dot{x}[/tex]
    [tex]y_m = a2x\dot{x}[/tex]

    So kinetic energy of the system is:
    [tex]T = \frac{m\dot{x_m}^2}{2} + \frac{M\dot{x_M}^2}{2}[/tex]
    [tex]T = \frac{m ( \dot{x_M}^2 + \dot{x}^2 + 2\dot{x}\dot{x_M} + 4a^2x^2\dot{x}^2 )}{2} + \frac{M\dot{x_M}^2}{2}[/tex]
    and potential is
    [tex]V = mgy = mgax^2[/tex]

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

    Now, Euler-Lagrange equations are:
    (1) [tex]\frac{d}{dt} \frac{\partial {\cal L}}{\partial \dot{x_M}} - \frac{\partial {\cal L}}{\partial x_M}[/tex]
    (2) [tex]\frac{d}{dt} \frac{\partial {\cal L}}{\partial \dot{x}} - \frac{\partial {\cal L}}{\partial x}[/tex]

    The solution of the first one yields (of course, if I done the math correctly):
    [tex]\ddot{x_M}(m+M) + \ddot{x}m = 0[/tex]
    and second is
    [tex]\ddot{x} + \ddot{x_M} + 4a^2x\dot{x}^2 4a^2x^2\ddot{x} + 2gax = 0[/tex]

    I tried combining two solutions and got this non-linear differential equation:

    [tex]\ddot{x}(1- \frac{m}{m+M} + 4a^2x^2) + 4a^2x\dot{x}^2 + 2gax = 0[/tex]

    Maybe there was a mathematical mistake in my solution, maybe this differential equation could be solved with some tricks, or maybe I'm downright wrong by choosing such coordinates. I don't know.

    Is there someone who can solve this question?
    Last edited: Jun 5, 2006
  2. jcsd
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