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Homework Help: Small oscillations (normal modes)

  1. Apr 16, 2006 #1
    Hi

    see the attached picture...
    2 coupled masses, each suspended from spring in gravitational field...
    also entire construction can vibrate only vertically...

    I need to write lagrangian for this system in the following form:

    [tex]
    L=\frac{1}{2}\sum_{i=1,j=1}^2m_{ij}\dot{x}_i\dot{x}_j-\frac{1}{2}\sum_{i=1,j=1}^2k_{ij}x_ix_j
    [/tex]

    [itex]x_i[/itex] and [itex]x_j[/itex] are displacements of masses from equalibrium positions and springs are identical


    the kinetic part of L is easy one but i stack with potential part...
    i get to this expression(i always confused about right signs):

    [tex]m_1g(l+x_1)+m_2g(l+x_1+l+x_2) + \frac{1}{2}m_1{x_1}^2+\frac{1}{2}m_2{x_2}^2[/tex]

    - positive direction taken along g direction
    - l is length of unstreched spring

    every potential term in L must in [itex]k_{ij}x_ix_j[/itex] form but i don't see a way how do i get it from my expression for potential energy because there are g force terms containing only [itex]x_i[/itex] or [itex]x_j[/itex]


    thanks
     

    Attached Files:

    Last edited: Apr 16, 2006
  2. jcsd
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