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Lagrangian (chain off spring connected masses)

  1. Mar 27, 2009 #1
    1. The problem statement, all variables and given/known data
    ______|equillibrium position________
    ______|__i_____________________
    m^^^^^m^^^^^m^^^^^m^^^^^m
    ____k_|qi|____k______ k________k


    A collection of particles each of mass m separated by springs with spring constant k. The displacement of the ith mass from its equilibrium position is q_i=q_i(t). Write the Langrangian for this one dimensional chain of masses.

    2. Relevant equations
    [tex]K=\frac{1}{2}mv^2[/tex]
    [tex]V=\frac{1}{2}kx^2[/tex]
    [tex]L=K-V[/tex]


    3. The attempt at a solution
    So we have [tex]K=\frac{1}{2}mv^2=\frac{1}{2}m(\frac{dx}{dt})^2[/tex]
    Can I now say
    [tex]K=\frac{1}{2}m\frac{dq_i}{dt}^2[/tex]
    ?
    (I think I have to because I then have to differentiate wrt q_i
    Also, the extension of the springs to begin with must matter because if they are taught then the potential energy must be higher. Am I right? Does it matter that there is a spring on either side? I guess it does, but as the mass is just moving between its equilibrium point maybe they kind of cancel out. There are N particles, so am I right in thinking that the energy required is the sum of all functions of [tex]q_i[/tex] for every [tex]i\leq N[/tex]?

    I haven't been given very long for this, and I've never come across this stuff before, so I'm having issues figuring it out. Thanks for the help.

    EDIT: Yeah, so I'm thinking [tex]V=\frac{1}{2}kq_i ^2[/tex] or [tex]V=kq_i ^2[/tex] but then I'm still not sure if the original taughtness of the spring comes into it.
     
    Last edited: Mar 27, 2009
  2. jcsd
  3. Mar 27, 2009 #2
    I think I would make a different coordinate for each mass. So mass 1-x1, mass 2-x2 and so on{at equilibrium position}. Get the potentials for each mass in terms of those different coordinates. Get kinetic energy for each mass, then do L=K-V. I think the typical convention though is L=T-U.
     
  4. Mar 29, 2009 #3
    Okay, thank you I will try that. I'm just using the notation that was given in the handouts. Not sure if there is any significance.
     
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