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Homework Help: Action in the continuum limit for an elastic medium.

  1. Apr 7, 2013 #1
    1. The problem statement, all variables and given/known data
    I'm stuck at the second part, not really sure what to here to be honest.

    2. Relevant equations
    Usually the Lagrangian is the function inside the integral for the action.
    L = T - U.
    Usually this becomes
    L = \frac{p^{2}}{2m} - U.

    3. The attempt at a solution
    I figured for a single particle the kinetic energy is:
    T = \sum_{i=1}^{3} \frac{1}{2} m (\delta_{t} u_{i})^{2}.
    The potential energy should be some force constant times the displacement squared:
    U = \sum_{i=1}^{3} k (a \delta_{i}u_{i})^{2}.
    So for N\end particles the Lagrangian would become:
    L_{discrete} = \sum_{n=1}^{N} \sum_{i=1}^{3} \left[ \frac{1}{2} m (\delta_{t} u_{i})^{2} - \frac{k}{3} (a \delta_{i}u_{i})^{2}\right].
    Where the factor 1/3 comes from the fact that I am counting each bond three times.
    Now this is where it gets a bit fuzzy for me; what exactly does it mean to take the continuum limit? I believe it has something to do with letting m and a approach zero in such a way that if N approaches infinity the mass stays constant, but I'm not exactly sure how to do that/what that means.

    I believe that I require the fact that this is a cubic lattice to be able to write the potential energy down the way I did. But how would change the potential energy if, say for instance, the particles were on a face centered cubic lattice? Or any other lattice for that matter.

    Any help would be appreciated.
    Last edited: Apr 7, 2013
  2. jcsd
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