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How to understand potential energy in Lagrangian

  1. Jan 27, 2015 #1
    Hi guys,

    So I'm trying to understand why the potential energy of a Lagrangian is the way it is.

    The system I'm considering is a closed necklace of N beads, each of mass m. Each bead interacts only with its nearest neighbour.

    First let me make some comments:
    1) Each bead is labeled with a generalised coordinate [itex]q_{i}[/itex]
    2) there is no explicit time dependence of the generalised coordinates
    3) the system is conservative, so the potential is a function only of the generalised coordinates: [itex]V=V(q_{1},q_{2},\dots q_{N})[/itex],

    The Lagrangian for this system is

    [itex]L=\frac{1}{2}\sum_{i=1}^{N}m\dot{q}_{i}^{2}-\frac{1}{2}\sum_{i=1}^{N}hq_{i}^{2}-k(q_{i}-q_{i+1})^{2}[/itex].

    I dont understand why the potential has this form. I think i know where the second term [itex]-k(q_{i}-q_{i+1})^{2}[/itex] comes from - its due to the harmonic approximation. But what about the first term?
     
  2. jcsd
  3. Jan 28, 2015 #2

    DrClaude

    User Avatar

    Staff: Mentor

    What is ##h##?
     
  4. Jan 28, 2015 #3
    To be honest I dont think [itex]h[/itex] has a particular meaning - it's just a constant for dimensional consistency perhaps? something like that
     
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