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How to integrate velocity squared to solve action integral?

  1. Feb 1, 2013 #1
    Consider a forcefree mass point in one-dimensional space.
    (a) Calculate the action S for the actual path of the mass point in the time interval
    [0, T] and for the boundary conditions x(0) = 0 and x(T) = d.

    I said the Lagrangian was just equal to L=1/2mv^2. I'm not sure if my reasoning for this is correct (I may have a conceptual error) but since there is no force acting on it, the potential is 0 (or a constant but it can be set to 0?)

    so the action integral is S=∫1/2mv^2dt where the limits are 0-T

    I'm not sure how to integrate v^2 with respect to t. Even if my approach is wrong, I would still like to know how that integral is done
     
  2. jcsd
  3. Feb 1, 2013 #2
    oops, velocity would be constant...so that solves that. However, if it wasn't constant, how would I go about solving that integral?
     
  4. Feb 1, 2013 #3

    haruspex

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    You cannot solve it without some information about how velocity varies with time.
     
  5. Feb 1, 2013 #4
    You need to either know v(t) or v(x). If you know the second then you can use the fact that v2dt=vdx.
     
  6. Feb 6, 2013 #5
    ahh, ok. Thank you
     
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