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Action integral

  1. Jun 9, 2009 #1

    CNX

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    The problem statement, all variables and given/known data

    Show the stationary value of,

    [tex]J = \int_{a}^{b} dt~L(...;x_i;\dot{x}_i;...;t)[/tex]

    subject to the constraint,

    [tex]\phi(...;x_i,\dot{x}_i;...;t) = 0[/tex]

    is given by the free variation of,

    [tex]I = \int_{a}^{b} dt~F = \int_{a}^{b}dt~\left[L(...;x_i;\dot{x}_i;...;t)-\lambda(t)\phi(...;x_i,\dot{x}_i;...;t)\right][/tex]

    The attempt at a solution

    Not sure where to start here; or really what's wanted... Do I start with [itex]J[/itex] and [itex]\phi[/itex] and get to the variation of [itex]I[/itex]?

    Is the free variation of [itex]I[/itex] given by,

    [tex]\delta I = \int_{a}^{b}dt~\left[\frac{\partial F}{\partial x} \delta x \frac{\partial F}{\partial \dot{x}} \delta{\dot{x}}\right][/tex] ?
     
    Last edited: Jun 9, 2009
  2. jcsd
  3. Jun 10, 2009 #2

    Matterwave

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    It's been a while since I did this, but you may want to take a look at "Lagrange multipliers", that might get you on track.
     
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