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Equation of motion of a Lagrangian density

  1. Dec 30, 2017 #1
    1. The problem statement, all variables and given/known data
    from the lagrangian density of the form : $$L= -\frac{1}{2} (\partial_m b^m)^2 - \frac{M^2}{2}b^m b_m$$
    derive the equation of motion. Then show that the field $$F=\partial_m b^m $$ justify the Klein_Gordon eq.of motion.
    2. Relevant equations
    bm is real.

    3. The attempt at a solution
    from the E-L equations I have reached the following results:

    $$\partial_m (\frac{\partial L}{\partial (\partial_m b^m)})= \partial_m \partial_m b^m$$

    which I find it to be problematic as I have three same indexes.
    The other derivative is : $$\frac{\partial L}{\partial b^m}= M^2 b_m$$

    So the eq.of motion is :

    $$\partial_m \partial_m b^m= -M^2b_m$$.

    I just find the above really odd so I believe I have made a mistake at my calculations.
     
  2. jcsd
  3. Dec 30, 2017 #2

    Orodruin

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    Indeed you should find it problematic. You should never have three of the same indices. Can you show us how you got this from the EL equations?
     
  4. Dec 31, 2017 #3
    The Lagrange function contain the maximum amount of "dynamical" information about any physical system(composed of particles and various Fields). So any physical system in the nature is "moving" according to
    Hamilton principle of extremum ( minimal) Action. Upon variation of the
    the previous mentioned functional(the
    Action) one can
    obtain the Euler-Lagrange equations
    which could lead to a Klein-Gordon
    non -homogeneous type of equation.
    The Green function of a Klein-Gordon
    Operator is well known.
    If I were in your place I would be more
    concerned about the integrals(involved) which contain a Green function multiply by
    the "source"-density of current of the
    Field. We can end up with a so -called
    Iterative method of solving it.
    Now at the iterative method we must work together and make improvements.
     
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