Equation of motion of a Lagrangian density

[radioactive8]

Homework Statement

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.

Homework Equations

bm is real.

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.

 

[Orodruin]

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.

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?

 

[Mike Stefan]

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.