(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Homework Help: Equation of motion of a Lagrangian density

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