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radioactive8
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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.