How can we determine lagrangian density?

[feynman60]

we know the lagrangian l=ke-pe right
in case of fields is called "lagrangian density"
let particle with mass "m" and position "x"
it kientic energy= 1/2(mv^2)
so lagrangian =1/2(mv^2)-v(x) , v(x)=potential energy
in case the field lagrangian density
how can i determine the lagragian density?
for klein gordon eq
dirac eq and any field
please anyone help

 

[montser]

Reply

we know the lagrangian l=ke-pe right
in case of fields is called "lagrangian density"
let particle with mass "m" and position "x"
it kientic energy= 1/2(mv^2)
so lagrangian =1/2(mv^2)-v(x) , v(x)=potential energy
in case the field lagrangian density
how can i determine the lagragian density?
for klein gordon eq
dirac eq and any field
please anyone help

note: I am not sure about my answer.
I think that if you have the Klein Gordon eq then you can find the lagrangian density by bulding lagrangian which the El eq for this lagrangian is exactly the Klein Gordon eq.
In the same way you can do for dirac eq

 

[masudr]

The Lagrangian density is the Lagrangian per unit volume.

As for spin-0 and spin-1/2 systems you need to first determine the object that satisfies the required transformation laws (e.g. spinor, wavefunction etc.) and then try and find suitable scalars that you can build out of these quantities.

This Wikipedia article presents some Lagrangians for the electromagnetic field, the electron field and the interaction between the two.

 

[jomoonrain]

It’s also my question。
It seems only for fields or continuous media we can talk about lagrangian density. for a particle it somewhat hard to do this.

but still， we can treat a particle as something with a considerable volume, and its lagrangian density is nothing but L/v,with L=lagrangian,v is it volume. and so lagrangian density in the "particle" keep unchanged.

it was only my opinion