Sunday, 5 March 2017

variational principle - Does a four-divergence extra term in a Lagrangian density matter to the field equations?


Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation:




L=12μAνμAν+12μAννAμ12μAμνAν
    =12μAνμAν+12μ[Aν(νAμ)(νAν)Aμ]


The last term is a four-divergence which has no influence on the field equations. Thus the dynamics of the electromagnetic field (in the Lorentz gauge) can be described by the simple Lagrangian


L=12μAνμAν



Yes, if it is a four-divergence of a vector whose 0-component doesn't contain time derivatives of the field, indeed according to the variational principle this four-divergence will not influence the field equation.


And actually I calculated the time-derivative dependence of 0-component of [Aν(νAμ)(νAν)Aμ], in which only [A0(0A0)(0A0)A0] could possibly contain time-derivative, which vanishes fortunately, so whatever the general case it doesn't matter in this present case.


But how can he seem to claim that it holds for a general four-divergence term,The last term is a four-divergence which has no influence on the field equations?


EDIT:
I only assumed the boundary condition to be Aμ=0 at spatial infinity, not at time infinity. And the variation of the action S=t2t1Ldt is due to the variation of fields which vanish at time, δAμ(x,t1)=δAμ(x,t2)=0, not having the knowledge of δ˙Aμ(x,t1) and δ˙Aμ(x,t2), which don't vanish generally, so the four-divergence term will in general contribute to the action, δSj=δt2t1dtd3xμj(A(x),A(x),˙A(x))μ=δt2t1dtd3x˙j0=d3x[δj(x,t2)0δj(x,t1)0]

which does not vanish in general!





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