Simplification of the Proca Lagrangian

[fabstr1]

Homework Statement

How do I simplify the Proca lagrangian for a spin-1 field

Relevant Equations

L = -(1/16*pi) * ( ∂^(μ)A^(ν) - ∂^(ν)A^(μ))(∂_(μ)A_(ν) - ∂_(ν)A_(μ)) + 1/(8*pi) * (mc/hbar)^2* A^ν A_ν

Hello,
I'm trying to figure out where the term (3) came from. This is from a textbook which doesn't explain how they do it.

∂_μ(∂L/(∂(∂_μA_ν)) = ∂L/∂A_ν (1)

L = -(1/16*pi) * ( ∂^(μ)A^(ν) - ∂^(ν)A^(μ))(∂_(μ)A_(ν) - ∂_(ν)A_(μ)) + 1/(8*pi) * (mc/hbar)^2* A^ν A_ν (2)

Here is Eq (1) the Euler-Lagrange equation and Eq (2) is the lagrangian for a vector field. In the textbook they just state the term

∂_μ(∂L/(∂(∂_μA_ν)) = -1/(4*pi)*(∂^(μ)A^(ν) - ∂^(ν)A^(μ)) (3)

Where does the term -1/(4*pi) come from, and how do I cancel out the rest of the term so that the equation becomes

∂_μ(∂^(μ)A^(ν) - ∂^(ν)A^(μ)) + (mc/hbar)^2* A^ν (4)

 

[Abhishek11235]

Hints:

$\mathcal{L}= -\frac{1}{16\pi}F^{\mu\nu}F_{\mu\nu}$

Now:
$F^{\mu\nu}= \eta^{\mu n}\eta^{\nu m}F_{nm}$

Now use this in above equation and:

$F_{nm}=\partial_n A_m- \partial_m A_n$
And
$F_{nm}= -F_{mn}$

 

[fabstr1]

But what happends to the rest of the term in Eq (10.17), where is the -1/(4*pi) term coming from.

L = - (1/(16*pi)) * η^(μν)η^(νμ)*F_(μν)

 

[Abhishek11235]

But what happends to the rest of the term in Eq (10.17), where is the -1/(4*pi) term coming from.

L = - (1/(16*pi)) * η^(μν)η^(νμ)*F_(μν)

Ok. I will do one more step for you.

$\mathcal{L}= \frac{-1}{16\pi} \eta^{\mu n}\eta^{\nu m} F_{nm}F_{\mu\nu}$

Now, taking the differential and applying product rule to above term throws a factor of 2(Due to symmetry of product) so this becomes (Dropping the metric tensors for brevity):

$\partial\mathcal{L}= \frac{-1}{8\pi} F_{nm}\partial F_{\mu\nu}$

Now, use the defination of $F_{nm}$

 

[fabstr1]

I haven't taken any tensor calculus before, so I'm not sure if I'm doing it right.

∂L/∂F_{ μν } = - 1/(16*pi) * η^(μν)η^(νμ)*F_(μν)* F_(μν) = - 1/(16*pi)*η^(μν)η^(νμ)* F_(μν)^2 = -1/(8*pi)*η^(μν)η^(νμ)* F_(μν) = - 1/(8*pi)*η^(μν)η^(νμ)*(∂^(μ)A^(ν) - ∂^(ν)A^(μ))

 

[fabstr1]

I have updated my solution shown below. Is it ok, or is there something that are missing ?