Mathematical Principles of Theoretical Physics

2.5. PRINCIPLE OF LAGRANGIAN DYNAMICS (PLD) 83

It is clear that

d

dλ

∣

∣

∣

λ= 0

L 2 (Aμ+λA ̃μ) =

1

c

∫

QT

d

dλ

(Aμ+λA ̃μ)Jμdxdt=

1

c

∫

QT

JμA ̃μdxdt.

We infer from (2.5.16) that

(2.5.17) δLμ 2 =

1

c

Jμ.

Noting thatgμ ν=gν μ, we have

d

dλ

∣

∣

∣

λ= 0

L 1 (Aμ+λA ̃μ) =

1

8 π

∫

QT

gμ αgν βFα β

d

dλ

∣

∣

∣

λ= 0

(Fμ ν+λF ̃μ ν)dxdt

=

1

8 π

∫

QT

gμ αgν βFα β

(

∂A ̃μ

∂xν

−

∂A ̃ν

∂xμ

)

dxdt

By the Gauss formula,

∫

QT

gμ αgν βFα β

∂A ̃μ

∂xν

dxdt=−

∫

QT

gμ αgν β

∂Fα β

∂xν

A ̃μdxdt,

∫

QT

gμ αgν βFα β

∂A ̃ν

∂xμ

dxdt=−

∫

QT

gμ αgν β

∂Fα β

∂xμ

A ̃νdxdt

=(by the permutation ofμandν)

=−

∫

QT

gν αgμ β

∂Fα β

∂xν

A ̃μdxdt

=

∫

QT

gμ αgν β

∂Fα β

∂xν

A ̃μdxdt,

where a permutation onαandβis performed, andFβ α=−Fα β.
Thus, we obtain that

d

dλ

∣

∣

∣

λ= 0

L 1 (Aμ+λA ̃μ) =−

1

4 π

∫

QT

gμ αgν β

∂Fα β

∂xν

A ̃μdxdt.

We infer then from (2.5.16) that

(2.5.18) δL 1 =−

1

4 π

gμ αgν β

∂Fα β

∂xν

=−

1

4 π

∂Fμ ν

∂xν

.

Hence it follows from (2.5.17) and (2.5.18) that

δL=−

1

4 π

∂Fμ ν

∂xν

+

1

c

Jμ,

which implies that the equationδL=0 takes the form:

(2.5.19)

∂Fμ ν

∂xν

=

4 π

c

Jμ.