Mathematical Principles of Theoretical Physics

82 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

2.5.3 Classical electrodynamics

Electrodynamics consists of two parts: the Maxwell field equations and the motion of charged
particles, each described by the related Lagrange actions.

Lagrange density of electromagnetic fields

In classical Maxwell theory, we usually take the electric fieldE, the magnetic fieldH, the
current density~Jand the electric charge densityρas state functions, because these physical
quantities are observable. However, the fieldsEandHare not fundamental physical quanti-
ties, and the basic fields describing electromagnetism are the 4-D electromagnetic potential
Aμand the current densityJμ:

(2.5.13)

Aμ= (A 0 ,A 1 ,A 2 ,A 3 ),

Jμ= (J 0 ,J 1 ,J 2 ,J 3 ).

Hence the Lagrange density for the Maxwell theory should be constructed with the fields in
(2.5.13).
Based on the Lorentz Invariance and theU( 1 )Gauge Invariance, the action of (2.5.13) is
uniquely determined and is given in the form (2.4.15):

(2.5.14) L=

∫T

0

∫

Ω

L(Aμ,Jμ)dxdt,

where

(2.5.15) L=

1

16 π

Fμ νFμ ν+

1

c

AμJμ Fμ νas in( 2. 4. 4 ).

We now derive the variational equation, also called the Euler-Lagrange equation, of
(2.5.14)-(2.5.15). SinceJμis an applied external field, we only take variation with respect to
the fieldAμ.
It is known thatδLis a 4-dimensional field

δL= (δL^0 ,δL^1 ,δL^2 ,δL^3 ),

and satisfies that for anyA ̃μwithA ̃μ|∂QT=0, we have

(2.5.16)

∫

QT

(δL)μA ̃μdxdt=

d

dλ

∣

∣

∣

λ= 0

L(Aμ+λA ̃μ),

whereQT=Ω×( 0 ,T)andλis a real parameter. By (2.5.14) we see that

L=L 1 +L 2 ,

L 1 =

∫

QT

1

16 π

gμagν βFμ νFα βdxdt, Fμ ν=∂νAμ−∂μAν,

L 2 =

∫

QT

1

c

AμJμdxdt.