Mathematical Principles of Theoretical Physics

100 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

2.6.3 PHD for Maxwell electromagnetic fields

PHD is also valid in the Maxwell electrodynamics. To establish the Hamiltonian dynamics
for electromagnetism, we first determine the conjugate fieldfunctions as follows

(2.6.41)

u=E= (E 1 ,E 2 ,E 3 ),

v=A= (A 1 ,A 2 ,A 3 ),

whereEis the electric field, andAis the magnetic potential. The Hamilton energyHis given
by

(2.6.42)

H=

∫

Ω

H(E,A)dx,

H=

1

8 π

(E^2 +|curlA|^2 )+

1

4 π

∇φ·E−

1

c

J·A,

whereφis the electric potential, andJis the current. Using

〈δH

δE

,E ̃

〉

=

d

dλ

∣

∣

∣

λ= 0

H(E+λE ̃,A),

〈δH

δA

,A ̃

〉

=

d

dλ

∣

∣

∣

λ= 0

H(E,A+λA ̃),

we can compute the derivatives of (2.6.42) as follows:

δH

δE

=

1

4 π

(E+∇φ),

δH

δA

=

1

4 π

curl^2 A−

1

c

J.

Thus, the Hamilton equations (2.6.31) are in the form:

(2.6.43)

1

c

∂E

∂t

= 4 π

δH

δA

=curl^2 A−

4 π

c

J,

1

c

∂A

∂t

=− 4 π

δH

δE

=−E−∇φ,

which are the classical Maxwell equations (2.2.33) and (2.2.35).

Remark 2.49.We note that the Lagrange action of electromagnetism definedby (2.4.15) can
be expressed as

(2.6.44) LEM=

∫

M^4

[

−

1

2

|curlA|^2 +

1

2

E^2 +

4 π

c

AμJμ

]

dxdt,

where

E=−

1

c

∂A

∂t

−∇A 0 , Aμ= (A 0 ,A 1 ,A 2 ,A 3 ).

The fields of PLD are given by

A 0 ,A 1 ,A 2 ,A 3 and A ̇ 0 ,A ̇ 1 ,A ̇ 2 ,A ̇ 3 (A 0 =φ),