Mathematical Principles of Theoretical Physics

2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 101

and the fields of PHD are

A 1 ,A 2 ,A 3 and E 1 ,E 2 ,E 3.

It is clear that
1
4 π

E(−A ̇)−

1

4 π

LEM=H(A,E)−

1

c

φJ^0 ,

andJ^0 =ρ. Hence, the relation (2.6.17) does not hold true in general.

Now, we consider the energy conservation. WhenH contains∇φandJ, which depend
ont, the HamiltonH=H(A,E,t)contains explicitlyt. It implies that the Maxwell fields
have energy exchange with other charged particles. ThenHis not conserved.
As there is no charged particles inΩ, then

(ρ,J) =0 inΩ.

In this case, ∫

Ω

∇φ·Edx=−

∫

Ω

φdivEdx=−

∫

Ω

4 π φ ρdx= 0.

Hence, the Hamilton energy (2.6.42) becomes

(2.6.45) H=

1

8 π

∫

Ω

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

By (2.6.37), the energy fluxPdefined in (2.6.45) reads

P=

1

4 π

curlA×

∂A

∂t

Note that
∂A
∂t

=−cE, curlA=B (Bthe magnetic field),

Then, by (2.6.34) we have that

dH

dt

=−

∫

∂Ω

P·nds=−

∫

∂Ω

(c

4 π

E×B

)

·nds.

The field
P=
c
4 π

E×B

is the Poynting vector, represents the energy flux density ofan electromagnetic field.

2.6.4 Quantum Hamiltonian systems

In quantum physics, the state functions are a set of complex valued wave functions:

ψ= (ψ 1 ,···,ψN)T, N≥ 1 ,

andψkare as

(2.6.46) ψk=ψ^1 k+iψk^2 for 1≤k≤N.