Mathematical Principles of Theoretical Physics

84 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

This is the second pair of the Maxwell equations (2.2.35) and (2.2.36). Since the first pair of
the Maxwell equations (2.2.33) and (2.2.34) are direct consequence of (2.4.4) and

(2.5.20) H=curl~A, E=∇A 0 −

1

c

∂~A

∂t

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

the action (2.5.14)-(2.5.15) completely determines the Maxwell equations (2.2.33)-(2.2.36).

Lagrangian for dynamics of charged particles

A particle with an electric chargeemoving in an external electromagnetic field(E,H)is
governed by

(2.5.21)

dp

dt

=eE+

e

c

v×H,

wherepis the momentum of the particle,vis the velocity, andf=ecv×His the Lorentz
force.
The action for the motion equation (2.5.21) is taken as

(2.5.22) L=

∫T

0

−mcds+

e

c

Aμdxμ.

It is clear that (2.5.22) is Lorentz invariant. By

ds=

√

1 −v^2 /c^2 cdt, dxμ= ( 1 ,v^1 ,v^2 ,v^3 )cdt,

the action (2.5.22) can be written as

(2.5.23)

L=

∫T

0

L(v,Aμ)dt,

L=−mc^2

√

1 −v^2 /c^2 +

e

c

Akvk+eA 0.

All physical properties of electromagnetic kinematics canbe derived from the action
(2.5.23).
We shall deduce (2.5.21) from (2.5.23). The Euler-Lagrange equation of (2.5.23) is given
by

(2.5.24)

d

dt

(

∂L

∂v

)

=∇L, ∇L=

∂L

∂x

forx∈R^3.

By (2.5.23), we have
∂L
∂v

=

mv

√

1 −v^2 /c^2

+

e

c

~A.

Then (2.5.24) becomes

(2.5.25)

dP

dt

+

e

c

d~A

dt

=∇L,

P=

mv

√

1 −v^2 /c^2

is the momentum.