L ¼ Àmc
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À v 2 =c 2
p
þ qA Á v À qϕ
ð5:2:13Þ
The principle of minimum action is used to derive the Lagrangian (5.2.13). In
(5.2.13), the last two terms represent the contribution by field and charged particle
interaction:
L int ¼ qA Á v À qϕ
ð5:2:14Þ
The equation of motion is derived from Euler-Lagrange equations with two
independent coordinate r and v ¼ dr/dt:
d
dt
∂L
∂v
¼
∂L
∂r
ð5:2:15Þ
From LHS of (5.2.15), the following Canonical momentum P
c is defined:
∂L
∂v
¼ p þ qA
P
c
¼ p þ qA
ð5:2:16Þ
where particle momentum p and Lorentz factor γ are
p ¼ γmv
ð5:2:17Þ
γ ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À v 2 =c 2
p
ð5:2:18Þ
Note that (r, P
c
) is the pair of generalized coordinate and momentum. Then, the
equation of motion is
d
dt
p þ qA
ð
Þ¼q v Á $ ∇A
½
ŠÀq∇ϕ
ð5:2:19Þ
where $ ∇A is matrix and (I, j) component is
$ ∇A ¼
∂A j
∂x i
ð5:2:20Þ
Using the mathematical formulae
v Á $ ∇A
½
Š¼v  ∇  A
ð
Þþ v Á ∇
ð
ÞA
ð5:2:21Þ
The equation of motion of (5.2.19) becomes
170
5 Relativistic Laser-Electron Interactions