ξ ¼ t À x
ð9:8:2Þ
The Canonical momentum conservation provides the relation for β ¼ 0 in (8.1.21):
p y ¼ a ¼ a 0 sin ξ
ð Þ
ð9:8:3Þ
Then, the basic equations are the following two for the case of incident laser only
d
dt
p x ¼
a
2
0
γ
sin ξ
ð Þ cos ξ
ð Þ ¼
a
2
0
2γ
sin 2ξ
ð Þ
ð9:8:4Þ
dx
dt
¼
p x
γ
ð9:8:5Þ
As already studied, (9.8.4) and (9.8.5) with (9.8.3) are integrable, because it has a
constant of motion. Note that α in this chapter is not the fractal index but α defined in
(8.1.20). This corresponds to the equation of radius in the motion of a planet or
asteroid around the sun for a given angular momentum. In this case, the second-order
differential equation to the radius of the planet is periodic oscillation in a nonlinear
potential, and total energy E is the constant of motion.
9.8.2 Adiabatic Constant
In order to study the property of the oscillating solution given in (8.2.2) and (8.2.7),
consider the similarity to the harmonic oscillation with Hamiltonian:
H ¼
1
2
p
2
þ
ω
2
2
q
2
ð9:8:6Þ
It is well-known that the action variable J is defined as
J ¼
1
2π
I
pdq
ð9:8:7Þ
This is an adiabatic constant and used by Bohr to introduce the discretization of
electron orbit of a hydrogen atom in the early time of quantum mechanics. It is easy
to integrate (9.8.7) to obtain the relation:
J ¼
E
ω
ð9:8:8Þ
9.8 Analytical Mechanics of Electron Motions
363