where E is the total energy of the oscillation. It is well-known that E/ω is the
adiabatic constant of a simple oscillation as described in Appendix 2. Then, the
angular frequency ω satisfies the relation:
∂E
∂J
¼ ω
ð9:8:9Þ
The angle variable θ is defined by
θ ¼ ωt
ð9:8:10Þ
Then, J and θ are new variables after Canonical transformation from (q, p). It is
convenient to use (J, ω) in such periodic motion, because J is an adiabatic invariant to
be kept constant under small perturbation to the system.
Analogously consider our nonlinear oscillation of electrons by relativistic laser. It
is a periodic oscillation, and both the action and angle variables are given in the
present two-dimensional case. The action variable in x-motion J x is given as
J x ¼
1
2π
I
p x dx ¼
I
p
2
x
2πγ
dt ¼
1
2πα
I
p
2
x dξ ¼
1
α
p
2
x
ð9:8:11Þ
where the relations (8.1.11) and (8.1.12) are used. In (9.8.11), < > means the average
value over the wave phase ξ. The action of the y-direction is also obtained in the
form:
J y ¼
1
2π
I
p y dy ¼
I p
2
y
2πγ
dt ¼
1
2πα
I
p
2
y dξ ¼
1
α
p
2
y
D E
ð9:8:12Þ
Setting E the average energy of the electron, the following relation is obtained:
E
2
À 1 ¼ α J x þ J y
À
Á
ð9:8:13Þ
The angle variable is given by ξ, and (8.1.11) and (8.1.12) reduce the relation:
ξ ¼ ατ
ð9:8:14Þ
Compared to the relation for the harmonic oscillation, the following analogy is
obtained:
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9 Theory of Stochasticity and Chaos of Electrons in Relativistic Lasers