Nonlinear Mechanics - Physics at Oregon State University

2.5. ACTION-ANGLE VARIABLES 29
2. We are using an F2 type generating function, which is a function of the
old coordinate and new momentum. Hamilton’s characteristic function
can be written as
W = W (q, I). (2.48)
The transformation equations are
Note that
so
dψ =
∂ψ
∂q
ψ = ∂W
∂I
∂ψ
∂q
∂ ∂W
dq =
∂I ∂q
p = ∂W
∂q
( )
∂ ∂W
=
∂I ∂q
∂
dq =
∂I
2.5.1 The harmonic oscillator (for the last time)
H = 1
2m (p2 + m 2 ω 2 q 2 )
p = ± √ 2mE − m2ω2q 2
I = 1
√2mE
− m2ω2q 2 dq
2π
(2.49)
p dq = ∂
(2πI) = 2π.
∂I
The integral is tricky in this form because p changes sign at the turning
points. We won’t have to worry about this if we make the substitution
q =
√
2E
sin ψ (2.50)
mω2 This substitution not only makes the integral easy and takes care of the sign
change, it also makes clear the meaning of an integral over a complete cycle,
i.e. ψ goes from 0 to 2π.
I = E
πω
cos 2 ψ dψ = E/ω
From this point of view the introduction of ψ at (50) seems nothing
more that a mathematical trick. We would have stumbled on it eventually,