Nonlinear Mechanics - Physics at Oregon State University

4.2. MANY DEGREES OF FREEDOM 51
θ =
√ 2I
mR 2 ω0
sin ψ (4.22)
The goal of the action-angle program is to express the original coordinates
and momenta in terms of the action-angle variables. This has now been
completed to zeroth order.
The first order correction is
H1(I, ψ) = − mR2 ω 2 0 θ4
24
= − I2
6mR 2 sin4 ψ.
We are now in a position to recast our Hamiltonian à la (4.1).
(
H(I, ψ) = Iω0 + ϵ − I2
6mR2 sin4 )
ψ
We have also obtained ω0 = √ g/R “for free.” The ϵ is there for bookkeeping
purposes only. We have no further need for it.
K0(J) = H0(J) = Jω0
K1(J) = H1(J) = 1
∫ 2π
2π 0
F1(J, ψ) = − 1
˜H = H1 − H1 =
ω0
∫
J 2
H1 dψ = −
16mR2 J 2
48mR 2 (3 − 8 sin4 ψ)
dψ ˜ J
H1 =
2
192 mR2 (sin 4ψ − 8 sin 2ψ)
ω0
ω = ω0 −
J
32mR 2
4.2 Many Degrees of Freedom
For systems of two or more degrees of freedom, canonical perturbation theory
is formulated in exactly the same way as before – but now profound
difficulties arise, even to first order in ϵ. The problem centers around equation
(4.16) repeated here for reference
ω0(J) ∂F1(ψ, J)
∂ψ
= − ˜ H1(ψ, J)
We were able to solve this with a simple integration (4.17). This is not
possible for more that one degree of freedom, so we must resort to Fourier