Nonlinear Mechanics - Physics at Oregon State University

3.2. GEOMETRY IN N DIMENSIONS: THE HAIRY BALL 41
6. There are no general criteria known for deciding whether or not a
system is integrable; however, if the Hamiltonian is separable, the
system is integrable.
3.2.1 Example: Uncoupled Oscillators
The Hamiltonian for two uncoupled harmonic oscillators (with m = 1) is
H = 1
2 (p2 1 + p 2 2 + ω 2 1q 2 1 + ω2q 2 2)
This is an important problem because every linear oscillating system can
be put in this form by a suitable choice of coordinates. 3 There are two
constants of motion
E1 = 1
2 (p2 1 + ω 2 1q 2 1) E2 = 1
2 (p2 2 + ω 2 2q 2 2)
In terms of action-angle variables, the constants are I1 and I2.
H = I1ω1 + I2ω2 = E1 + E2 = E
Every integrable system can be put in this form, although in general the
ω’s will be functions of the I’s. Here they are just parameters from the
Hamiltonian.
This is a simple problem, but the phase space is four dimensional. Let’s
think about all possible ways we might visualize it. In the q1 - p1 or (q2 -
p2) plane the trajectories are ellipses with
qk(max) = √ 2Ek/ωk
pk(max) = √ 2Ek,
where k = 1, 2. The area enclosed by each ellipse is significant, because
∫
area = dq dp = p dq = 2πI (3.24)
s
The first integral is a surface integral over the area of the ellipse. The second
is a line integral around the ellipse. This identity is a variant of Stokes’s
theorem. It’s useful to rescale the variables so that they both have the same
units and the trajectory is a circle. An natural choice would be
q ′ k
√
= qk ωk = √ 2Ik sin ψk
p ′ k = pk/ √ ωk = √ 2Ik cos ψk
3 This comes under the heading of “theory of small oscillations.” Most mechanics texts
devote a chapter to it.