Nonlinear Mechanics - Physics at Oregon State University

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42 CHAPTER 3. ABSTRACT TRANSFORMATION THEORY The trajectories are now circles with radius √ 2Ik. The area enclosed is 2πIk, as required by (3.24). The motion in the q1 - q2 plane is more complicated. It depends on the ratio ω1/ω2 called the winding number. If this is a rational number, say N1/N2 then after N1 cycles of q1 and N2 cycles of q2, the trajectory will come back to its starting point. This is called a Lissajou figure. If the winding number is irrational, the trajectory will be confined to a limited area but will never return to its starting point. It will eventually “color in” all available space. In the next chapter we will be concerned with systems that are “almost” integrable. For such systems the winding number is all-important. Systems with irrational winding numbers tend to be stable under perturbation. Those with rational winding numbers disintegrate at the slightest push! The centerpiece of this chapter is the torus. The trajectories spiral around the donut. If the winding number is rational they “wear a path” around the donut. If it’s irrational they cover the donut evenly. A useful way of visualizing this was invented by Poincaré. Imagine a flat plane cutting through the donut in such a way that every point on the plane has the angle ψ1 = 0. Place a dot on the plane at he point where each trajectory passes through it. If the winding number is a rational fraction, there will be a finite number of points. Each time a trajectory passes through ψ1 = 0 it will pass through one of the dots. If the winding number is irrational the crossings will mark out a continuous circle. The Poincaré section as it is called (some books call it the surface of section) is a useful diagnostic tool. Suppose you have a system of equations that are not integrable (so far as you know) but is amenable to computer calculation. Take various Poincaré sections. If they are circles then the system is at least approximately integrable and can be described with action-angle variables. As we will see, there are often regions of phase space, “islands” as it were, where motion is simply periodic and other regions that are wildly chaotic. Pictures of this motion appear in all the standard texts. I have yet to see a clear explanation of the coordinates involved, however. What does it mean really to say that the donut is a 2-d surface in a 4-d space? Your breakfast donut, after all, is imbedded in 3-d space. If we take a Poincaré section through the donut at the plane ψ2 = 0 and plot q ′ 1 versus p′ 1 , we will get either a circle of dots or a continuous circle with a radius equal to √ 2I1, or we can take a slice through ψ1 = 0 and get a circle with radius √ 2I2. Put it this way, any point on the torus has four (polar) coordinates, ( √ 2I1, ψ1, √ 2I2, ψ2), but in 3-d space, only three of them are independent. When the torus is in 4-d space, all four of them are independent. If we really
3.2. GEOMETRY IN N DIMENSIONS: THE HAIRY BALL 43 lived in 4-d space, we would label the axes of the donut plot (q ′ 1 , p′ 1 , q′ 2 , p′ 2 ). This is impossible for us to imagine. The donut is easy; just remember that there is no equation of constraint among the four variables. 4 3.2.2 Example: A Particle in a Box Consider a particle in a two-dimensional box with elastic walls. 0 ≤ x ≤ a 0 ≤ y ≤ b H = 1 2m (p2x + p 2 y) = π2 ( I2 1 2m a2 + I2 2 b2 ) I1 = 1 px dx = 2π a π |px| I2 = b π |py| ω1 = ∂H ∂I1 = π2 I1 ω2 ma2 = π2 I2 mb2 There are several interesting points about this apparently trivial problem. The Hamiltonian looks linear, but in fact it contains an invisible nonlinear potential that reverses the particle’s momentum when it hits the wall. One symptom of this is that the frequencies depend on I. This looks odd, but it’s just the action-angle way of saying that the particle makes a round trip (in the x direction) in a time T = 2am/px. The loop integral in this context is an integral over one “round trip” of the particle. ∫ a ∫ 0 pxdx = |px| dx + (−|px|) dx = 2a|px| 0 My real point in showing this example is to call your attention to the angle variable. I will work through the calculation for the x variable. This same thing holds for y of course. 1 2m a ( ) 2 dWx = E1 dx ∫ Wx = (±) √ ∫ 2mE1 dx = ψ1 = ∂Wx ∂I1 = ± π ∫ a (±) πI1 a dx dx = ± π a x + ψ10 = ψ1 4 Of course, I1 and I2 are constant for any given set of initial conditions. It is this sense in which the torus is a 2-d surface.
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