Statistical Mechanics - Physics at Oregon State University

146 CHAPTER 7. CLASSICAL STATISTICAL MECHANICS.
7.4 What is chaos?
One dimension is not enough.
The nature of chaos is best understood by considering some simple examples.
First, we investigate a one-dimensional harmonic oscillator. The Hamiltonian
is given by
with solutions
H(x, p) = 1
2 (p2 + x 2 ) (7.24)
x(t) = x0 cos(t) + p0 sin(t) , p(t) = −x0 sin(t) + p0 cos(t) (7.25)
Phase space is two-dimensional, and the surface of states with energy U is
a circle with radius √ 2U. Each orbit (x(t), p(t)) samples all states on this
available surface. These orbits are not chaotic, since a small change in the
initial conditions gives only a small change in the state at arbitrary times t.
Also, the time needed to sample all of the available states is the period of the
oscillator, and this is a long time for the system. Any average over a complete
period essentially throws away all information about details of the system! In
a case like this one would like to measure over a much shorter time. Hence a
one-dimensional harmonic oscillator is an exceptional case, it is too simple!
Two dimensions will do.
A two-dimensional harmonic oscillator is described by the Hamiltonian
H(x, y, px, py) = 1
2 (p2 x + p 2 y + x 2 + y 2 ) (7.26)
and the solutions for the orbits are
x(t) = x0 cos(t) + px0 sin(t) , y(t) = y0 cos(t) + py0 sin(t)
px(t) = −x0 sin(t) + px0 cos(t) , py(t) = −x0 sin(t) + py0 cos(t) (7.27)
In this case phase space is four-dimensional, and the surface of constant energy
is a three-dimensional hyper-sphere.
Poincaré surface.
In order to present results for systems with many degrees of freedom we
often use a simple representation. An arbitrary orbit is represented by its crosssections
with a number of two-dimensional cuts through the constant energy