Chapter 27

The topic of Hamiltonian chaos is a whole course in itself, and we will only touch on a few highlights. For further information see Ott chapter 7, and review papers by Berry [1] and Helleman [2]. The ideal pendulum discussed in chapter 2 is an example of a Hamiltonian system. In many ways this system is typical of the ones taught in elementary courses on mechanics: the orbits are periodic and can be constructed analytically or by perturbation theory. Other familiar examples are a collection of simple harmonic oscillators, or a planet revolving around the sun. All these systems are “integrable” and can be solved by various analytic means. However these examples taught are selected because they are solvable, rather than because they represent the typical behavior of Hamiltonian systems. Integrability is in fact the exception, and most Hamiltonian systems will show complex, chaotic dynamics. A simple modification to the planetary problem illustrates the complexity of the motion in nonintegrable systems. Consider a “solar system” consisting of a binary pair of identical suns (coordinates E R1, E R2, momenta E P1, E P2) of large mass M orbiting in the x − y plane. Now add a planet of small mass m with an initial condition on the z axis above the centre of mass of the two suns and with a velocity in the z direction; by the symmetry of the system the small mass will remain on the z axis and its dynamics is defined by the coordinate z and z-momentum pz. There are 6 independent dynamical variables: X1, Y1, P1, P2, z, pz (thenX2 = −X1 etc. and the z coordinate and momenta of the heavy masses are fixed by the stationarity of the centre of mass). The energy and angular momentum are constants, leaving a 4-dimensional dynamical system. The dynamics of this system turns out to by quite complex. Consider the crossing times τ1,τ2 . . . for which the “planet” crosses the plane of the orbit of the “suns”. It can be shown that for any chosen