Darwin's paper was entitled Periodic Orbits (see Barrow-Green (1997), Section 8.4.1 for more details). Darwin's results were discussed by Poincaré in the third volume of Les methodes nouvelles, Sections 381-4. Until the advent of computers, this type of work was incredibly time-consuming. An attempt to trace a continuous family of periodic orbits from those encircling both primaries to those just orbiting one primary was made by Moulton (1920). In the Introduction he wrote: 'The amount of labor [Chapter XVI] cost can scarcely be overestimated'.

that S = U,in which case the stable and unstable manifolds represent a smooth connection between the unstable fixed point and itself.

The consequences of S and U actually crossing each other leads to some very odd behaviour indeed, and Poincare believed initially that this was impossible. In his original prize entry he had shown that if the Hamiltonian for the restricted three-body problem written in the form of Eqn (11.11) was suitably truncated, then the resulting system was completely integrable and its Poincare map possessed unstable fixed points, the stable and unstable manifolds of which had to coincide. Then, he claimed a stability result on the assumption that S and U must still coincide when the truncated terms in the Hamiltonian were restored. The entries to the prize competition were read initially by Lars Edvard Phragmen, an editor of Acta mathematica, and he corresponded with Poincare (via Mittag-Leffler) over parts of the manuscript he considered unclear. Sometime during this process, perhaps prompted by Phragmen, Poincare realized he had made a serious error.

Poincare realized suddenly that so-called transverse homoclinic points could exist, with devastating consequences for the dynamics of the three-body problem. The existence of a single such point implies the existence of infinitely many such points all intertwined in an extremely complicated fashion, known as a 'homoclinic tangle':

When we try to represent the figure formed by these two curves and their infinitely many intersections, each corresponding to a doubly asymptotic solution, these intersections form a type of trellis, tissue, or grid with infinitely fine mesh. Neither of the two curves must ever cut across itself again, but it must bend back upon itself in a very complex manner in order to cut across all of the meshes in the grid an infinite number of times.

The complexity of this figure is striking, and I shall not even try to draw it. Nothing is more suitable for providing us with an idea of the complex nature of the three-body problem

In correcting his error, Poincare changed completely the nature of his original memoir. Instead of demonstrating stability, he was now opening up the possibility of all sorts of weird and wonderful behaviour. The nature of the trajectories in a homoclinic tangle showed that solutions corresponding to initial conditions that were very close together could separate extremely quickly; here was the first glimpse of what we now term 'chaos'. He had understood that, in systems modelled by differential equations (e.g. the equations of celestial

0 0

Post a comment