In Pursuit of the Unknown by Ian Stewart

this question. Instead, his memoir on the topic, published in 1890, provided evidence that it might not possess that kind of answer, even for just three bodies – star, planet, and dust particle. By thinking about the geometry of hypothetical solutions, Poincaré discovered that in some cases the orbit of the dust particle must be exceedingly complex and tangled. He then, in effect, threw up his hands in horror and made the pessimistic statement that ‘When one tries to depict the figure formed by these two curves and their infinity of intersections, each of which corresponds to a doubly asymptotic solution, these intersections form a kind of net, web or infinitely tight mesh. . . One is struck by the complexity of this figure that I am not even attempting to draw.’
    We now see Poincaré’s work as a breakthrough, and discount his pessimism, because the complicated geometry that led him to despair of ever solving the problem actually provides powerful insights if it is properly developed and understood. The complex geometry of the associated dynamics turned out to be one of the earliest examples of chaos: the occurrence, in non-random equations, of solutions so complicated that in some respects they appear to be random, see Chapter 16 .
    There are several ironies in the story. Mathematical historian June Barrow-Green discovered that the published version of Poincaré’s prizewinning memoir was not the one that won the prize. 4 This earlier version contained a major error, overlooking the chaotic solutions. The work was at proof stage when an embarrassed Poincaré realised his blunder, and he paid for a new printing of a corrected version. Almost all copies of the original were destroyed, but one remained tucked away in the archives of the Mittag-Leffler Institute in Sweden, where Barrow-Green found it.
    It also turned out that the presence of chaos does not, in fact, rule out series solutions, but these are valid almost always rather than always. Karl Frithiof Sundman, a Finnish mathematician, discovered this in 1912 for the three-body problem, using series formed from powers of the cube rootof time. (Powers of time won’t hack it.) The series converge – have a sensible sum – unless the initial state has zero angular momentum, but such states are infinitely rare, in the sense that a random choice of angular momentum is almost always nonzero. In 1991 the Chinese mathematician Qiudong Wang extended these results to any number of bodies, but did not classify the rare exceptions when the series fail to converge. Such a classification is likely to be very complicated: it must include solutions where bodies escape to infinity in finite time, or oscillate ever faster, both of which can happen for five or more bodies.
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    Newton’s law of gravity is routinely applied to design orbits for space missions. Here even two-body dynamics is useful in its own right. In its early days, the exploration of the Solar System mainly used two-body orbits, segments of ellipses. By burning its rockets the spacecraft could be switched from one ellipse to a different one. But as the aims of space programmes got more ambitious, more efficient methods were needed. They came from many-body dynamics, usually three bodies but occasionally as great as five. The new methods of chaos and topological dynamics became the basis of practical solutions to engineering problems.

    Fig 14 Hohmann transfer ellipse from low-Earth orbit to lunar orbit.
    It all started with a simple question: What is the most efficient route from the Earth to the Moon or the planets? The classic answer, known as a Hohmann transfer ellipse ( Figure 14 ), starts from a circular orbit round the Earth, and then follows part of a long, thin ellipse to join up with a second circular orbit round the destination. This method was employed for the Apollo missions of the 1960s and 1970s, but for many types of mission it has one disadvantage. The