2.3 Is the solar system stable?
As well as working on probability, Laplace worked extensively on celestial mechanics – the branch of astronomy which deals with the motion of bodies in space – and was the author of a monumental five-volume work, Mécanique Céleste, published between 1799 and 1825, which was the intellectual successor to Newton’s Principia. The first two volumes of it were expanded and translated into English as the Mechanism of the Heavens (1831) by the Scottish mathematician, Mary Somerville. This translation was highly acclaimed and established Somerville’s reputation as a mathematician at a time when very few women had the opportunity to study mathematics.
The problem that particularly interested Laplace relates to the question of whether the solar system is stable. Will the planets continue to follow similar paths to the ones they’ve travelled along in the past? Could something catastrophic occur, such as a collision or an escape?
The stability of the solar system can be considered a mathematical problem because it can be modelled by what is called the n-body problem: given n objects or ‘bodies’ interacting gravitationally with known initial positions and velocities, can you predict their individual motions? Simply put, can you determine their positions and velocities at any time of your choosing? (The way the problem is modelled, the only forces acting on the bodies are the forces of gravity. All other forces, such as those generated by solar winds, are ignored.)
The n-body problem, which originated with Newton’s law of gravity, is a notoriously hard problem and was attempted by many distinguished mathematicians. Little wonder then, that it was set as a prize competition problem in 1885. The occasion for the competition was the 60th birthday of King Oscar II of Sweden and Norway, who had himself studied mathematics at university. The King’s birthday was due to take place in 1889, when the winner would be announced.
The competition was won by one of the most talented mathematicians of the day, the French mathematician Henri Poincaré – even though he hadn’t actually solved the problem! Which begs the question: what did Poincaré do that was so good that he won the competition anyway?
He began by doing what mathematicians often do when they are struggling to solve a problem: he attacked a simpler version of the problem in the hope that if he found a solution, he would be able to generalise it. The version he initially attacked was the ‘three-body problem’. But even this version of the problem turned out to be too difficult – there are more variables than there are equations to describe them – and so Poincaré turned to an even simpler version: the ’restricted three-body problem’. (The two-body problem, by the way, had been solved by Newton.)