Two orbiting bodies are easy to predict, but add a third and nobody can
Newton showed that a planet and its star follow tidy, predictable paths. Add one more body and the neat formulas vanish. Henri Poincaré proved in the 1890s that no general exact solution exists for three gravitating objects, and in doing so planted the seed of chaos theory.
Celestial mechanics studies how objects in space move and pull on one another, traditionally applying classical physics to stars and planets to produce ephemerides, tables of where things will be. Choosing a suitable reference frame, such as one centred on the Sun, simplifies the arithmetic. For a pair of bodies, Newtonian physics yields orbital elements that forecast their positions well and confirm Kepler's laws; if one partner is massive enough, general relativity is needed to account for the slow rotation of the orbit, called apsidal precession.
Before Kepler, predicting planetary positions and explaining why planets move were largely separate pursuits. His 1609 book, often translated as New Astronomy, merged the geometric tradition running from Ptolemy to Copernicus with physical causes, and Tycho Brahe's observations let him derive elliptical orbits that sharpened predictions dramatically. Newton's Principia of 1687 then placed falling apples, cannonballs, the Moon and the planets under one set of laws, deriving Kepler's ellipses from gravity. Newton himself preferred the phrase rational mechanics; Leibniz later spoke of dynamics, and the label celestial mechanics came from Pierre-Simon Laplace more than a century after Newton.
The three-body problem resisted everyone. In 1762 Leonhard Euler found three balance points lying on a line through two large masses, where a tiny object could sit in a stable orbit. Joseph-Louis Lagrange added two more in 1772, at the corners of equilateral triangles formed with the two masses; together they are now the Lagrange points. Lagrange also recast mechanics around energy rather than force and wrote a single polar equation covering any orbit, including parabolic and hyperbolic ones, which later proved handy for comets and spacecraft.
Poincaré's monographs, published between 1892 and 1910, showed that the three-body problem is not integrable, the biggest advance since Newton. They introduced bifurcation points and ideas that grew into chaos theory and the study of dynamical systems, and earned him the Royal Astronomical Society's Gold Medal in 1900. Today perturbation theory supplies the approximate answers.
Source: Celestial mechanics
