Integrability and Nonintegrability

The development of classical mechanics accompanies the understanding of the motion of celestial bodies. The starting point of the great tradition of classical mechanics can be traced back to Galileo, and the tradition of studying mechanics in terms of differential equations started from Newton, who successfully solved the two-body problem and proved Kepler’s three laws. Systems like the two-body problem that can be solved exactly are now called integrable systems. In the eighteenth century, mathematicians including Euler, Lagrange, Laplace, Hamilton etc, laid the foundation of classical mechanics. The classical mechanics was then written in terms of Lagrangian and Hamiltonian formalisms. In the 1880s, when studying the three-body problem, Poincaré discovered chaos and realized the complexity of the N-body problem.