KAM theorem
Most invariant tori of a near-integrable Hamiltonian system survive small perturbations. Proves the solar system is mostly stable.
Definition
The Kolmogorov-Arnold-Moser theorem states that when a small perturbation is added to an integrable Hamiltonian system, most of the invariant tori in phase space (those with sufficiently irrational frequency ratios) survive as slightly deformed tori. Only a measure-zero set of tori is destroyed, and the dynamics on the surviving tori remains quasi-periodic.
The theorem was sketched by Kolmogorov in 1954, proved in full by Arnold in 1963 (analytic case) and by Moser in 1962 (smooth case). It is the mathematical reason the solar system has remained approximately stable for 4.5 billion years despite the planets continuously perturbing one another — the KAM tori that describe the planets' near-regular motion survive these perturbations.
History
Sketched by Kolmogorov in 1954; proven in full by Arnold (analytic case, 1963) and Moser (smooth case, 1962).