Three-Body Problem: Mathematical Status — Solvable but Computationally Useless
Sundman proved in 1912 that a three-body solution EXISTS as a power series — but it needs ~10^8,000,000 terms, making it useless. The problem is solvable in theory, intractable in practice, and chaotic in behavior.
The three-body problem has a nuanced mathematical status that is often oversimplified. Sundman's proof (1912): Karl Fritiof Sundman proved that an analytic solution EXISTS as a convergent power series (Puiseux series in powers of t^1/3). However, this series converges so slowly that useful astronomical calculations would require approximately 10^8,000,000 terms — making it theoretically solved but practically useless. So the three-body problem is: - Mathematically solvable: a solution formula exists (Sundman) - Computationally intractable: the formula requires impossibly many terms - Chaotic: sensitive dependence on initial conditions makes numerical simulation the only practical approach, but errors accumulate over time Known exact special-case solutions: - Euler's collinear solutions (1767): three masses aligned on a line - Lagrange's equilateral triangle solution (1772): three masses forming a stable triangle - Lagrange points L1-L5: equilibrium positions in the restricted three-body problem (one mass negligible) - Figure-eight orbit (discovered 1993, proven 2000): three equal masses tracing a figure-8 - Thousands of additional periodic solutions have been discovered for specific configurations The distinction matters: saying "no solution exists" is technically wrong. Saying "no practical general solution exists" is accurate.