Why the three-body problem has no general closed-form solution

We review the results that delimit the Newtonian three-body problem. The state of the system has 18components, and the symmetries provide only 10 classical first integrals. The theorems of Bruns (1887) andPoincaré (1890) exclude additional algebraic and uniform analytic integrals, but not other representations.Sundman’s series (1912) solves the problem for nonzero total angular momentum, with convergence too slow forpractical use. In the nonhierarchical regime, the exponential divergence of trajectories makes the predictabilityhorizon grow only with the logarithm of the initial precision, a limit that no exact formula would remove. Weconclude that the open question is not that of a closed-form formula, but rather that of understanding the structuresthat make the motion regular and of describing the chaotic regime statistically, and we indicate the correspondingresearch directions.