The Three-Body Problem
The two-body problem is tidy: two masses trace ellipses you can write down in a line of algebra. Add a third gravitating body of comparable mass, and that tidiness vanishes. There is no general closed-form solution — no formula into which you plug the time and read off the positions. The three bodies tug on one another in a feedback loop that, for almost all starting conditions, never settles and never repeats.[1]
Three equal masses, pulling on each other by gravity. At nudge = 0 they run the famous figure-eight — one of the few exact, repeating solutions, where all three chase each other along a single looping path. Nudge the starting velocity by even a few thousandths and the pattern shatters into ever-changing loops that never repeat: the three-body problem has no general closed-form solution, and tiny differences in the start blow up into totally different futures — chaos. (The figure-eight itself will eventually wander here too, as rounding error acts like an infinitesimal nudge.) Restart to rerun from the same setup.
What makes it hard is chaos: sensitive dependence on initial conditions. Two setups that differ by a millionth diverge into completely different histories after enough time, so even though the physics is perfectly deterministic, long-term prediction becomes practically impossible — you would need infinitely precise starting values.[2] The best we can usually do is integrate the equations forward numerically, step by step, exactly as the visualization above does.
That is not to say nothing is knowable. Special, delicately balanced solutions do exist — like the figure-eight orbit above, where three equal masses chase each other along one shared loop — and the restricted three-body problem (one mass negligibly small) is tame enough to give us the stable Lagrange points. But these are rare islands of order in an ocean of chaos, and finding or engineering them is a whole field of celestial mechanics.