Computation ①: solvable with force → Computation ②: the law of force is simple, yet unsolvable
In Computation ① we saw that "nature solves problems with force." This time is the flip side ── no matter how simple the law of force is, trying to solve it can turn out to be unsolvable. The law of universal gravitation is \(F=Gm_1m_2/r^2\), just one line. For two bodies, it solves cleanly as a Keplerian elliptical orbit. But with three, it can no longer be written in closed form (a formula). And the "three-body problem" is not merely hard: the tiniest difference in initial conditions opens up explosively over time (chaos). So the distant future is unpredictable in principle, and all we can do is compute step by step. From a simple force, uncomputable complexity is born.
The two-body problem ── just the Sun and a planet ── was solved by Newton. The answer is an ellipse (Kepler's laws). With a single equation, you can compute the position a million years from now. But with three bodies (Sun, Earth, Moon, or three stars), the situation changes completely. At the end of the 19th century, Poincaré showed that ── the three-body problem has no general formula like the two-body case (no solution expressible in elementary functions). The law of force is the same one line, yet just by gathering three bodies, the closed-form answer vanishes.
Law of force: \(F=Gm_1m_2/r^2\) (one line, completely deterministic).
Two bodies: solvable (an ellipse, a formula exists). Three bodies: in general not solvable in closed form (Poincaré).
However simple the law, its consequence (the motion) can become tremendously complex.
The true terror of "unsolvable" is chaos. Chaos is when the tiniest difference in initial conditions opens up exponentially over time (the butterfly effect). Two motions that were identical at first become completely different things after a while. Even though it is deterministic (the law is completely fixed), knowing the initial conditions with infinite precision is impossible, so the distant future is unpredictable in principle. The figure below shows two pendulums released at almost the same initial angle (a double pendulum = a familiar, gravity-driven chaos). At first they move in lockstep, but eventually they scatter completely apart.
If there's no formula, and it's chaotic on top of that, what do you do? The answer is ── compute numerically, one step at a time. Get the acceleration from the force (\(F=ma\), Episode 1), advance time a little, compute the force again… repeating this enormously many times on a computer. A space probe's trajectory, the weather forecast, a collision of galaxies ── these are all this kind of numerical computation. But as long as there is chaos, there is a limit to how far ahead you can predict (the Lyapunov time). The reason the weather is only accurate a few days out is exactly this. Paired with Computation ①'s "solvable with force," here is Computation ②'s "even a simple force can't be known without computing, and prediction has a horizon."
The conclusion of Force and Computation ②. The simplicity of a law of force and the complexity of its consequences are entirely different things. From the one line of universal gravitation are born the unsolvability of the three-body problem and the unpredictability of chaos. So even knowing the force, to know the future we can only compute, and even that computation has a horizon of prediction. Computation ① (solvable with force) and Computation ② (even with force, unsolvable, and prediction has limits) ── force and computation had both faces.
"The three-body problem is unsolvable" means, precisely, "the general solution cannot be written in elementary functions (or a finite formula)." Exact solutions for special initial conditions (the Lagrange points, the figure-eight orbit, and so on) do exist, and Sundman gave a convergent series solution in 1912 (though it converges hopelessly slowly and is useless in practice). Chaos is not randomness but deterministic: with the same initial conditions the result is always the same ── what differs is that "a tiny difference expands rapidly." The predictable timespan (the Lyapunov time) depends on the system.
The figure is a schematic of a numerically integrated double pendulum (a representative gravity-driven chaotic system), not the three-body problem itself (used as a familiar, easy-to-see example of chaos). Because of numerical error, over long times the display too drifts from the true solution ── which is itself a manifestation of the difficulty of chaos.
Universal gravitation is one line. Two bodies are solvable as an ellipse, but three are generally unsolvable in closed form (Poincaré). Further, chaos ── because the tiniest difference in initial conditions opens up exponentially, the distant future is unpredictable even though it is deterministic. So even knowing the force, to know the future we can only compute numerically step by step, and even that prediction has a horizon (the Lyapunov time). The reason the weather only reaches a few days out is this.
Computation ① ("nature solves with force") and Computation ② ("even with force, unsolvable, prediction has limits"). The simplicity of a law of force and the complexity of its consequences were different things. Next time (Computation ③) closes the force-and-computation cluster ── confirming, by computation, a force we cannot prove. Episode 11's "confinement" (whose mathematical proof is unsolved = a Millennium Problem) told through "lattice QCD," which carves spacetime into a lattice and computes on a supercomputer. It shakes hands with the sister series "The Universe Is a Computer."
Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, the slider lets you watch two nearly identical pendulums eventually scatter into completely different motions. "See the answer" opens each solution.