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How can they simplify the three-body problem enough to be used by modern computers?

horizon zero dawn

Warning, I only got through first-year calculus and it was many years ago. I watched that short video explaining what makes the 3 body problem so hard. Can they reduce it to something more like a two-body problem by acting as if the center of mass between star A and B is one body and the center of mass between star B and C is a second body to help get it closer to solvable (for example)? I’m just wondering if there is a way to explain how it gets simplified enough for modern computers to attempt to solve it. A way that a nongrad student+ in physics/math might be able to understand.

Ans by phiwong

Your idea (although good) doesn’t work. As a good approximation (say consider the earth and moon as a “single” body in relation to the sun), then it might be “good enough”. But if the three bodies are relatively significant in mass (and distance) to each other this is not a general approach that works.

Computers don’t “solve” the problem algebraically but (generally) are used to simulate by stepping through time in small time increments. Given an “n-body” problem (position and velocity) at time 0, it is possible to calculate using a very small “delta t” a good idea of what the position and velocities of the n bodies are at t = 0 + delta t. Then using this new data as the initial set, calculate the next set of positions and velocities at t = delta t + delta t. The results can be arbitrarily accurate for arbitrarily small delta t. Of course the smaller the delta t the more calculations need to be done so this is really only practicable using computers.

TL;DR the computer doesn’t simplify – it just does time simulation and derives a solution using brute force calculation.