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I’ve done a bit of research, and have learned that computers “solve” the three-body-problem by using “Numerical methods for ordinary differential equations”, but I can’t really find anything about it other then Wikipedia. Does anyone have any good sources about this topic that isn’t any kind of Wikipedia? Currently I’m using simulations of three bodies flying around each other, using Newton’s gravitational law, and at a random time in the simulation, everything goes chaotic. I though that this was the only way to kind of “solve” it, but how does this “Numerical methods for ordinary differential equations” method work? And what does the computer actually do?
Do you want to learn how to write your own gravity sim software? What are these “simulations” fat you’re currently using? In addition to the answers that mention Verlet, the computational approach for high-precision integration of the n-body problem is described in the article High precision Symplectic Integrators for the Solar System. Numerical analysis is used to calculate approximations to things: the value of a function at a certain point, where a root of an equation is, or the solutions to a set of differential equations. If you try it for a two body problem it will make the orbiting masses perform a precessing rosette orbit because of the error build-up, especially when they get close to each other. There is a menagerie of methods for solving ODEs numerically. One can use higher order methods that sample the functions in more points and hence approximate them better.
As I said, this is a big topic. Perhaps the simplest is the semi-implicit Euler method. You calculate the updated velocity first, and then use it to update the positions – a tiny trick, but suddenly 2-body orbits are well behaved. The three body problem is chaotic in a true mathematical sense. So even with an arbitrarily fine numerical precision there will be a point in time where our calculated orbits will be totally wrong. I don’t recall the mathematical process used to constrain the total energy to a constant value. There are some difficult cases, such as when one of the bodies has a relatively huge orbit, such as a sun, a planet, and an asteroid.