Q: What is the three body problem?

Posted on October 3, 2011 by The Physicist

Physicist: The three body problem is to exactly solve for the motions of three (or more) bodies interacting through an inverse square force (which includes gravitational and electrical attraction).

The problem with the 3-body problem is that it can’t be done, except in a very small set of frankly goofy scenarios (like identical planets following identical orbits).

The unsolvableness of the 3-body problem, rather than being an embarrassing hole in physics; an obvious but unsolved problem, is actually the norm.  In physics, the number of not-baby-simple, exactly solvable problems can be counted on the fingers of one hand (that’s missing some fingers), and that includes the 2-body problem.

The dynamics of one body is pretty straight forward, in as much as it travels straight forward.

The dynamics of two bodies, while not trivial, can be reduced by pretending that one body is sitting still, and then restricting all of your attention to the other body.  Using that technique, you find (or, at least, Newton found) that the motion of a body under gravity is an ellipse.  The same idea can be applied to the quantum mechanics of electrons and protons to find the exact structure of the electron shells in hydrogen (1 proton + 1 electron = 2 bodies).  In that case you’re not talking about actual orbits, but the idea is similar.

But, for three bodies, there doesn’t seem to be a fancy trick for finding solutions.  As a result, the exact behavior of 3 or more bodies can’t be written down.  The exact energy levels and orbital shell shapes in anything other than hydrogen is impossible to find.  Even deuterium (hydrogen with one extra neutron)!  Can’t be done.

Despite that, we do alright, and happily, reality doesn’t concern itself with doing math, it just kinda “does”.  For example, quantum field theory, despite being the most accurate theory that ever there was, never involves exactly solving anything.  Once a physicist gets a hold of all the appropriate equations and a big computer, they can start approximating things.  With enough computing power and time, these approximations can be made amazingly good.  Computer simulation and approximation is a whole science unto itself.

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