Tuesday, July 11, 2017

The Physics Problem that Isaac Newton Couldn't Solve

I spent the year before graduate school doing research at the Institute of Astronomy in Cambridge. While I was there, Martin Rees (later Sir Martin, and now Lord Rees) gave a talk to the new graduate students on the best way to choose a Ph.D. dissertation topic. I remember him saying, "Don't choose a problem that Poincare couldn't solve. Choose a problem that Poincare never heard of."

But let me dial up the challenge even more. There's a physics problem so difficult, so intractable, that even Isaac Newton, undoubtedly the greatest physicist who ever lived, couldn't solve it. And it's defied everyone else's attempts ever since then.

This is the famous three-body problem. When Newton invented his theory of gravity, he immediately set to work applying it to the motions of the planets in the solar system. If you have a planet orbiting a much larger body, like the sun, and the orbit is circular, then the problem is easy to solve -- it's something that's done in a high school physics class.

But a circular orbit isn't the most general possibility, and sometimes one body isn't much smaller than the object it orbits (think of the Moon going around the Earth). This more complicated case can still be solved -- Newton showed that the two bodies orbit their common center of mass in elliptical orbits. In fact, this prediction of elliptical orbits really cemented the case for Newton's theory of gravity. The calculation is a lot trickier than for circular orbits, but we still throw it at undergraduate physics majors in their second or third year.

Now add a third body, and everything falls apart. The problem goes from one that a smart undergraduate can tackle to one that has defied solution for 400 years. There are a few special cases that are much easier to solve. If the third body is so small that it doesn't affect the motion of the other two, then it just moves around in the gravitational field of the first two bodies as they orbit each other. This can give rise to some complex orbits for the third body, and it produces the famous "Lagrange points" illustrated here for the Earth-Sun system. These are points at which the smallest body can orbit in sync with the two bigger bodies without being pulled away.

But when the three bodies are all roughly equal in size, there's no way to solve the problem. We can use computers to visualize the motion, but that doesn't always give great insight into what's going on. And some truly bizarre motions are possible. Cristopher Moore at the Santa Fe Institute has compiled movies of some of them, which you can view here. Perhaps the most unusual is the figure-8 trajectory, in which the three masses all chase each other around a figure 8. Weird! The three-body problem figures prominently in the novel of the same name by Liu Cixin, which I thought was pretty good but not great.

The conclusion: don't choose the three-body problem for your dissertation topic. Just don't.


Anonymous said...

does the three body problem apply to quarks in neutrons and protons of which there are three?

wac dw2hkb95@gmail.com

Robert Scherrer said...

It's really a different situation with quarks. Gravity is a long-range force, so objects can orbit each other at great distances. The strong force that binds together the quarks in a proton or neutron behaves very differently - it gets stronger with distance instead of weaker! But that creates complexities of its own that physicists have not yet entirely solved.