We come from the future
We come from the future

# Newtonian physicists do it with five bodies

That is, until Einstein comes along and ruins it. Check out this thought experiment that shows how, under Newtonian physics, five bodies can complete a task an infinite number of times in a finite amount of time.

The physics of two planets orbiting each other in a stable system is easy to work out with math. Simply take the speed of the planets, and their relative masses, and you can find a stable orbit for them. Introduce a third planet, and things get tricky. Although relatively stable systems of multiple masses are possible - look at the solar system - they're tought to puzzle out on paper. With gravity, speed, and location changing with every motion of three different bodies, they're very volatile. Forces pull on each body in surprising ways. Often, one of the objects gets kicked out of system, taking energy with it, leaving the other two objects to drop into a lower, but stable, orbit.

Ah, but what happens if there are five bodies? This is an idea that physicist Jeff Xia thought of in the late nineties. First, take two stable systems. Each consists of two orbiting bodies of equal weight. One system spins one direction, while the other spins the other direction. They both lie in planes that are parallel to each other, as if they were rolling in circles on separate shelves of a bookcase. Now take a fifth body, a considerably lighter one, and set it moving back and forth between the two systems. As it moves through the center of mass of one system, it destabilizes the whole thing and gets kicked back towards the other system. Once it reaches the other system, the same thing happens.

Since both of these systems push the fifth body towards each other, the systems themselves get kicked away from each other to conserve momentum. Each oscillation pushes the bodies apart farther. Eventually, after infinite oscillations, they expand to an infinite distance apart. Here's the twist; work out the math and they do it in a finite amount of time. There are some restrictions. These bodies, rather than being planets, are particles, and they have mass but no real size, since the systems oribit tighter with each oscillation. But looking at Newtonian physics, this thing actually works. Physically distinct oscillations would occur an infinite number of times in a finite amount of time.

Until Einstein screws everything up. Newton assumes that gravity acts instantaneously. It does not. Instead, it travels the same way every other awesome thing travels through the universe - no faster than the speed of light in a vacuum. The problem also assumes that particles can get infinitely close to each other, since the orbits of both two-body systems would get smaller and the fifth body would still have to pass between them. If they actually got that close, they'd form a black hole. And so, although in the philosophic science of Newton's time a task could be accomplished an infinite number of times with a certain set-up and physical conditons, practically, it can't be done. Which is a shame because it would be an awesome lab experiment.

Via New Scientist, and Maths.com. 