What makes the three body problem intractable?

Answer 1

Two things. Firstly, you can write down a simple equation that can be solved analytically for two bodies orbiting each other; for three or more bodies, you have to solve the problem numerically, except for a few special cases, like two very close bodies and one much further away. Like the Alpha Centauri star system, our closest stellar neighbour. The two close stars don't notice the gravity of the distant one, so they behave like a two-body system, and to the distant one they're like a single object. Secondly, a three or more body system is highly chaotic. A very small change in initial conditions soon gives very different results, very quickly. And the system can change from one seemingly simple, stable state, to another suddenly. For instance if you've got the Earth, Moon and your spacecraft, it's possible to get into an orbit that looks like a stable orbit around Earth, but which suddenly switches to a lunar orbit.

Answer 2

Well, the basic problem is, for say a collision experiment,where does the energy in the system go? When we have a two body problem, the energy goes with one body or the other. The sum of the final energies must be the same as the sum of the initial energies. Simple enough.

The problem with the three body problem is that there is no distinct answer like there is with the two body problem. Since there are two different bodies that can have energy imparted to, then who's to say what amount goes to which? What happens is that a distribution of energies becomes apparent. You can tell what range of energies are imparted to each ball, but you cannot predict exactly how much each time.

One great example of this is found in radiation decay physics. A U-238 atom decays via an alpha particle emission to Th-234. There are only two bodies to consider here: the thorium and the alpha. Bingo, we can solve for the momentum, kinetic energy, speed, etc. for each one. It's a two body problem. However, K-39 decays to Ca-39 and emits a beta particle and an antineutrino. There can then be a distribution to the energy of the beta depending on how much energy is carried away by the antineutrino. This is shown quite easily if you look up any beta decay information; you will see a spectrum of energies that encompass the possible energies of the beta. This is hardly a solved case as there are a great number of possible energy states.

Answer 3

There is no coordinate system which separates the variables describing the positions of the 3 bodies, so the equations of motion cannot be separated into independent equations.

The equations of motion for the 2 body system can be greatly simplified by adopting a rotating coordinate system whose origin is the center of mass of the two bodies.

The restricted 3 body problem, in which the third body is much smaller in mass than the other two bodies, can be solved approximately by solving the 2 body problem first, as above, and then solving for the motion of the much smaller third body in the gravitational field of the other two bodies.

Answer 4

2 men cannot have intercourse with one woman at the same time!

Answer 5

It's not real