# Three body (or more) gravitation

If I understand this correctly, Newton’s law of gravitation turns out to only be accurate for calculating the gravitational force between two and only two objects. And in a broader sense, two massive objects like the earth and the sun, for example, are composed of many smaller particles. So it’s not even truly 100% accurate for planetary attraction, even if you make adjustments for relativity. As far as I can tell, no one has been able to “solve” a gravitational system with three or more masses in a purely analytical way, right?

Can someone perhaps explain why this isn’t a simple vector problem? No one ever cites exactly where the impossibility arises. The most specific answer I can find is that they interact with one another in a “chaotic way”. What do astronomers do when they want to predict the behavior of a trinary star system, for example?

Also, by extension, shouldn’t this problem be true for other forces like the electromagnetic force? Granted, at a quantum scale you run into intrinsic uncertainty, so maybe it’s a moot point, but it’s interesting nonetheless. Do we have difficulty calculating the probabilites for 3 or more electrons interacting with one another, too?

Two and only two? Newtonian mechanics is pretty good for multi-body systems. The key is that you can prove that a massive body acts as though all its mass is concentrated at the centre, so you can treat all the bodies in the problem as point masses, even if they’re stars. If you couldn’t do a three-body problem, we would never have been able to send anyone to the Moon.

The Coulomb force (repelling/attracting like/unlike charged particles) can handle multi-body systems as well. Last year in my symbolic computation course we had to simulate a field of charged particles, and send another charged particle through it. It was quite fun to watch the test particle getting flung around and slingshot by the fixed charges. IIRC Newton’s laws predict fine the attraction between any number of bodies. That is, you get equations giving the accelerations in terms of the positions and masses, that sort of thing.

The problem is that you can’t always solve them to say where what’ll be in the future, except in special cases, like there being only two bodies, or all the bodies being very different sizes.

More specifically, in some special cases (eg. circular orbit) you can solve the equations explicitly, and get a formula where you plug in current position and time and get an answer. In others, you can solve it approximately by computer or whatever. In general, iirc, it’s chaotic, ie. very sensitive to how it starts, so however well you think you’ve measured it, some n’th decimal place turns out to be important and it does something else.

Hold on, I’ll try to think of good analogy.

Repetition to drive home the point: Newton’s law works for any number of objects.

The issue is coming up with “closed form” equations to describe the solutions. For two bodies, you get simple equations that reduce to either exponential or trigonometric forms (both instrinsicly related). It’s only a little harder than solving a quadratic equation. Both for 3+ objects, you get equations that have no closed form. Much like (and for very similar reasons) there are no closed form equations for roots of polynomials of degree 5 or higher.

So, in Real Life, you approximate things, do a lot of calcs. using computers (which are inherently discrete rather than continuous), etc. In some sense, this is a solution, just not a closed form and also much more sensitive to initial condition measurement errors.

Almost. Any spherically symmetric body acts as though all its mass is concentrated at the center, and fortuneately, most objects in the Universe are approximately spherically symmetric. But to the extent that an object is not spherically symmetric, there are corrections that you need to make to the orbits.

To sum up: For two spherically symmetric objects, the equations are easy to solve exactly. If you have more bodies, or the bodies aren’t spherical, then (except for a very few special cases, like Trojan orbits), you can’t solve the equations exactly, but might be able to solve them well numerically (i.e., by computer simulation). For example, our Solar System has the Sun, nine planets, and a host of moons, comets, asteroids, etc., but it’s still non-chaotic (for the major bodies at least). But in some cases, you do get chaos. You can still run your computer simulations, but they’ll only really be good predictions for a very short time. Sooner or later (generally sooner), the inevitable small errors you started with will build up enough to send your predictions to hell in a handbasket.

Incidentally, General Relativity makes the situation even worse. Effectively, the gravitational field itself acts as another body in the system, so it’s possible to get chaos even in two-body systems.