the three body problem

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the three body problem

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The three-body problem can be summarized like so:

Given the initial positions and velocities of three given *point masses* — that is, bodies of nearly zero size — find out how they move over time using Newton’s laws of motion and gravitation.

No general solution for this problem exists.

Take two objects in space, so far out that the only gravitational effects they experience are from each other. It is now easy to calculate what force each exerts on the other and what those forces do to the objects to determine their paths in space as they orbit/flyby/collide/whatever. You can do this equation once and it tells you where the objects will be, how fast in relationship to each other they are traveling, and in what direction, for all time so long as nothing changes.

Now, add a third body.

It turns out that the third body complicates things so much that, most of the time, it is impossible to do this. If you want to calculate the trajectory of, say, a lunar module that is being affected by both the Earth and the Moon, you cannot use one straightforward equation. You can calculate its current velocity and direction, but you cannot calculate its future velocity and direction.

Instead, you jump a short distance into the future in your calculating. For example, your calculation for these factors now will probably not change a lot in one second. So, I use those figures to figure out where the object will be, how fast it will be going, and in what direction, in one second.

I then repeat the calculation for the new position and circumstances.

And then I repeat once a second until I have the entire path worked out. If working out the entire path is impossible I just do this forever to find out the next step in the path.

Whether you do this by the second, the hour, the year, or the millisecond is entirely dependent upon the three bodies involved and the precision that you require.

There are some special circumstances where the equations work out so neatly that the 3-body problem ceases to exist. If you take three bodies and put them into just the right orbits then the equations do predict future velocity, vector, and location. The Lagrange points for the Earth-Moon system give us places where we can put a satellite in space that is influenced by the Earth and the Moon at the same time, but where the forces work out for an easy solution.

We can come up with a mathematical formula for where objects will be as they go through all kinds of different situations. We have a formula defined for how far something will fall *x* seconds after being released, another for how fast something will be moving *x* seconds after being struck by another object, and another for where a spacecraft will be after orbiting for *x* minutes. We even have formulas that describe where two planetary bodies (like a moon-planet pair) will be after they orbit each other for *x* years.

We don’t have a formula for precisely where a system of 3 or more bodies (planets, moons, stars, etc) will be after *x* years. To calculate that, we need to go step-by-step through time, simulating where they’ll be at each small slice of time from now until *x* years from now.

How does the solar system stay stable enough for us to keep living?

You want equations to give you the position and velocity of 3 planets orbiting each other

But the three planets’ gravities influence each other’s movement in dynamic ways too complex for basic equations to represent

Thus approximate simulations are needed instead of truly-perfect equations