# Problematic

18 October 2013

Newton’s law of gravity takes a bit of calculus to really wrap your head around it, but the basic relation is very simple. Every pair of masses in the universe experiences a mutual gravitational attraction, each feeling the pull of the other’s gravitational field. The force of attraction is mutual, which leads to some interesting consequences. For example, when you step on the scale in the morning, you are weighing yourself in the Earth’s gravitational field. You are also weighing the Earth in your gravitational field. Your weight is also the Earth’s weight in your gravity.

The strength of that attraction depends on the size of the two masses and the distance between them, following a relation known as the inverse square law. Suppose two masses feel a force of 100 Newtons between them. If you double their distance of separation, then they would feel a force of only 25 Newtons, a fourth of their initial force. It’s a remarkably simple relationship.

But there’s a gravitational force between every pair of masses in the universe, which is where things can get complicated. In order to calculate the motion of an object in general, you would need to determine the gravitational attraction from all the other masses in the area, and you need to determine how those masses are gravitationally affected by the object. There are lots of ways to simplify things a bit. After all, the largest and closest masses will have the greatest effect on an object, so usually you can ignore the gravity of distant objects. When calculating the motion of the Moon, for example, you don’t have to worry about the gravity of the Andromeda galaxy.

But even with simplifications there is still a problem. While we have a simple general solution for the motion of two masses acting under their own gravity, we don’t have a general solution for three masses. We just can’t determine the exact motion for three masses moving in general ways. This is known as the 3-body problem, and it is problematic.