Three body problem

The three body problem is an important problem in classical mechanics, with many applications, particularly in astrophysics. Despite being a simply defined, well-posed problem, it does not have an exact analytic solution, but rather must be approximated or solved numerically.

The three body problem involves, naturally, three bodies, at positions r₁, r₂, and r₃. These bodies have masses m₁, m₂, and m₃, and are attracted together by a central gravitational force. It is a closed system, so there are no other forces. The problem is this; given the masses, initial positions, and initial velocities, what are the positions and velocities of the bodies for all subsequent times? The usual solution involves considering conservation of energy and conservation of momentum, or, equivalently, the Lagrangian.

The most common approximation applied for the three body problem is for the very common case where one of the three bodies is much further away from the other two bodies than they are from each other. A frequently cited example is the system of the Earth, the Moon, and the Sun. The Sun is so much further away than the other two bodies that you can first consider the two body problem of the Sun and the Earth-Moon centre of mass. Then, a second two body problem involving the two other bodies can be solved, treating the gravitational force from the Sun as negligible. The Sun's influence can be applied as a perturbation to increase the accuracy of this result. Another common approximation is for the case where two of the bodies are comparable in mass but the third is much lighter, for example, the Earth, the Moon, and an Apollo spacecraft. In this situation, we can assume that the third body has a negligible effect on the dynamics of the two other bodies. Therefore, the dynamics of the two larger bodies can be calculated, and the resulting gravitational field used to find the motion of the small body. One of the most famous features of this solution is the presence of five equilibrium points called Lagrangian points. (The case where the three bodies are coplanar and one is much smaller than the others is called the "restricted three-body problem")

Outside of these circumstances, it is not possible to approximate with a simpler problem. Thus the conservation of energy, momentum, and angular momentum equations, or the Lagrange equations must be applied. There are eighteen unknowns in the problem, the three components of each of velocity and position for each of the three bodies. Conservation of energy supplies one equation, conservation of momentum three, and conservation of angular momentum three more. The definition of velocity gives nine equations, one for each component. Finally, since the system is closed, the position of the centre of mass does not change, which gives three more equations. This leaves two degrees of freedom, further complicating the problem beyond simple non-integrability.