Three-body problem

In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.^[1] The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,^[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.

Historically, the first specific three-body problem to receive extended study was the one involving the Moon, Earth, and the Sun.^[2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.

The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions of three gravitationally interacting bodies with masses :${\displaystyle \mathbf {r_{i}} =(x_{i},y_{i},z_{i})}$${\displaystyle m_{i}}$

where is the gravitational constant.^[3][4] This is a set of nine second-order differential equations. The problem can also be stated equivalently in the Hamiltonian formalism, in which case it is described by a set of 18 first-order differential equations, one for each component of the positions and momenta :${\displaystyle G}$${\displaystyle \mathbf {r_{i}} }$${\displaystyle \mathbf {p_{i}} }$

where is the Hamiltonian:${\displaystyle {\mathcal {H}}}$

In this case is simply the total energy of the system, gravitational plus kinetic.${\displaystyle {\mathcal {H}}}$

Approximate trajectories of three identical bodies located at the vertices of a scalene triangle and having zero initial velocities. It is seen that the center of mass, in accordance with the law of conservation of momentum, remains in place.

The circular restricted three-body problem is a valid approximation of elliptical orbits found in the Solar System, and this can be visualized as a combination of the potentials due to the gravity of the two primary bodies along with the centrifugal effect from their rotation (Coriolis effects are dynamic and not shown). The Lagrange points can then be seen as the five places where the gradient on the resultant surface is zero (shown as blue lines), indicating that the forces are in balance there.

While a system of 3 bodies interacting gravitationally is chaotic, a system of 3 bodies interacting elastically isn't.

An animation of the figure-8 solution to the three-body problem over a single period T ≃ 6.3259.^[11]