n-Body Problem

The -body problem is the initial value problem for bodies interacting through their mutual Newtonian gravitational attraction. For positive masses and position vectors , the equations away from collisions are

for , where is the gravitational constant. Initial positions and velocities specify the initial value problem (Heggie 2006).

These nonlinear ordinary differential equations form a Hamiltonian system. The total energy

is an integral of motion, as are total linear and angular momentum. The center of mass moves with constant velocity. The virial theorem relates time-averaged kinetic and potential energies when its averaging hypotheses hold.

The two-body problem reduces to a single equation for the relative position, with nonradial trajectories that are conic sections. The three-body problem already has substantially richer dynamics, including special motions generated by central configurations.