n-body problem

Remarks

The equations are a system of ordinary differential equations of order two. They form a Hamiltonian system with Hamiltonian $$H = \sum_i \frac{\norm{p_i}^2}{2 m_i} - \sum_{i < j} \frac{G m_i m_j}{\norm{x_i - x_j}}, \qquad p_i = m_i x_i'.$$ There are ten classical first integrals. They are the energy, the three components of the total momentum, the three components of the angular momentum, and three from the uniform motion of the centre of mass.

For $n = 2$ the problem reduces to the Kepler problem and is solved completely: the relative motion follows a conic section. For $n \ge 3$ there is no general solution by elementary formulas, and the three-body problem already shows chaotic behaviour. A solution can stop existing in finite time at a singularity. Collisions are one kind of singularity. Xia proved in 1992 that for $n = 5$ there are also singularities without collision, in which bodies escape to infinity in finite time.