n-body Problem - Mathematical Formulation of The n-body Problem

Mathematical Formulation of The n-body Problem

The general n-body problem of celestial mechanics is an initial-value problem for ordinary differential equations. Given initial values for the positions and velocities of n particles (j = 1,...,n) with for all mutually distinct j and k, find the solution of the second order system

where are constants representing the masses of n point-masses, are 3-dimensional vector functions of the time variable t, describing the positions of the point masses, and G is the gravitational constant. This equation is Newton's second law of motion; the left-hand side is the mass times acceleration for the jth particle, whereas the right-hand side is the sum of the forces on that particle. The forces are assumed here to be gravitational and given by Newton's law of universal gravitation; thus, they are proportional to the masses involved, and vary as the inverse square of the distance between the masses. The power in the denominator is three instead of two to balance the vector difference in the numerator, which is used to specify the direction of the force.

For every solution of the problem, not only applying an isometry or a time shift but also a reversal of time (unlike in the case of friction) gives a solution as well.

For n = 2, the problem was completely solved by Johann Bernoulli (see Two-body problem below).