The physical problem can be informally stated as:

More precisely,

In mathematical terms, this means to find a global solution of the initial value problem for the differential equations describing 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 (unlike in the case of friction) also a reversal of time gives also a solution.

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

In the physical literature about the n-body problem (n ≥ 3), sometimes reference is made to the impossibility of solving the n-body problem. However care must be taken when discussing the 'impossibility' of a solution, as this refers only to the method of first integrals (compare the theorems by Abel and Galois about the impossibility of solving algebraic equations of degree five or higher by means of formulas only involving roots).

The n-body problem contains 6n variables, since each point particle is represented by three space (displacement) and three velocity components. First integrals (for ordinary differential equations) are functions that remain constant along any given solution of the system, the constant depending on the solution. In other words, integrals provide relations between the variables of the system, so each scalar integral would normally allow the reduction of the system's dimension by one unit. Of course, this reduction can take place only if the integral is an algebraic function not very complicated with respect to its variables. If the integral is transcendent the reduction cannot be performed.

The n-body problem has 10 independent algebraic integrals