In order to progress from qualitative arguments and toy models it is necessary to set up apparatus for describing a gravitational N-body system. There are several ways in which this can be done.

One common approach is to employ the N position vectors ri and the N velocity vectors vi of the stars at some time. Each of these vectors has three components, and so the entire system can be described by a single 6N-dimensional vector, i.e. a single point in a 6N-dimensional space Г. This is a useful description, because it is sufficient to specify uniquely the entire subsequent evolution of the system, as the equations of motion are of second order; they describe the motion of this point through Г. Implicitly, therefore, this is the description adopted in N-body methods, even though it is more natural to think of N particles moving in a six-dimensional phase space.

This description in a 6N-dimensional space can be turned into a statistical one if we imagine a collection of stellar systems, each described by a distinct point in Г. If their distribution is described by a probability density function f, the evolution of f is determined by the equations of motion, and indeed is equivalent to them. This description is almost never used in stellar dynamics.

Another way of describing a stellar system is to represent each star by a single point in a six-dimensional space with coordinates r and v.