Consider a system of $$N$$ classical particles. The particles a confined to a particular region of space by a container'' of volume $$V$$. The particles have a finite kinetic energy and are therefore in constant motion, driven by the forces they exert on each other (and any external forces which may be present). At a given instant in time $$t$$, the Cartesian positions of the particles are $$r_1(t), \cdots , r_N(t)$$﻿) ) . The time evolution of the positions of the particles is then given by Newton's second law of motion:
$m_i \ddot {r} _i = F_i ( r_1, \cdots , r_N )$
where $$F_1, \cdots , F_N$$ are the forces on each of the $$N$$ particles due to all the other particles in the system. The notation $$\ddot {r} _i = \frac {d^2 r_i}{dt^2}$$.
N Newton's equations of motion constitute a set of $$3N$$ coupled second order differential equations. In order to solve these, it is necessary to specify a set of appropriate initial conditions on the coordinates and their first time derivatives, $$\{r_1 (0), \cdots , r_N(0), \dot {r} _1 (0), \cdots , \dot {r} _N (0) \}$$. Then, the solution of Newton's equations gives the complete set of coordinates and velocities for all time $$t$$.