Here we are, at the beginning of the second leg of our journey, the second of the three–volume sequence. In Volume I, we have been uncovering some of the secrets of ordinary differential equations and their initial conditions—the so–called initial value problems. This was not done systematically though, but by pulling white rabbits out of a magician’s top hat, e.g., by saying: “Let us try this,” or “Empirical evidence indicates that,” and so on. In this volume, we present a systematic mathematical treatment of initial value problems, with applications to n-particle dynamics.