In basic physics, the emphasis is calculating total forces, and going from that to the position function of the body as a function of time by integration. Now, from this prespective, how did people reach the idea to look at phase spaces?; What is the historical motivation for this?
Since the initial conditions of a mechanical system involve positions (configuration space) and velocities (or momenta), the space of initial conditions is the phase space. This was realized by Hamilton who rewrite the second order equations of motion as a first order system of double size.