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?
$\begingroup$ Would "analytical mechanics" be enough of a response? Other than that I think the development of Liouville's theorem (1838) is an event where the importance of phase space became definitive $\endgroup$– MauricioApr 14 at 22:06
$\begingroup$ People did not reach the idea from the perspective of single body dynamics, not even Hamiltonian dynamics. Although it is implicit in the work of Liouville and Jacobi, the concept of phase space was only introduced by Boltzmann in his 1872 work on statistical mechanics of gases. It was developed by Gibbs, also in the context of statistical mechanics, who came up with the metaphor of phase flow, see What is the etymology of "phase space" of a dynamical system? $\endgroup$– ConifoldApr 14 at 23:25
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.