Intuitive

A phase space is a mathematical tool that allows to grasp important aspects of complicated systems.

Each point of the phase space represents one specific configuration a given system can be in.

The state of a system is recorded in a phase space point through all the location and all the momenta the objects in the system have at a given point in time.

The time evolution of a system can then be represented as a path in phase space.

Concrete

Each point of the phase space corresponds to one particular state of the system.

$\{q_i \}$ defines a point in n-dimensional configuration space $C$. Time evolution is a path in $C$. However, the complete state of the system is defined by $\{q_i \}$ and $\{p_i \}$. Only with this information, we are able to determine the state at all times in the future. The pair $\{q_i, p_i \}$ defines a point in a $2n$-dimensional space we call phase space.

Since each point in phase space is sufficient to determine the time evolution of the system, paths in phase space can never cross. Otherwise, the time-evolution would not be unique.

It is conventional to say that the time evolution is governed by a flow in phase space.

Abstract

The space of states in classical mechanics is modeled as a manifold $M$ equipped with a symplectic structure: $(M,ω)$. This manifold is what we usually call phase space. The phase space is a symplectic manifold which simply means that is a manifold equipped with a symplectic structure. A symplectic structure is a distinguished 2-form $(\omega)$.

Such a 2-form is an object that eats two vector fields on our manifold and returns another function on the manifold. Functions on the manifold are smooth maps $f \ : \ M \longarrow R$. These functions are what we call “the observables of our classical system”. So in words, this means that the observables of our classical system map each state to a real number.

One of the most important function on our phase space manifold is the Hamiltonian function. This function represents the energy of the system and describes the time-evolution of phase space points.

Why is it interesting?

Hamiltonian Mechanics is geometry in phase space. […]

Mathematical Methods of Classical Mechanics Vladimir Arnold

While it is true that the primary perception we, human beings, have of our world privileges positions, and their evolution with time, this does not mean that we have to use only, mathematics in configuration space. As Basil Hiley puts it “…since thoughts are not located in space-time, mathematics is not necessarily about material things in space- time”. Hiley is right: it is precisely the liberating power — I am tempted to say the grace — of mathematics that allows us to break the chains that tie us to one particular view of our environment." The Principles of Newtonian and Quantum Mechanics: by Gosson

History

See: The tangled tale of phase space by David D. Nolte https://works.bepress.com/ddnolte/2/download/