How to go from the configuration space to the phase space?

[aries0152]

How can we go from the configuration space of the system to the phase space when velocity can be expressed in terms of momenta?

 

[mathfeel]

By a Legrendre transformation. Suppose $f(q, \dot{q})$ is a function of configuration space variable, then there is a corresponding function in phase space is:$g(q, p) = \dot{q}(p) p - f(q, \dot{q}(p))$.