Phase space

Phase space is the means by which the mechanical problem is converted in to geometry. Phase space, or $\Gamma$-space, is a Euclidean space in $2s$ dimensions (i.e. $E^{2s}$), where $s$ is the number of degrees of freedom. Thus our description of our system in terms of positions and velocities now becomes a point in phase space. Changes in our system now trace out a trajectory in phase space.