Phase space
Phase space is the name given to a coordinate-momentum space. It is the means by which a mechanical problem can be converted in to a geometrical problem. Phase space, sometimes written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} -space, is an Euclidean space in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2s} dimensions (i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E^{2s}} ), where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} is the number of degrees of freedom. Thus a description of a system in terms of positions and velocities now becomes a point in phase space. Changes in the system now trace out a trajectory in phase space.
One of the most important properties of phase space is that, for a long period of time, the phase-trajectory will spend an equal amount of time in equal volume elements.