# Phase space

We construct a Cartesian space in which each of the $$6N$$ coordinates and momenta is assigned to one of $$6N$$ mutually orthogonal axes. Phase space is, therefore, a $$6N$$ dimensional space. A point in this space is specified by giving a particular set of values for the $$6N$$ coordinates and momenta. Denote such a point by

$x = (p_1, \cdots , p_N, r_1, \cdots , r_N )$

$$x$$ is a $$6N$$ dimensional vector. Thus, the time evolution or trajectory of a system as specified by Hamilton's equations of motion, can be expressed by giving the phase space vector, $$x$$ as a function of time.

Figure: Evolution of an ensemble of classical systems in phase space (top). The systems are a massive particle in a one-dimensional potential well (red curve, lower figure). The initially compact ensemble becomes swirled up over time.

The law of conservation of energy, expressed as a condition on the phase space vector:

$H(x(t)) = \text {const} = E$

defines a $$6N - 1$$ dimensional hypersurface in phase space on which the trajectory must remain.