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
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, as a function of time.
The law of conservation of energy, expressed as a condition on the phase space vector:
defines a 6N-1 dimensional hypersurface in phase space on which the trajectory must remain.