Note 7
The phase space of a given dynamical system is, to begin with, the set of all states relevant to, and consistent with, that system. These states are given by the points of, generally, a multidimensional space (phase space), where each dimension represents the range of possible values of a particular variable of a particular particle of the system. Said differently : If we have a system consisting of, to begin with, just one (moving) particle, then the dynamic characteristics of this particle at a particular instant in time can be given by its position, which needs to be specified by, generally, three space coordinates, and its momentum, of which the variable part consists in the velocity of the particle, which must be specified by three vectors (themselves specifying the particle's speed and direction). So six variables in all specify the state of the one-particle system. And this state can now be represented by a point in a six-dimensional space (phase space). And when the system proceeds, i.e. when one or more values of the particle's variables change continuously, then our point traces a trajectory through phase space, reflecting the system's time evolution If the system consists of two particles (instead of just one), then we need 2 x 6 = 12 variables to characterise the state of the system (at a particular instant in time). And the value of these 12 variables can now be represented by a point in a 12-dimensional space (phase space). And, again, when the system proceeds ('runs'), this point traces a trajectory through this 12-dimensional space (phase space), reflecting the system's time evolution.
In the same way, to characterize the state of a system consisting of N interacting particles, we need a 6N-dimensional space (phase space) to represent the state of the system by a single point (and represent the system's history by a trajectory through this 6N-dimensional phase space).
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