Trajectory

in the original sense of the term

A curve described by the motion of a point (the trajectory of this point). In the motion of a system of material points, each point moves along its trajectory. At the same time, the state of the entire system is depicted by a point in the phase space, which also moves along some trajectory in this space. When it is necessary to emphasize the fact that one is dealing with the latter trajectory, one talks about the phase trajectory. (For a "system" consisting of one material point $M$, the difference between its "geometric" trajectory in ordinary space and the phase trajectory is still retained, since the state is not just the geometric position of $M$ but includes its velocity as well.)

In the more abstract theory of dynamical systems (cf. Dynamical system), a trajectory usually means the phase trajectory (the more so because in the general case it is not appropriate to talk about the trajectory in any other sense; the word "system" need not have the physical sense of a system of material points). Strictly speaking, the phase trajectory need not be a curve but can reduce to a single point (an equilibrium position). Finally, the use of the term "trajectory" as a synonym of orbit (a trajectory that is not necessarily a curve) corresponds to the abstract concept of a dynamical system as a group or semi-group of transformations, not necessarily a one-parameter group (semi-group).

For a cascade, obtained by the iteration of a non-invertible mapping $S$, by the trajectory (or complete trajectory) of a point $x$ one sometimes means the collection of all its images $S^nx$, $n=0,1,\dots,$ and all the pre-images of these images under the mappings $S^m$, $m=1,2,\dots$.

References

[a1]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a2]	V.I. Arnol'd, V. Avez, "Ergodic problems of classical mechanics" , Benjamin (1968) (Translated from Russian)