States and Vectors in Phase Space

This line of thinking started for me upon reading _The Emperor's New Mind_ by Roger Penrose. I found the idea of phase spaces very fascinating. They give the state of a system, and its evolution all in one concise geometry. Breaking this system down into short intervals, and addressing a system as a collection of subsystems, I came upon the idea of using points in space as anchors for vectors, giving a "point-and-vector" system: find a space in which all of the features of the various parts of the system can be qualified and quantified, give each subsystem its own evolutionary path, and then watch as the relationships between subsystems change; they may produce an entirely new sort of system, one which may not have been observed before, and may not even have existed before. This by itself is kind of an interesting way to see the world, but what if you could improve upon it further?

Magnitude -- the magnitude of the vector over a given interval of time; the quantity of change

Origin -- the state of the entity at any moment in time

Direction -- the qualitative nature of the change that occurs over the given interval; this may be a vector in any number of measurable or quantifiable dimensions

Evolution -- the tendency of the vectors to change as well, over a given interval

Its interesting to note at this point that "Evolution" points to another dimension of a phase space in its own right, giving the possibility of an evolving phase space, containing a system which is itself evolving. The modeling system itself may have to evolve alongside the system it is modeling. A good example would include the history of the automobile, civic engineering, environmental impacts, and traffic density and patterns; each affects the other a great deal, and none of the developments is truly reversible. Emerging considerations would include speed limits, safety restraints for adults and children, gas stations, hybrid and electric vehicles, recharging stations and availability, etc. One day personal vehicles may come in a form unrecognizable to us at present, or cease to exist altogether for a considerable period of time.

Over the short term, however, certain theories and developments are more likely than others, and if we can model how these smaller elements interact to shape the future of the larger system, we may understand its future much better than a more global but distant perspective. In order to see the future of the forest, you have to watch the trees grow. P;D

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