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Phase Space
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By now, you have learned about one-dimensional maps, and about
two-dimensional maps, both uncoupled and coupled. So far, however, we
haven't said much about what the maps actually represent. That is, what
do the mathematical
transformations of the map tell us about the physical world?
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Phase Space
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In the example of the two-dimensional map, for example, it was implied that the
variables x and y that were transformed by the map might represent the horizontal and vertical positions of a point in space.
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Phase Space
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That being so, one might expect that the interation of the map,
which causes x and y to change in value, is representative of some
physical process in which the horizontal and vertical locations of an
object are changed repeatedly in time. The change in location of the object
in time is
mirrored by the change in the mathematical values under iteration of
the map.
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Phase Space
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Each iteration of the map would represent the change in the physical
location of the object over a set time interval. That is, the time elapsed
between each iteration of the map is always the same. For example, if the
first iteration of the map represents the change in location after exactly one second, then the next iteration must represent the change after another
one second, and so on. | |
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Phase Space
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But although physicists usually use maps to represent physical processes
in time, the two variables in two-dimensional maps don't usually represent
the horizontal and vertical positions of the particles.
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Phase Space
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Instead, the two-dimensional maps that interest physicists are the ones
in which one variable of the map represents a position, but the other
variable of the map represents a speed associated with the position.
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Phase Space
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In other words, if one variable of two-dimensional map (x, for example) represents horizontal
position, the other variable (y, for example) does not represent an actual
vertical position, but instead it represents the horizontal speed of the
particle at the same moment that the position is measured.
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Phase Space
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Another example might be where one variable of the two-dimensional map
represents the angular position of an object which is moving around in
a circle. In this case, the other variable of the two-dimensional map
would represent the angular speed, that is, the speed at which the object
is moving around the circle.
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Phase Space
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Like any two variables, the position and its associated speed can be plotted
on a two-dimensional graph. But in this case, the two-dimensional graph does
not represent a real two-dimensional area, since only one variable is a
real position.
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Phase Space
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The two-dimensional space formed by a real position and its associated
speed is an abstract space that is called phase space by physicists.
A phase space is any abstract space formed by a position (either linear
or angular) and its associated speed.
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Phase Space
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Phase space maps are the maps that physicists use to study the behavior of
physical systems, including chaotic systems. The Standard Map, for example,
is a phase space map. This means that in any graphical representation of
the map, the plot does not actually represent what you would see with your
eyes, since only one variable is a real position. Instead, the plots of
phase space maps depict a general view of the motion, a view that is abstract,
yet very powerful as tool for studying motion.
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