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What is a Map? | ||

The word "map" in the phrase "Standard Map" has a different meaning than
it does in everyday life. It doesn't mean that the Standard Map is a chart
or a guide to a location. In physics and mathematics, a "map" is any mathematical transformation that is applied over and over again in sequence. Sometimes the "mapping" is used instead of "map." |

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What is a Map? | ||

The word "map" is used as a verb too. A physicist might say that under
a particular mathematical transformation, a certain point is "mapped" to
another point at a different location.
The "map" equations can be used over and over again on a single point, so
that the point gets "mapped" repeatedly to new locations. Each implementation
of the mathematical equations is called an |

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What is a Map? | ||

The best way to see this is by an example. First let's make up a starting
condition, then we'll make up a "map."
Suppose that you are keeping track of the location of a point by
the variable x = 2The value of the starting point isn't important here. We just picked "2" to give an example. |

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What is a Map? | ||

Now that we have our starting condition, let's make up a map.
An example of a very simple map might be expressed as: "To get the location of
the new point, add three to location of the old point." The way you would write
this map in mathematical terms is:
x' = x + 3The map above is not a very exciting map, to be sure, but is useful for showing how all maps work, including the Standard Map.
Notice the way that this map is written in mathematical form. There are two
variables present, Using primed variables like this is a very common way of writing the mathematical equations of maps, by the way. |

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What is a Map? | ||

So now we have a starting condition, and the map to go with it. Let's see what
happens when we apply the mapping equation to the starting condition.
We started out with:
x = 2 (Starting condition) x' = x + 3 (The map)When we iterate the map, the effect can be shown by a column of numbers as follows:
x ---> x' 2 ---> 5 (First iteration) 5 ---> 8 (Second iteration) 8 ---> 11 (Third iteration) 11 ---> 14 (Fourth iteration) . (etc.) . .Notice that as promised, the effect of this particular map is to add 3 to value of the old point to get the new point. Notice another important point: after the map is iterated, the "new" value (x-prime) becomes the "old" value for the next time that the map is iterated. The dots at the bottom of the column mean that you could keep on iterating the map forever, adding three to the old value over and over again. |

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What is a Map? | ||

So that's basically how maps work in general.
In the next section, you will learn how |