Tuesday, May 20, 2014

Rectangle billiard solution

Suppose you have a table and a linear paramterization for a ball on the table...

These equations tell you where on the table you'll be after time t, if the ball is thought to bounce off the walls as on a billiard table, where X and Y are the width and height of the table.

So it's interesting that with these equations, one can calculate directly the x,y position of the ball at, say,  time = 12845 without figuring out where it was at every time before that. 
Here is a plot of one example:

There is a field called dynamical billiards that studies more complicated ones, but for some reason they didn't give this nice solution to the rectangular one on wikipedia:, I guess physicists are more interested in the ones that become chaotic.

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