Hamiltonian Map

Consider a one-dimensional Hamiltonian map of the form

(1)

which satisfies Hamilton's equations

			(2)
			(3)

Now, write

(4)

where

			(5)
			(6)

Then the equations of motion become

			(7)
			(8)

Note that equations (7) and (8) are not area-preserving, since

			(9)
			(10)
			(11)

However, if we take instead of (9) and (10),

			(12)
			(13)

			(14)
			(15)

which is area-preserving.