Hamiltonian Map

A Hamiltonian map is a map on phase space obtained by evolving a Hamiltonian system for a fixed time. It is a symplectic diffeomorphism wherever the flow is defined (Hairer 2010, p. 7). For example, consider a system with one degree of freedom and Hamiltonian

(1)

where has two continuous derivatives. Then Hamilton's equations give

			(2)
			(3)

For a fixed time step , let denote numerical approximations at . The Euler forward method gives

			(4)
			(5)

This update is not area-preserving in general, since its Jacobian is

			(6)
			(7)

Replacing the explicit update by

			(8)
			(9)

the chain rule gives the Jacobian

			(10)
			(11)

so this numerical map is area-preserving.