Liouville's Phase Space Theorem

States that for a nondissipative Hamiltonian system, phase space density (the area between phase space contours) is constant. This requires that, given a small time increment ,

			(1)
			(2)
			(3)
			(4)

the Jacobian be equal to one:

			(5)
			(6)
			(7)
			(8)

Expressed in another form, the integral of the Liouville measure,

(9)

is a constant of motion. Symplectic maps of Hamiltonian systems must therefore be area preserving (and have determinants equal to 1).

REFERENCES:

Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994.