## 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)

the Jacobian be equal to one:
 (3)

Expressed in another form, the integral of the Liouville Measure,
 (4)

is a constant of motion. Symplectic Maps of Hamiltonian Systems must therefore be Area preserving (and have Determinants equal to 1).