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 $dt$,

$$q_1\equiv q(t_0+dt) = q_0+ {\partial H(q_0,p_0,t)\over\partial p_0} dt+{\mathcal O}(dt^2)$$ (1)

$$p_1\equiv p(t_0+dt) = p_0- {\partial H(q_0,p_0,t)\over\partial q_0} dt+{\mathcal O}(dt^2),$$ (2)

the Jacobian be equal to one:

$${\partial(q_1,p_1)\over\partial(q_0,p_0)} = \left\vert\begin{array}{ccc}{\partial q_1\over\partial q_0} & {\p... ...r\partial p_0} & {\partial p_1\over\partial p_0}\end{array}\right\vert\nonumber$$

$$= \left\vert\begin{array}{ccc}1+{\partial^2 H\over\partial q_0\part... ...\partial q_0\partial p_0}\,dt\end{array}\right\vert+{\mathcal O}(dt^2)\nonumber$$

$$= 1+{\mathcal O}(dt^2).$$ (3)

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

$$\prod_{i=1}^N \int dp_i\,dq_i,$$ (4)

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

See also Liouville Measure, Phase Space

References

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