Analytical mechanics

Liouville’s theorem

Hamiltonian flow preserves phase-space volume. For an integrable system with n degrees of freedom, regular invariant level sets are organized as n-dimensional tori.

Hamilton’s equations define an incompressible flow in phase space. In canonical coordinates, the phase volume of a bundle of states remains constant in time.

\frac{\partial \rho}{\partial t}+\nabla_{(q,p)}\!\cdot(\rho\,\dot z)=0, \qquad \nabla_{(q,p)}\!\cdot\dot z=0

Definition: Conservation of phase volume

For $z=(q_1,\ldots,q_n,p_1,\ldots,p_n)$ , the Hamiltonian vector field is $\dot q_i=\partial H/\partial p_i$ , $\dot p_i=-\partial H/\partial q_i$ . Its divergence cancels pairwise, so the flow Jacobian is one and phase volume is preserved. This is Liouville’s theorem for Hamiltonian mechanics; it does not say every distribution has uniform density.

A trajectory winds on the invariant torus of a two-degree-of-freedom integrable system; vary the frequency ratio to see a closed orbit or one that progressively covers the torus.

Connection to integrability

The phase-volume theorem applies to general Hamiltonian flows. The Liouville–Arnold theorem adds a structural result: an autonomous system with n independent, mutually commuting integrals and a compact regular level set moves on an n-torus; in action–angle coordinates the angles advance linearly in time.

Example: Example

On what surface can an integrable two-degree-of-freedom system move in four-dimensional phase space when the independent integrals are fixed and the level set is compact?

Solution

Correct answer: a two-dimensional torus. By the Liouville–Arnold theorem, when an integrable system with $n$ degrees of freedom has a compact, regular level set $\{F_1=c_1,\dots,F_n=c_n\}$ , that set is an $n$ -torus $T^n$ — here $n=2$ , so the orbit lies on $T^2$ , not on a sphere or a line.

Quick check

In canonical coordinates, the divergence of the phase-space velocity field is $\sum_i(\partial\dot q_i/\partial q_i+\partial\dot p_i/\partial p_i)=\sum_i(\partial^2H/\partial q_i\partial p_i-\partial^2H/\partial p_i\partial q_i)=0$ . By the divergence theorem, a small phase volume carried by Hamiltonian flow keeps its volume. This does not mean each trajectory is stationary or that a distribution stays uniform: its density may deform, but is transported so that phase volume is preserved. The result underlies equilibrium statistical mechanics and the Liouville equation for phase-space density.

For an integrable system with two degrees of freedom and a compact regular level set, on what surface does Liouville–Arnold place the motion?

For an autonomous system, which quantity is conserved along Hamiltonian motion?

References

Herbert Goldstein, Charles Poole, John Safko (2002). Classical Mechanics
L. D. Landau, E. M. Lifshitz (1976). Mechanics