Liouville’s theorem


Let

x˙=f(x) (1)

be a autonomousMathworldPlanetmath ordinary differential equationMathworldPlanetmath in n defined by a smooth vector field f:nn and the Jacobian of f is denoted fx. Also let Φt(x) be the flow (http://planetmath.org/Flow2) associated with (1). Let

V(t)=Φt(D)𝑑x

be the volume of the image of D under this flow after a time t.

Theorem 1 (Liouville’s theorem).

If DRn is a bounded measurable domain. Then

V˙(t)=Φt(D)divf(x)𝑑x
Proof.

Let V(t) be defined as above then

V(t0+h) = Φt0+h(D)𝑑y
= Φh(Φt0(D))𝑑y
= Φt0(D)det(Φhx(x))𝑑x.

We claim that, for xΦt0(D),

Φtx(x)=I+tfx(x)+o(t)

as t0.

In fact,

Φt(x)=x+0tf(Φs(x))𝑑s,

and by the Leibniz integral rule

Φtx(x)=I+0txf(Φs(x))𝑑s,

so that

tΦtx(x)=xf(Φt(x))

and evaluating at t=0 we get

tΦtx(x)|t=0=xf(Φ0(x))=fx(x).

Our claim follows from this and from the definition of derivative.

Hence

det(Φtx(x)) = det(I+tfx(x))+o(t)
= i=1n(1+fixi(x))+o(t)
= 1+ti=1nfixi(x)+o(t)
= 1+tdivf(x)+o(t)

as t0. It follows that

V(t0+h)=Φt0(D)1+hdivf(x)+o(h)dx

and

V˙(t0) = limh0V(t0+h)-V(t0)h
= Φt0(D)1+hdivf(x)+o(h)dx-V(t0)h
= V(t0)+hΦt0(D)divf(x)𝑑x+o(h)-V(t0)h
= Φt0(D)divf(x)𝑑x+limh0o(h)h
= Φt0(D)divf(x)𝑑x.

Corollary 1.

The flow of an Hamiltonian system (http://planetmath.org/HamiltonianEquations) preserves volume.

Proof.

It follows directly since the vector field of an Hamiltonian system has divergence equal to zero. Hence V˙=0 implies that the volume is constant. ∎

References

  • TG Teschl, Gerald: Ordinary Differential Equations and Dynamical SystemsMathworldPlanetmathPlanetmath. http://www.mat.univie.ac.at/ gerald/ftp/book-ode/index.htmlhttp://www.mat.univie.ac.at/ gerald/ftp/book-ode/index.html, 2004.
Title Liouville’s theorem
Canonical name LiouvillesTheorem
Date of creation 2013-03-22 15:14:55
Last modified on 2013-03-22 15:14:55
Owner Koro (127)
Last modified by Koro (127)
Numerical id 20
Author Koro (127)
Entry type Theorem
Classification msc 34A34