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Liouville's Theorem Consider the following linear system of ordinary differential equations: $$\tag{1} \dot{\mathbf{x}}=A(t)\mathbf{x}(t).$$ Let $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n$ be solutions of (1). Define the Wronskian determinant to be $$W(t)=\det \begin{bmatrix} \mathbf{x}_1 & \mathbf{x}_2 & \ldots & \mathbf{x}_n\end{bmatrix}.$$ Then we have the following differential relation for $W(t)$: $$\tag{L} \dot{W}(t)=\verb+trace+\, A(t)\,\cdot\,W(t).$$

Compare with a lemma due to Euler which I encountered in a course of fluid mechanics I am attending. Here $\mathbf{x}$ refers to Eulerian coordinates and $\mathbf{y}$ refers to Lagrangian coordinates.

Euler's Lemma Let $\mathbf{u}(\mathbf{x}, t)$ be the velocity field of a fluid flow $\mathbf{\Phi}(\mathbf{y}, t)$, meaning that: $$\begin{array}{cc} \displaystyle \begin{cases} \dot{\mathbf{x}}(t)=\mathbf{u}(\mathbf{x}(t), t) \\ \mathbf{x}(0)=\mathbf{y}\end{cases}, & \mathbf{x}(t)=\mathbf{\Phi}(\mathbf{y}, t)\end{array}.$$ Denote with $J$ the Jacobian of the deformation gradient, that is $$J(\mathbf{y}, t)=\det D_{\mathbf{y}}\mathbf{\Phi}(\mathbf{y}, t).$$ Then we have the following differential relation for $J$: $$\tag{E}\frac{dJ}{dt}(t)=(\verb+div+\,\mathbf{u})\,J.$$

Even if it is formulated with the language of fluid mechanics, this lemma is essentially a result in ordinary differential equations, just like Liouville's theorem. My question is if Euler's lemma can be viewed as a nonlinear version of Liouville's theorem and if either one of the two results can be derived from the other.

Thank you for reading.

Giuseppe Negro
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1 Answers1

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Nonlinear form of Liouville's theorem is also found in the literature under the name of Liouville's theorem. See these notes, where Example 3.1 (the linear form that you stated) is obtained as a special case of Lemma 3.2 (which is what you call Euler's lemma above). Another, very thorough, source is Hartman's book Ordinary differential equations, chapters IV and V.

Indeed, one of its principal applications of Liouville's theorem is to Hamiltonian systems which are typically nonlinear.

Giuseppe Negro
  • 34,508