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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 $W(t)$: $$\tag{L} \dot{W}(t)=\verb+trace+\, A(t)\,W(t).$$

The following is 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, the second lemma is essentially a result in ordinary differential equations. My question is if the second lemma can be viewed as a nonlinear version of the first and if either one of the two lemmas can be derived from the other.

Thank you for reading.

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1 Answer 1

up vote 2 down vote accepted

Nonlinear form of Liouville's theorem is also found in the literature under the name of Liuoville'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.

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