# Is Euler's lemma of fluid mechanics a nonlinear version of Liouville's theorem of ODEs?

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.