I feel as if I understand the conceptual difference between Lagrangian and Eulerian descriptions. But I can't seem to follow the math. I have one example that I would greatly appreciate an explanation or full calculation for

The motion can be expressed as: $$\left\{\begin{align} x_1(X_1,X_2,X_3,t) &= X_1 \\ x_2(X_1,X_2,X_3,t) &= X_2 + \dfrac{X_2 v_0 t}{h_0} \\ x_3(X_1,X_2,X_3,t) &= X_3 \\ \end{align}\right.$$ velocity in Lagrangian description: $$v_i = \begin{bmatrix} 0 \\ \left(\dfrac{X_2v_0}{h_0}\right) \\ 0 \end{bmatrix}$$ in Eulerian description: $$v_i = \begin{bmatrix} 0 \\ \left(\dfrac{x_2v_0}{h_0 + v_0 t}\right) \\ 0 \end{bmatrix}$$

I am entirely on board on the velocity in the Lagrangian description. I cannot for the life of me figure out how the Eulerian description is computed nor why.

Can someone show me how the Eulerian description of the velocity is calculated?

The motion is $$ x = \varphi(X, t) \quad \leftrightarrow \quad X = \varphi^{-1}(x, t) $$ The Lagrangian velocity is $$ V(X, t) = \dot{\varphi}(X, t) = \frac{\partial \varphi (X, t)}{\partial t} $$ The spatial (Eulerian) velocity is defined as $$ v(x, t) = V(\varphi^{-1}(x, t),t) = \dot{\varphi}(\varphi^{-1}(x, t),t) = \frac{\partial \varphi ( \varphi^{-1}(x, t), t)}{\partial t} $$ In your case $$ \varphi(X, t) = \begin{bmatrix} X_1 \\ \frac{h_0 + v_0 t}{h_0} X_2 \\ X_3 \end{bmatrix} \quad \implies \quad V(X, t) = \begin{bmatrix} 0 \\ \frac{v_0}{h_0} X_2 \\ 0 \end{bmatrix} $$ and $$ \varphi^{-1}(x, t) = \begin{bmatrix} x_1 \\ \frac{h_0}{h_0 + v_0 t} x_2 \\ x_3 \end{bmatrix} \quad \implies \quad v(x, t) = V(\varphi^{-1}(x, t),t) = \begin{bmatrix} 0 \\ \frac{v_0}{h_0} \frac{h_0}{h_0 + v_0 t} x_2 \\ 0 \end{bmatrix} $$

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    Nice. What are X1, X2 and X3 relatively to x1, x2 and x3? The two descriptions for the velocity are clear (the Eulerian - with respect to spatial position and the Lagrangian with respect to the particle frame of reference). But the position description (you name it the motion function) are quite confusing. Can you shed some light about their physical meaning? – Yaniv Ben David Sep 15 at 7:47
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    @YanivBenDavid: The $X_i$ values indicate a reference positions from which the motion is measured. The $x_i$ values indicate the current positions of points which were initially at $X_i$. If there is no motion $x_i = X_i$. In continuum mechanics, a "motion" of a body is a smooth map between $x$ and $X$. If we examine the motion at a fixed $t$, the motion is called a "deformation". – Biswajit Banerjee Sep 16 at 21:05

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