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There is a similar Phys.SE question here, but I still didn't get the idea. The problem is:

Write down the equations for one-dimensional motion of an ideal fluid in terms of the variables $a$ and $t$, where $a$ (called a Lagrangian variable) is the $x$ coordinate of a fluid particle at some instant $t=t_0$.

$x$ is clearly a function of $a$ and $t$, and so $x=x(a,t).$ One of the equations of motion is the equation of continuity: $$\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v= 0$$

I can't derive the solution, which was supposed to be $$\rho\frac{\partial x}{\partial a}=\rho_0$$

Where $ρ_0$ is the density at a at $t_0$.

This is what I've done so far: I have that $\frac{d\rho}{dt}=0$, and (I suppose) $$d\rho=\frac{\partial\rho}{\partial t}dt+\frac{\partial\rho}{\partial a}da$$

as well as

$$dx=\frac{\partial x}{\partial t}dt+\frac{\partial x}{\partial a}da$$

Aparently the main problem for me is that I don't know how to get to $\rho_0$...

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In the end this is physics, so maybe you should try it less formal? (and not confuse eulerian and langrangian methods)

Initially the mass element at $a$ had mass ( I use $\delta$ to make clear that we really should consider finite differences and then perform a limit at the end)

$$m = \rho(a,0) \delta a$$

now, we follow its motion and after some time t we should find that the mass is conserved, but we are now working in the distorted material frame thus

$$ m = \rho(a,t) \delta x$$

Equating these gives

$$ \rho(a,0) = \rho(a,t) \frac{\delta a}{\delta x} \rightarrow \rho(a,t) \frac{\partial x}{\partial a}$$


edit: just to clarify things: the limit considered here is in the size of the initial fluid element (not in time differences or sth like that), this equation is valid for all t


I never really liked Langrangian formalism (maybe you can estimate, how ugly it will become in higher dimensions.. you possibly will get there), so I looked up a reference to make sure, I had it right. Maybe these lecture notes: http://www.whoi.edu/science/PO/people/jprice/class/ELreps.pdf will be of some value to you

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