fluid dynamics - Lagrangian to Eulerian coordinates

I have a 1D fluid and at a time t0 I know the speed of all its particles. So I can know the future position of every particle. Given a particle at position x0 at time t0, i can know its position at any future time. If I am not wrong this is called the lagrangian coordinates.

My problem is that with this information I want to know the speed of the fluid at any time in any position. I don´t want to be following the particles but I want to be able see what happens at a fixed position. I think this is called the eulerian coordinates.

How do I make this change of variables?

Answer

To get the Eulerian velocity you must first know the full Lagrangian description. The Lagrangian description means you know the trajectory $\vec X$ of the particles, $$\vec X(\vec x_0, t),$$ from which you can derive the particles velocity: $$\vec V(\vec x_0, t)=\frac{d}{dt}\vec X(\vec x_0, t).$$ To get the Eulerian velocity at a certain position $\vec x$, you set it equal to $\vec X$ and solve for $x_0$. You will get $x_0$ as a function of $x$ and $t$. $$\vec x=\vec X(\vec x_0, t)\rightarrow\vec x_0=\vec x_0(x, t)$$ What you are doing here is basically asking the question 'what was the starting point of the particle that is currently at $x$? '. This is useful because now you can plug $x_0$ in $\vec V$. Now you are done because you know the velocity of the particle currently at $x$, so you can just set $$\vec u(\vec x,t)=\vec V(\vec x_0,t)$$