# Which are the exact definitions of Lagrangian and Eulerian descriptions of motion?

Trying to reply to myself to the question 'Which are the exact definitions of Lagrangian and Eulerian descriptions of motion?' My attempts go in three directions.

• Lagrangian description of motion is defined by the use of some reference frame of coordinates $$(X_0, Y_0, Z_0, \tau)$$, while Eulerian description by the use of some reference frame $$(x, y, z, t)$$. The passage from one reference frame to the other is accomplished by the transformation

$$\left\{\begin{matrix}x=\chi_1(X_0, Y_0, Z_0, \tau) \\y=\chi_2(X_0, Y_0, Z_0, \tau) \\z=\chi_3(X_0, Y_0, Z_0, \tau) \\t=\tau\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \end{matrix}\right.$$

• Lagrangian description is defined as the transformation itself from $$(X_0, Y_0, Z_0, t)$$ to $$(x, y, z, t)$$, while Eulerian as the backward of it

• Lagrangian description is defined as the transformation itself from $$(X_0, Y_0, Z_0)$$ to $$(x, y, z, t)$$, while Eulerian as the backward of it

So first question is

1. Is any of these attempts correct? If not, what would be the correct definitions, and where these others would fail?

And secondly

1. Is correct to say that Lagrangian reference frame moves with the body while Eulerian reference frame is fixed in space? Is possible to deduce this from model statement?

2. Lagrangian reference frame is the same concept as non-inertial reference frame? Or, which is the difference between the two?

Any help is really appreciated, thanks

• Feedback to the post (v4): OP's first sentence in point 2 is correct. The rest seems incomplete at best. Commented Jan 23, 2020 at 19:57