lagrangian and eulerian coordinates

This is a simple analogy that I learnt in Dr. Caglar Oskay's class on Advanced Solid Mechanics.

Consider a horse race in progress. Say you have bet on one horse and you are only in interested in making money- then you will be following only that particular horse. This is an Eulearian description where you are following only one point in the system.

However, if you are enjoying the race in general, you will be looking at all the horses (lets say at the place where where you are seated) and not concentrating on only one. This is the Lagrangian description where you are looking at every point in a given space and not focussing on a single point. So, the difference is between observing a parcel of material through space and time (Lagrangian) vs looking at a specific location through time. (Eulerian)

In continuum mechanics, one assumes that the body starts from an undeformed configuration that is mapped into its deformed configuration (an unstressed initial configuration may not exist but even in that case one can define a reference/material manifold). The reference manifold (space) can always be locally described by material (Lagrangian) coordinates. Similarly in the ambient space the deformed body can always be locally described by spatial (Eulerian) coordinates.

As an example, one can write the balance of linear momentum either with respect to the reference configuration or the current configuration. In either case, all the quantities live in the ambient space. Cauchy stress acting on a unit normal of a deformed surface gives the traction. First Piola-Kirchoff stress acting on the unit normal of the corresponding undeformed surface would give exactly the same traction in the deformed configuration. Choosing a measure of deformation is a matter of choice, though things should be done consistently.

If I remember correctly, in Malvern's book it is mentioned that a third approach would be to consider an intermediate configuration as the reference configuration. I don't think there is anything profound in doing so and in some sense that would still be a spatial description of motion.

It is misleading to refer to these as "coordinates". A coordinate system (patch) is a choice for locally describing a manifold. I would rather say material and spatial (or Lagrangian and Eulerian) descriptions of motion.

In linearized elasticity (or any linearized theory) one cannot distinguish between the two. However, there is always an underlying reference motion with respect to which things are linearized. That reference motion explicitly affects everything.

Hope this helps.

Regards,
Arash

http://en.wikiversity.org/wiki/Nonlinear_finite_elements/Lagrangian_and_Eulerian_descriptions

Lagrangian approach can
also be referred as system approach in which we follow the material
point/object whereas  Eulerian approach refers to investigation on
open systems i.e control volume approach.

The mathematical formulations for each of the same basic laws of nature will look different  in the two approaches.

<>It would be helpful if you go through the first few chapters of "Computational Fluid Dynamics" by John D Anderson