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This is in continuation/related to the following question asked in the below link

Eulerian and Lagrangian descriptions of velocity

According to my understanding Lagrangian approach is an approach where you track the individual object of which you want to take measurement of for eg in a fluid flow you track a particular fluid particle and measure the physical quantity of that particular fluid particle.(leading to pathlines)

But in the Eulerian Approach, you basically try to measure the physical properties around a particular area/surface.

In application to continuum mechanics in structure, it is mentioned that the initial/reference position of a body is called as the Lagrangian Configuration and the current position(Mapped from Initial Configuration) is called as Eulerian Approach.

Q1) I am not able to understand why is that so? Is it because of the fact the in Lagrangian Configuration/Initial Condition the body is supposed to be rigid and it becomes easier to understand rigid body motion and when there is no rigid body motion ie when deformation starts to occur we shift to Eulerian Approach to study deformation?

Q2) If the answer to the above question is yes, why is the Eulerian approach used to study deformation and if not why do we use two approaches??

EDIT 1:

Thanks for your reply. I have been using this particular reference. http://web.mit.edu/abeyaratne/Volumes/RCA_Vol_II.pdf in which Eulerian and Lagrangian configurations are explained.(Page 6, 1.3). Here it describes Eulerian configuration as

"the representation (1.6) which deals with the positions of the particles in the deformed configuration, (the configuration in which the physical quantity is being characterized,) is called the Eulerian or spatial description."

and for lagrangian

"If a reference configuration has been introduced we can label a particle by its position x = χref(p) in that configuration, and this, in turn, allows us to describe physical quantities in Lagrangian form."

This is why I am getting confused with Eulerian and Lagrangian. Is there any specific reason so as to why its named like that?

EDIT 2:

Adding on more reference for the same with the same query(Why are these configurations called Eulerian or Lagrangian ? )

enter image description here

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I think it is quite not clear for you what is an Eulerian and Lagrangian description.
As you briefly said at the beginning of your post, in the Eulerian approach, you consider a small volume that is fixed is time and space, and study what happens to this volume: what enters or goes out (convection), if there are some forces applied to it, if it is heated or not etc.
In the Lagrangian approach, you consider a material particle an follows its evolution. That is why for example there is no convection term in a Lagrangian description of the motion, since there is no material motion to consider within a material particle.
Lagrangian and Eulerian approaches are thus two different way to describe the evolution of physical quantities, and the equations that result are different.
In solids mechanics, the Lagrangian description of the motion is very widely used, while the Eulerian approach is usually preferred in fluid mechanics.

EDIT:
I went through the chapter you mention in the reference you sent. It is a very good book, but quite involved if you start studying continuum mechanics. Section 1.3 is very clear about what is the difference between the Eulerian and Lagrangian approaches (or description, as written in the book): They are two different "mindsets" to describe the evolutions of physical quantities.
Either you decide that you decide that you sit in a specific location, and look what happens around you. You observe that particles are moving, the temperature is changing etc. You observe the evolution of the physical quantities from that specific location point of view. This is an Eulerian approach.
Otherwise you decide to follow a specific particle and move with it. You position is changing with time, and you observe the evolution of the physical quantities from that specific particle point of view. This is a Lagrangian approach.

I do not know the reference you are using, but having quite extensively studied continuum mechanics, these are the most common appellations for the various configurations:

  • Initial configuration = reference configuration = undeformed configuration
  • Current configuration = deformed configuration

EDIT:
I was quite surprise you wrote about "Eulerian configuration" and "Lagrangian configuration" since I never encountered this terminology before. Actually, they are not in the reference you provided neither. The terms that I introduced (reference/current/(un)deformed) are the one that are used as well.
You have however Eulerian and Lagrangian coordinates, as introduced in the snippet you added in your post. The Lagrangian coordinates $\mathbf{X}$ are measured in the undeformed configuration, and the Eulerian coordinates $\mathbf{x}$ are measured in the deformed configuration. You thus have the displacement $\mathbf{u}$ of a material point, which is the difference between its current position and its original position, that is given by $\mathbf{u} = \mathbf{x}-\mathbf{X} $.

Be careful : in your post you say that the current configuration is the Eulerian approach. "Approach" here means a way to describe the motion. "Configuration" relates to the state of the body at a certain time.
When you decide to write your equations that describe the motion of the domain you want to study, you chose one or the other approach. You cannot switch from one to the other.

I think that the questions you asked result from a misunderstanding of these notions, which, I agree, are quite complex.

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  • $\begingroup$ Seems like I have mixed the meaning of approach and configuration. I have added the source where I found this definition. Am I missing something ? $\endgroup$ – daniel Apr 15 '20 at 17:33
  • $\begingroup$ Thanks a lot. Any specific reason why the coordinates in undeformed configuration called Lagrangian and coordinates in deformed configuration called Eulerian ? $\endgroup$ – daniel Apr 16 '20 at 17:08
  • $\begingroup$ Never Mind. I found this youtube.com/watch?v=qpA8X5Gw2gc and I think it's an appropriate definition for different coordinates? Thanks a lot, professor!!!! $\endgroup$ – daniel Apr 16 '20 at 17:17
  • $\begingroup$ It is a pretty nice video indeed to explain the two approaches. You're very welcome! $\endgroup$ – ClariB Apr 17 '20 at 11:16

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