vector fields: Lagrangian and Eulerian description

vector fields: Lagrangian and Eulerian description

When we deal with

vector fields

defined over a

continuum

media, a suitable choice of

coordinate systems

becomes indispensable in produce on the continuum and the subsequent physical behavior that such a material media experiences.  We shall discuss two possible methods that allow to approach the mentioned phenomena.

Some basic

concepts

and definitions

Let us consider a continuum embedded in an

Euclidean vector space

(

ℝ

3

,

∥

⋅

∥

)

.  This continuum, alternatively called

body

, can be either deformable or undeformable. Let

ℜ

0

be the region initially accupied by the body and

ℜ

any subsequently space occupied by that continuum. Each region of the

Euclidean space

filled by the body shall be called

configuration

.

In a system of

coordinates

arbitrarily chosen, every particle of the body is identified in

ℜ

0

as a point

P

0

of coordinates

(

a

,

b

,

c

)

, and let this point be carried over to a point

P

in

ℜ

, being its coordinates

(

x

1

,

x

2

,

x

3

)

. Thus, the

transformation

of all the points

(

a

,

b

,

c

)

in

ℜ

0

into the points

(

x

1

,

x

2

,

x

3

)

in

ℜ

, indeed corresponds to a

mapping

which it can be expressed by

x

1

=

x

1

⁢

(

a

,

b

,

c

)

,

x

2

=

x

2

⁢

(

a

,

b

,

c

)

,

x

3

=

x

3

⁢

(

a

,

b

,

c

)

,

(1)

provided

ℜ

0

and

ℜ

are subsets of

ℝ

3

and the mapping represented by

x

i

:

ℜ

0

→

ℜ

,

i

=

1

,

2

,

3

,

occupying the body different regions or configurations in the space.  The change of configuration that the body experiences we shall call it

deformation.

It is essential to understand that it is the body that deforms, not the space the body occupies.

Let us assume that Eq.(1) is a smooth continuous transformation (or deformation) and it can be inverted to the equations

a

=

a

⁢

(

x

1

,

x

2

,

x

3

)

,

b

=

b

⁢

(

x

1

,

x

2

,

x

3

)

,

c

=

c

⁢

(

x

1

,

x

2

,

x

3

)

,

(2)

then the equations (1) and (2) define the continuum or body on study.
So far, we have introduced some basic concepts in to give a physical description of a continuum adopting different configurations as it experiences a deformation.  We shall now consider a more general and formal description.

Material and spatial coordinates

Let

X

α

be the coordinates defining the points

P

0

of a continuum initially located in the region

ℜ

0

and let

G

α

⁢

β

be the respective

metric tensor

. The

X

α

are called

material

or

Lagrangian

coordinates which allow a description of the configuration

ℜ

0

. Analogously, let

x

i

be the coordinates defining the position of points

P

in the configuration

ℜ

, once a deformation of body ocurrs, and let

g

i

⁢

j

be the correspondent metric tensor. The

x

i

are called

spatial

or

Eulerian

coordinates which describe the space occupied for the continuum in the configuration

ℜ

.  Thus, for

instance

, the squares of

line elements

in those regions

ℜ

0

and

ℜ

are given by

d

⁢

S

0

2

=

G

α

⁢

β

⁢

d

⁢

X

α

⁢

d

⁢

X

β

,

d

⁢

s

2

=

g

i

⁢

j

⁢

d

⁢

x

i

⁢

d

⁢

x

j

respectively. Consequently, whereas the Lagrangian coordinates describe an initial configuration of the body in

ℜ

0

,  the Eulerian coordinates describe the region

ℜ

of the space occupied by the continuum once the deformation takes place. That very general scheme, in which the choice of material system is

independent

of the choice of spatial coordinates, was introduced by Murnagham

[

1

]

.

Indeed those so-called descriptions are current erroneous German terminology, which refers as Lagrangian to the material coordinates that were introduced by Euler

[

2

]

and spatial coordinates as Eulerian that were introduced by D’Alembert

[

3

]

.

Possibly a more useful scheme is that introduced by Brillouin

[

4

]

, as it allows a suitable definition of

motion of a continuum.

In that case a parameter is introduced (usually the time

t

) and the approach requires that metric tensors coincide, i.e.

g

i

⁢

j

⁢

(

x

k

)

=

G

i

⁢

j

⁢

(

X

α

⁢

(

x

k

,

t

)

)

,

considering the motion as a transformation of coordinates. (See the motion of continuum for more details.)

It is relevant to mention that certain quantities which are relative

invariants

in Brillouin’s scheme, are absolute in Murnagham’s. In particular, if

ρ

0

is the

density

of the media in

ℜ

0

and

ρ

in

ℜ

, considering the

Jacobian

of the transformation, we have

ρ

0

=

J

⁢

ρ

.

With respect to the transformations of either spatial (Eulerian) or material (Lagrangian) alone, all of those quantities are absolute scalars, but if

ρ

is considered as the transformed value of

ρ

0

, then both values must be the densities in the

x

i

and

X

α

systems, respectively, therefore showing a fundamental

difference

between the mentioned schemes.

References

1

F. D. Murnagham,

Finite deformations of an elastic solid

, Amer. J. Math.

59,

235-260, 1937.

2

L. Euler, Lettre de M. Euler à M. de Lagrange,

Recherches sur la propagation des ÃÂ©branlements dans une milieu ÃÂ©lastique,

Misc. Taur.

𝟐

2

(1760-1761), 1-10 =

Opera

(2)

10,

255-263 =

Oeuvres de Lagrange

14,

178-188, 1762.

3

J. L. D’Alembert,

Essai d’une Nouvelle ThÃÂ©orie de la

Resistance

des Fluides,

Paris, 1752.

4

L. Brillouin,

Les lois de l’ÃÂ©lasticitÃÂ© en coordonnÃÂ©es quelconques,

Proc. Int. Congr. Math. Toronto (1924)

2,

73-97

(a preliminary version of [1925,

1

]), 1928.

Title

vector fields: Lagrangian and Eulerian description

Canonical name

VectorFieldsLagrangianAndEulerianDescription

Date of creation

2016-05-24 22:37:07

Last modified on

2016-05-24 22:37:07

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

17

Author

perucho (2192)

Entry type

Definition

Classification

msc 53A45

Defines

continuum

Defines

body

Defines

material coordinates

Defines

spatial coordinates

Defines

configuration

Defines

deformation

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