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# 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 order to accurately describe the action that such fields 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 $(\mathbb{R}^{3},\lVert\cdot\rVert)$. This continuum, alternatively called body, can be either deformable or undeformable. Let $\Re_{0}$ be the region initially accupied by the body and $\Re$ 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 $\Re_{0}$ as a point $P_{0}$ of coordinates $(a,b,c)$, and let this point be carried over to a point $P$ in $\Re$, being its coordinates $(x_{1},x_{2},x_{3})$. Thus, the transformation of all the points $(a,b,c)$ in $\Re_{0}$ into the points $(x_{1},x_{2},x_{3})$ in $\Re$, indeed corresponds to a mapping which it can be expressed by

$\displaystyle x_{1}=x_{1}(a,b,c),\quad x_{2}=x_{2}(a,b,c),\quad x_{3}=x_{3}(a,% b,c),$ | (1) |

provided $\Re_{0}$ and $\Re$ are subsets of $\mathbb{R}^{3}$ and the mapping represented by

$x_{i}:\Re_{0}\to\Re,\quad i=1,\,2,\,3,$ |

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

$\displaystyle a=a(x_{1},x_{2},x_{3}),\quad b=b(x_{1},x_{2},x_{3}),\quad c=c(x_% {1},x_{2},x_{3}),$ | (2) |

then the equations (1) and (2) define the continuum or body on study. So far, we have introduced some basic concepts in order 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 $X^{\alpha}$ be the coordinates defining the points $P_{0}$ of a continuum initially located in the region $\Re_{0}$ and let $G_{{\alpha\beta}}$ be the respective metric tensor. The $X^{\alpha}$ are called material or Lagrangian coordinates which allow a description of the configuration $\Re_{0}$. Analogously, let $x^{i}$ be the coordinates defining the position of points $P$ in the configuration $\Re$, once a deformation of body ocurrs, and let $g_{{ij}}$ 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 $\Re$. Thus, for instance, the squares of line elements in those regions $\Re_{0}$ and $\Re$ are given by

$\displaystyle dS_{0}^{2}=G_{{\alpha\beta}}dX^{\alpha}dX^{\beta},\quad ds^{2}=g% _{{ij}}dx^{i}dx^{j}$ |

respectively. Consequently, whereas the Lagrangian coordinates describe an initial configuration of the body in $\Re_{0}$, the Eulerian coordinates describe the region $\Re$ 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.

$\displaystyle g_{{ij}}(x^{k})=G_{{ij}}(X^{\alpha}(x^{k},t)),$ |

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 $\rho_{0}$ is the density of the media in $\Re_{0}$ and $\rho$ in $\Re$, considering the Jacobian of the transformation, we have

$\displaystyle\rho_{0}=J\rho.$ |

With respect to the transformations of either spatial (Eulerian) or material (Lagrangian) alone, all of those quantities are absolute scalars, but if $\rho$ is considered as the transformed value of $\rho_{0}$, then both values must be the densities in the $x^{i}$ and $X^{\alpha}$ 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©branlements dans une milieu Ã©lastique, Misc. Taur. ${\bf 2}^{2}$ (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©orie de la Resistance des Fluides, Paris, 1752.
- 4
L. Brillouin, Les lois de l’Ã©lasticitÃ© en coordonnÃ©es quelconques, Proc. Int. Congr. Math. Toronto (1924) 2, 73-97

(a preliminary version of [1925, 1]), 1928.

## Mathematics Subject Classification

53A45*no label found*

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