vector fields: Lagrangian and Eulerian description

When we deal with vector fields defined over a continuumMathworldPlanetmath media, a suitable choice of coordinate systemsMathworldPlanetmath 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(3,).  This continuum, alternatively called body, can be either deformable or undeformable. Let 0 be the region initially accupied by the body and any subsequently space occupied by that continuum. Each region of the Euclidean space filled by the body shall be called configurationMathworldPlanetmathPlanetmath. In a system of coordinatesMathworldPlanetmathPlanetmath arbitrarily chosen, every particle of the body is identified in 0 as a point P0 of coordinates (a,b,c), and let this point be carried over to a point P in , being its coordinates (x1,x2,x3). Thus, the transformationMathworldPlanetmath of all the points (a,b,c) in 0 into the points (x1,x2,x3) in , indeed corresponds to a mapping which it can be expressed by

x1=x1(a,b,c),x2=x2(a,b,c),x3=x3(a,b,c), (1)

provided 0 and are subsets of 3 and the mapping represented by

xi:0,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(x1,x2,x3),b=b(x1,x2,x3),c=c(x1,x2,x3), (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 Xα be the coordinates defining the points P0 of a continuum initially located in the region 0 and let Gαβ be the respective metric tensor. The Xα are called material or Lagrangian coordinates which allow a description of the configuration 0. Analogously, let xi be the coordinates defining the position of points P in the configuration , once a deformation of body ocurrs, and let gij be the correspondent metric tensor. The xi are called spatial or Eulerian coordinates which describe the space occupied for the continuum in the configuration .  Thus, for instance, the squares of line elements in those regions 0 and are given by


respectively. Consequently, whereas the Lagrangian coordinates describe an initial configuration of the body in 0,  the Eulerian coordinates describe the region 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.


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 invariantsMathworldPlanetmath in Brillouin’s scheme, are absolute in Murnagham’s. In particular, if ρ0 is the density of the media in 0 and ρ in , considering the JacobianMathworldPlanetmath of the transformation, we have


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 ρ0, then both values must be the densities in the xi and Xα systems, respectively, therefore showing a fundamental difference between the mentioned schemes.


  • 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 ébranlements dans une milieu élastique, Misc. Taur. 𝟐2 (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 Théorie de la Resistance des Fluides, Paris, 1752.
  • 4 L. Brillouin, Les lois de l’élasticité en coordonnées 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