Mechanics of deformable solids Definition

Lagrangian and Eulerian description

Builds on Continuum mechanics Deformation

Definition

Let $B \subseteq \R^3$ be the reference configuration of a continuum body, and let $\chi \colon B \times I \to \R^3$ be a motion. So each $\chi(\cdot, t)$ is a deformation of $B$ onto $\Omega_t = \chi(B, t)$ . A physical quantity of the body can be given in two ways.

The Lagrangian or material description gives it as a function $f_L(X, t)$ of the material point $X \in B$ and the time $t$ .
The Eulerian or spatial description gives it as a function $f_E(x, t)$ of the place $x \in \Omega_t$ and the time $t$ .

The two describe the same quantity when $f_L(X, t) = f_E(\chi(X, t), t)$ for all $X \in B$ and $t \in I$ .

Remarks

The velocity has the Lagrangian form $V(X, t) = \partial_t \chi(X, t)$ and the Eulerian form $v(x, t) = V(\chi^{-1}(x, t), t)$ , where $\chi^{-1}(\cdot, t)$ is the inverse of $\chi(\cdot, t)$ . For the motion $\chi(X, t) = e^t X$ in $\R^3$ , one has $V(X, t) = e^t X$ and $v(x, t) = x$ : the Eulerian velocity field is steady although every particle other than the one at the origin accelerates. The time derivative at fixed $X$ is the material derivative; in Eulerian terms it is $\partial_t f_E + (v \cdot \nabla) f_E$ by the chain rule.

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Continuum mechanics concept
Deformation concept
Lagrangian and Eulerian description definition

Tag	`2EDJ3YJCP0`
URI	`m2://2EDJ3YJCP0`
Kind	definition
Writing	W0 stub
MSC	74A05, 76A02
Defines	Lagrangian and Eulerian description
Uses	Continuum mechanics, Deformation
Changed	3 November 2026

Lagrangian and Eulerian description

Definition

Remarks

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