Page 234 - Intro to Tensor Calculus

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Lagrangian and Eulerian Systems

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Let x denote the initial position of a material particle in a continuum. Assume that at a later time the
i
particle has moved to another point whose coordinates are x . Both sets of coordinates are referred to the
same coordinate system. When the final position can be expressed as a function of the initial position and
i
i
3
2
1
time we can write x = x (x , x , x ,t). Whenever the changes of any physical quantity is represented in terms
of its initial position and time, the representation is referred to as a Lagrangian or material representation of
x
the quantity. This can be thought of as a transformation of the coordinates. When the Jacobian J( ) of this
x
i
1
i
3
2
transformation is different from zero, the above set of equations have a unique inverse x = x (x ,x ,x ,t),
where the position of the particle is now expressed in terms of its instantaneous position and time. Such a
representation is referred to as an Eulerian or spatial description of the motion.
Let (x 1 , x 2 , x 3 ) denote the initial position of a particle whose motion is described by x i = x i (x 1 , x 2 , x 3 ,t),
then u i = x i − x i denotes the displacement vector which can by represented in a Lagrangian or Eulerian
form. For example, if
t
x 1 =2(x 1 − x 2 )(e − 1) + (x 2 − x 1 )(e −t − 1) + x 1
t
x 2 =(x 1 − x 2 )(e − 1) + (x 2 − x 1 )(e −t − 1) + x 2
x 3 = x 3
then the displacement vector can be represented in the Lagrangian form
t
u 1 =2(x 1 − x 2 )(e − 1) + (x 2 − x 1 )(e −t − 1)
t
u 2 =(x 1 − x 2 )(e − 1) + (x 2 − x 1 )(e −t − 1)
u 3 =0

or the Eulerian form
−t
−t
u 1 = x 1 − (2x 2 − x 1 )(1 − e ) − (x 1 − x 2 )(e −2t − e ) − x 1 e −t
−t
−t
u 2 = x 2 − (2x 2 − x 1 )(1 − e ) − (x 2 − x 1 )(e −2t − e ) − x 2 e −t
u 3 =0.
Note that in the Lagrangian system the displacements are expressed in terms of the initial position and
time, while in the Eulerian system the independent variables are the position coordinates and time. Euler
equations describe, as a function of time, how such things as density, pressure, and fluid velocity change at
a fixed point in the medium. In contrast, the Lagrangian viewpoint follows the time history of a moving
individual fluid particle as it moves through the medium.