Displacement Vector

By definition, the displacement vector $u_i$ of a particle is the difference between its position vectors at time $t$ and at time $t = t_0$ (or 0).

$$u_i = x_i - X_i.$$ (3.3.1)

In the Lagrangian description, the displacement $u_i$ is specified as a function of $X_i$ and $t$. For example, consider the motion

$$x_1 =\ X_1t^2 + 2X_2t + X_1,$$

$$x_2 =\ 2X_1 t^2 + X_2t + X_2,\hspace*{1in}$$ (3.3.2)

$$x_3 =\ \frac{1}{2} X_3t + X_3.$$

The corresponding components of the displacement are given by

$$u_1 =\ X_1t^2 + 2X_2t,$$

$$u_2 =\ 2X_1t^2 + X_2t,$$

$$u_3 =\ \frac{1}{2}X_3t.$$

In the Eulerian description, $u_i$ will be expressed as a function of $x_i$ and $t$. Solving (3.3.2) for $X_i$ in terms of $x_i$ and $t$ and substituting that in (3.3.1) we obtain

$$u_1 =\ x_1-X_1 = x_1 - \frac{x_1(1 + t) -2x_2t}{-3t^3 + t^2 + t + 1},$$

$$u_2 =\ x_2 - X_2 = x_2 - \frac{x_2(1 + t^2) - 2x_1t^2}{-3t^3 + t^2 + t + 1},$$

$$u_3 =\ x_3 - X_3 = x_3 - \frac{2x_3}{t+2}.$$