Mechanics of Continua and Structures
Cartesian vector basis is of the form {$\hat{\boldsymbol{e}}_i$}$_{i=1,2,3}$. These vectors are fixed, having unit magnitude and always pointing in the same direction. They also form a mutually orthogonal trial \(\begin{align} \hat{\boldsymbol{e}}_i \cdot \hat{\boldsymbol{e}}_j = \delta_{ij} \end{align}\)
In Cartesian coordinate system, the position vector is represented as \(\begin{align} \boldsymbol{r} = x_1\hat{\boldsymbol{e}}_1 + x_2\hat{\boldsymbol{e}}_2 + x_3\hat{\boldsymbol{e}}_3 \end{align}\) where $x_1,x_2,x_3$ are the Cartesian co-ordinates.
When the particle is in motion, the Cartesian co-ordinates are a function of time and hence the position vector is given as a time dependent function $\boldsymbol{r}(t)$. It is represented as \(\begin{align} \boldsymbol{r}(t) = x_1(t)\hat{\boldsymbol{e}}_1 + x_2(t)\hat{\boldsymbol{e}}_2 + x_3(t)\hat{\boldsymbol{e}}_3 \end{align}\)
Using the Einstein Summation Convention, above equations can be written as: \(\begin{align} \boldsymbol{r} = x_i\hat{\boldsymbol{e}}_i \end{align}\)
The velocity vector of a particle is defined as, \(\begin{align} \boldsymbol{v} = \frac{d\boldsymbol{r}}{dt} \end{align}\)
Therefore, in terms of Cartesian components, it is: \(\begin{align} \boldsymbol{v}(t) &= \frac{d}{dt}(x_i\hat{\boldsymbol{e}}_i) \\ &= \frac{dx_i}{dt}\hat{\boldsymbol{e}}_i + x_i\frac{d\hat{\boldsymbol{e}}_i}{dt} \\ &= \frac{dx_i}{dt}\hat{\boldsymbol{e}}_i \end{align}\) Note: $\boldsymbol{e}_i$ being fixed as per the definition of coordinate system
In order to denote differentiation w.r.t. time, we will use an overhead dot \(\begin{align} \frac{dx}{dt} = \dot{x} \end{align}\)
Using (9), we get \(\begin{align} \boldsymbol{v} = \dot{x}_i\hat{\boldsymbol{e}}_i \end{align}\)
The acceleration vector of a particle is defined as, \(\begin{align} \boldsymbol{a} = \frac{d\boldsymbol{v}}{dt} = \frac{d^2\boldsymbol{r}}{dt^2} \end{align}\) Component wise, we have \(\begin{align} \boldsymbol{a} = \ddot{x}_i\hat{\boldsymbol{e}}_i \end{align}\)