3.3: Coordinate systems
In this textbook we deal with two main types of coordinate systems for determining position and analyse motion in three dimensional (3D) space: Cartesian coordinates and cylindrical coordinates. We will not discuss spherical coordinates.
3.3.1 Cartesian coordinates
In Cartesian coordinates, space is spanned by \(x, y\) and \(z\) coordinate axes, which determine the position of objects with respect to the origin \(O\), as is shown in Figure 3.5. Along each of the axes a unit vector with length 1 is defined, respectively \(\hat{\boldsymbol{\imath}}, \hat{\boldsymbol{\jmath}}\) and \(\hat{\boldsymbol{k}}\), which point in the positive axis directions.
Note. When drawing a coordinate system in 3 dimensions, the convention is to always use right-handed axis system, since otherwise the vector rules for taking cross-products will fail. For Cartesian coordinates this means that you should make sure, e.g. using the right-hand rule, that for a fixed choice of the order of coordinates \(x, y, z\), we have the relation \(\hat{\boldsymbol{\imath}} \times \hat{\boldsymbol{\jmath}}=\hat{\boldsymbol{k}}\) between the respective unit vectors.
To describe the position vector \(\overrightarrow{\boldsymbol{r}}_{A}\) of a point mass \(A\) with respect to the origin in Cartesian coordinates you can use one of the following 3 notations:
\[\begin{align} \overrightarrow{\boldsymbol{r}}_{A} & =x_{A} \hat{\boldsymbol{\imath}}+y_{A} \hat{\boldsymbol{\jmath}}+z_{A} \hat{\boldsymbol{k}} \tag{3.13} \label{3.13}\\[4pt] & =\left[\begin{array}{l} x_{A} \\[4pt] y_{A} \\[4pt] z_{A} \end{array}\right]=\left[\begin{array}{lll} x_{A} & y_{A} & z_{A} \end{array}\right]^{\mathrm{T}} \tag{3.14} \label{3.14}\end{align}\]
Where the superscript \(\mathrm{T}\) in (3.14) stands for the transpose of the matrix, converting the row vector to a column vector. We use square brackets and no commas to indicate vectors, and round brackets with commas for the coordinates \(\left(x_{A}, y_{A}, z_{A}\right)\).
3.3.2 Cylindrical coordinates
Cylindrical coordinates \((\rho, \phi, z)\) are often useful, for example to analyse the dynamics of a point mass \(A\) that rotates around an axis (the \(z\)-axis). An example of such a coordinate system in 2D is shown in Figure 3.6 and in 3D Figure 3.7. Ideally the \(z\)-axis of the coordinate system is chosen parallel to the rotation axis, and the origin of the coordinate system is chosen to be located at a suitable position on the axis. Then the position of the point mass \(A\) can be described by 3 coordinates: the radial coordinate \(\rho_{A}\) that determines the shortest distance to the axis, the azimuthal angle \(\phi_{A}\) with respect to a reference line (polar axis) and the axial coordinate \(z_{A}\). The unit vectors \(\hat{\boldsymbol{\rho}}, \hat{\boldsymbol{\phi}}\) and \(\hat{\boldsymbol{k}}\), which are shown in the figure, point in the direction the tip of the position vector would move if you would increase 1 of the coordinate values by a small amount. Mathematically the unit vector at the position \(\hat{\boldsymbol{r}}_{A}\) is \(\hat{\boldsymbol{\phi}}_{A}=\frac{\partial \overrightarrow{\boldsymbol{r}}_{A}}{\partial \phi_{A}} /\left|\frac{\partial \overrightarrow{\boldsymbol{r}}_{A}}{\partial \phi_{A}}\right|\), and similarly for \(\hat{\boldsymbol{\rho}}_{A}\) and \(\hat{\boldsymbol{k}}_{A}\). In contrast to Cartesian vectors, the direction of unit vectors in cylindrical coordinates can depend on the position vector \(\overrightarrow{\boldsymbol{r}}_{A}\) of the object. So if there are multiple objects it can be useful to label the relevant unit vector with a subscript \(A\).
Note. Like in Cartesian coordinates one should adhere to the convention that a cylindrical coordinate system \(\rho, \phi, z\) is a right-handed axis system with \(\hat{\boldsymbol{\rho}} \times \hat{\boldsymbol{\phi}}=\hat{\boldsymbol{k}}\), by choosing \(\phi\) to increase in the anticlockwise direction, as observed when looking from the positive \(z\)-axis towards the origin. This is consistent with the convention to take the anticlockwise direction as the positive direction for measuring angles.
A force vector \(\overrightarrow{\boldsymbol{F}}_{A}\) that acts on a particle \(A\) at cylindrical coordinates \(\left(\rho_{A}, \phi_{A}, z_{A}\right)\) can be expressed in the following 3 ways:
\[\begin{align} \overrightarrow{\boldsymbol{F}}_{A} & =F_{A \rho} \hat{\boldsymbol{\rho}}+F_{A \phi} \hat{\boldsymbol{\phi}}+F_{A z} \hat{\boldsymbol{k}} \tag{3.15} \label{3.15}\\[4pt] & =\left[\begin{array}{l} F_{A \rho} \\[4pt] F_{A \phi} \\[4pt] F_{A z} \end{array}\right]=\left[\begin{array}{lll} F_{A \rho} & F_{A \phi} & F_{A z} \end{array}\right]^{T} \tag{3.16} \label{3.16}\end{align}\]
One needs to be careful when working with vectors in cylindrical coordinates, because the vector component values depend on the coordinates \(\left(\rho_{A}, \phi_{A}, z_{A}\right)\). Standard vector operations, like addition, between multiple vectors in cylindrical coordinates can therefore only be carried out if the vectors refer to the same coordinates, such that their unit vectors are identical. In other cases the vectors should first be converted to Cartesian coordinates, which can be done using the following relation between the unit vectors:
\[\begin{align} \hat{\boldsymbol{\imath}} & =\cos \phi_{A} \hat{\boldsymbol{\rho}}-\sin \phi_{A} \hat{\boldsymbol{\phi}} \tag{3.17} \label{3.17}\\[4pt] \hat{\boldsymbol{\jmath}} & =\sin \phi_{A} \hat{\boldsymbol{\rho}}+\cos \phi_{A} \hat{\boldsymbol{\phi}} \tag{3.18} \label{3.18}\\[4pt] \hat{\boldsymbol{k}} & =\hat{\boldsymbol{k}} \tag{3.19} \label{3.19}\end{align}\]
With these relations the components of a Cartesian vector like \(\overrightarrow{\boldsymbol{F}}_{A}\) can be related to its cylindrical vector components \(F_{A \rho}, F_{A \phi}\) and \(F_{A z}\) by taking its dot product with the unit vectors:
\[\overrightarrow{\boldsymbol{F}}_{A}=\left[\begin{array}{l} F_{A x} \tag{3.20} \label{3.20}\\[4pt] F_{A y} \\[4pt] F_{A z} \end{array}\right]=\left[\begin{array}{l} \overrightarrow{\boldsymbol{F}}_{A} \cdot \hat{\boldsymbol{\imath}} \\[4pt] \overrightarrow{\boldsymbol{F}}_{A} \cdot \hat{\boldsymbol{\jmath}} \\[4pt] \overrightarrow{\boldsymbol{F}}_{A} \cdot \hat{\boldsymbol{k}} \end{array}\right]=\left[\begin{array}{c} F_{A \rho} \cos \phi_{A}-F_{A \phi} \sin \phi_{A} \\[4pt] F_{A \rho} \sin \phi_{A}+F_{A \phi} \cos \phi_{A} \\[4pt] F_{A z} \end{array}\right]\]
where we replaced the unit vectors \(\hat{\boldsymbol{\imath}}, \hat{\boldsymbol{\jmath}}\) and \(\hat{\boldsymbol{k}}\) by Equation 3.17-(3.19).
3.3.3 Position vector in cylindrical coordinates
It is important to note that whereas in Cartesian coordinates the components of the position vector \(\overrightarrow{\boldsymbol{r}}_{A}=x_{A} \hat{\boldsymbol{\imath}}+y_{A} \hat{\boldsymbol{\jmath}}+z_{A} \hat{\boldsymbol{k}}\) are identical to the coordinates \(x_{A}\), \(y_{A}, z_{A}\), the position vector in cylindrical coordinates is given by \(\overrightarrow{\boldsymbol{r}}_{A}=\rho_{A} \hat{\boldsymbol{\rho}}+z_{A} \hat{\boldsymbol{k}}\) and does not provide information about the angle \(\phi_{A}\). Motion in cylindrical coordinates can therefore better by expressed by giving the 3 coordinate functions \(\left(\rho_{A}(t), \phi_{A}(t), z_{A}(t)\right)\) than using the position vector.
\[\begin{align} \overrightarrow{\boldsymbol{r}}_{A} & =\rho_{A} \hat{\boldsymbol{\rho}}+z_{A} \hat{\boldsymbol{k}} \tag{3.21} \label{3.21}\\[4pt] & =\left[\begin{array}{c} \rho_{A} \\[4pt] 0 \\[4pt] z_{A} \end{array}\right] \neq\left[\begin{array}{l} \rho_{A} \\[4pt] \phi_{A} \\[4pt] z_{A} \end{array}\right] \tag{3.22} \label{3.22}\end{align}\]
It is incorrect to have \(\phi_{A}\) in the vector notation as shown on the right side of Equation 3.22, as is also evident from the units which should be equal for every vector component.