2.1: Notation

In this section we discuss the used notation, labelling and typesetting in this textbook. Sec. 2.1 and Sec. 2.2 are an adaptation of the text of Vallery and Schwab \([9]\).

2.1.1 Typesetting of scalars, vectors, and matrices

There are several ways to typeset scalars, vectors, unit vectors, and matrices as depicted in Table 2.1. In particular for vectors and unit vectors there are multiple methods applied. Although in many works vectors are only written with boldface italics, it is hard to clearly write a boldface letter in handwriting. Therefore we here use a combination of boldface and vector arrow above the letter for clarity and similarity to the handwritten vector. Similarly, for unit vectors (vectors with a magnitude of 1 ), we use a boldface letter with a hat ^ above it. All variables are italics, also if they appear in subscripts or superscripts. Matrices are non-italic and also names and texts in sub or superscripts are non-italic except if they are single letters[8].

Table 2.1: Notation of scalars, vectors, unit vectors and matrices

This book Other notations

Scalars	\(F\)
Vectors	\(\overrightarrow{\boldsymbol{F}}\)	\(\vec{F}, \underline{F}, \boldsymbol{F}, \mathbf{F}\)
Unit vectors	\(\hat{\boldsymbol{\imath}}, \hat{\boldsymbol{\rho}}\)	\(\mathbf{u}_{x}, \mathbf{e}_{1}, \mathbf{i}, \hat{\boldsymbol{x}}\)
Matrices	\(\mathbf{R}\)	\(\mathcal{R}\)

2.1.2 Quantities, labels and subscripts

In this textbook we generally follow the guidelines set out in the Red Book [2]. To label objects or points we either use letters like \(A, B, \ldots\), numbers \(i=1,2, \ldots\) or short words like rope or ball. For adding extra information and distinguishing quantities, we can add subscripts like in \(m_{\text {rope }}\) and \(m_{\text {ball }}\). Multiple labels can be separated in the subscript by commas. Italic integer variables, like \(i\) in \(F_{i}\) (with \(i=1,2, \ldots\) ) can indicate that multiple quantities \(F_{1}, F_{2}, \ldots\) exist, each with a different value of \(i\). Although labels are not always essential, make sure to always use sufficient labels to uniquely identify each quantity. So, if there is only one mass in your problem, it is fine to just \(m\) to identify it. However, if there are two point-masses, subscripts \(m_{A}\) and \(m_{B}\) are needed for unique identification. With an axis coordinate as a subscript, like \(x\) in \(F_{x}\), we indicate a projection of a vector \(\overrightarrow{\boldsymbol{F}}\) along the \(x\)-axis.

A special notation for relative position, velocity and acceleration vectors is used (see also Sec. 3.1.2). A position vector that points to a point \(A\) from a point \(P\) will be denoted as \(\overrightarrow{\boldsymbol{r}}_{A / P}=\overrightarrow{\boldsymbol{r}}_{A}-\overrightarrow{\boldsymbol{r}}_{/ P}\), which reads: "position of \(A\) with respect to \(P\) ". Analogous notation will be used for other relative quantities, such as angles, velocities and accelerations (see Sec. 3.1).

For components of 3-dimensional vectors, there are three possible ways of notation of which we will usually use the last one because it is the shortest to write:

\[\overrightarrow{\boldsymbol{r}}=\left(\begin{array}{l} r_{x} \tag{2.1} \label{2.1}\\[4pt] r_{y} \\[4pt] r_{z} \end{array}\right)=\left(\begin{array}{lll} r_{x} & r_{y} & r_{z} \end{array}\right)^{\mathrm{T}}=r_{x} \hat{\boldsymbol{\imath}}+r_{y} \hat{\boldsymbol{\jmath}}+r_{z} \hat{\boldsymbol{k}}\]

The superscript \({ }^{\mathrm{T}}\) indicates the transpose of the vector (or matrix), and is sometimes used to convert the column vector to a row vector to save space. For components of \(m\)-dimensional vectors, one can also use the index notation, with unit vectors written like \(\hat{\boldsymbol{e}}_{1}=\hat{\boldsymbol{\imath}}\), which has the advantage that it can be written even more compactly as a sum:

\[\overrightarrow{\boldsymbol{r}}=\left(\begin{array}{l} r_{1} \tag{2.2} \label{2.2}\\[4pt] r_{2} \\[4pt] \ldots \\[4pt] r_{m} \end{array}\right)=\left(\begin{array}{llll} r_{1} & r_{2} & \ldots & r_{m} \end{array}\right)^{\mathrm{T}}=\sum_{i=1}^{m} r_{i} \hat{\boldsymbol{e}}_{i}\]

Elements of \(m \times n\)-dimensional matrices receive two indices, for row and column:

\[\mathbf{A}=\left(\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1 n} \tag{2.3} \label{2.3}\\[4pt] a_{21} & a_{22} & \ldots & a_{2 n} \\[4pt] \ldots & \ldots & \ldots & \ldots \\[4pt] a_{m 1} & a_{m 2} & \ldots & a_{m n} \end{array}\right)\]

Note that in dynamics, we use Newton’s notation (also called dot notation or fluxions) for time derivatives, indicating them by a dot above the variable:

\[\begin{align} \dot{x} & =\frac{\mathrm{d} x}{\mathrm{~d} t} \tag{2.4} \label{2.4}\\[4pt] \ddot{x} & =\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}\end{align}\]

In mathematics the prime symbol ’ is often used for derivatives like \(f^{\prime}=\frac{\mathrm{d} f}{\mathrm{~d} x}\). In dynamics it is advisable not to use the prime symbol to avoid confusion, and instead clearly indicate with respect to what variable the derivative is taken e.g. \(\frac{\mathrm{d} f}{\mathrm{~d} x}\) or \(\frac{\mathrm{d} f}{\mathrm{~d} s}\). Also clearly indicate if it is a total derivative \(\frac{\mathrm{d} f}{\mathrm{~d} x}\) or partial derivative \(\frac{\partial f}{\partial x}\). In this textbook the prime symbol’ will not be used for derivatives but is sometimes used as a label. Finally, when making multiplications of scalar quantities, both \(\cdot\) and \(\times\) are used, such that \(a \cdot b=a \times b\). For vectors these symbols indicate the dot and cross product which clearly are different: \(\vec{a} \cdot \vec{b} \neq \vec{a} \times \vec{b}\)