4.2: Newton's laws of motion
In his Principia Newton introduced three laws of motion that can mathematically be expressed in vector format as follows:
\[\begin{align} & \text { 1. If } \sum_{j} \overrightarrow{\boldsymbol{F}}_{i j}=\overrightarrow{\mathbf{0}} \text { then } \overrightarrow{\boldsymbol{a}}_{i}=\overrightarrow{\mathbf{0}} \tag{4.1} \label{4.1}\\[4pt] & \text { 2. } \sum_{j} \overrightarrow{\boldsymbol{F}}_{i j}=m_{i} \overrightarrow{\boldsymbol{a}}_{i} \tag{4.2} \label{4.2}\\[4pt] & \text { 3. } \overrightarrow{\boldsymbol{F}}_{i j}=-\overrightarrow{\boldsymbol{F}}_{j i} \tag{4.3} \label{4.3}\end{align}\]
These equations and the variables will be explained and defined in more detail in the next two chapters. It can be seen that Newton’s first law is a direct consequence of Newton’s second law when setting \(\sum \overrightarrow{\boldsymbol{F}}_{i j}=\overrightarrow{\mathbf{0}}\). Newton’s third law (action=-reaction), describes a property of all known fundamental forces of nature, which states that for every force \(\overrightarrow{\boldsymbol{F}}_{i j}\) that acts on point mass \(i\) there is another collinear force \(\overrightarrow{\boldsymbol{F}}_{j i}\) that acts on another point mass \(j\) with equal magnitude and opposite direction. In other words if point mass \(j\) generates a force \(\overrightarrow{\boldsymbol{F}}_{i j}\) on object \(i\), then point mass \(i\) generates a force \(\overrightarrow{\boldsymbol{F}}_{j i}=-\overrightarrow{\boldsymbol{F}}_{i j}\) on point
mass \(j\). The points of action of both forces coincide with the positions of the corresponding point masses. The first and third laws should be familiar if you followed a course in statics. Newton’s second law \(\sum \overrightarrow{\boldsymbol{F}}=m \overrightarrow{\boldsymbol{a}}\) is the essential equation in dynamics.
Note. Newton’s laws are strictly only valid for objects that are point masses, which are so small that they can be described by a single point (position vector) in space. By applying Newton’s laws to each point mass in an object or system that consists of many point masses, new laws can be derived for these larger systems. Much of this textbook is dedicated to deriving and discussing these new laws and using them to analyse more complex dynamical systems, like rigid bodies.