Newtonian Formalism

Introduction

The Newtonian formalism of classical mechanics is a model of the causes behind movement. It is ubiquitously known and you have been taught it at school in one form or another using “Newton’s laws of motion”. Its popularity is due to several reasons:

• Historically, Newtonian formalism, described in Isaac Newton’s Principia Mathematica, was the first such model to be rigorously defined and yielded predictions which agreed experimentally with observations.
• It is built on our direct observations of how solid objects move when we push or pull them and is therefore very intuitive.

Although the Newtonian formalism is very intuitive and fairly simple, this does not make it the only true formalism. There are alternative ways to phrase classical mechanics which do not make any use of the notions that the Newtonian formalism relies on. However, all of these are equivalent in the sense that they both yield the same predictions, agree equally well with experiments and can be mathematically derived from one another.

Newtonian Formalism

At the heart of the Newtonian formalism lie the notions of an inertial reference frames and forces.

Definition: Inertial Motion

An object is in inertial motion if it is not interacting with any other objects.

Inertial motion is an idealization because for an object not to interact with anything else would require that the Universe be empty except for that single object. Nevertheless, we can still use this model as a very good approximation when an object interacts with other objects very weakly.

Newton’s First Law of Motion

Definition: Inertial Reference Frame

A reference frame is inertial if a physical object under inertial motion retains a constant velocity.

We can think of an inertial reference frame as the point of view of a ghost which interacts with nothing but is perfectly capable of perceiving the entirety of the Universe. No reference frame can be truly inertial though because inertial motion is an idealization. However, we can often approximate a reference frame as inertial when the motion of objects closely approximating inertial motion deviates very slightly from motion with constant velocity.

The definition of an inertial reference frame is also known as Newton’s First Law of Motion or the Law of Inertia. It states that, when viewed in an inertial reference frame, the velocity of objects which do not interact with other objects does not change. This manifests in one of two ways:

• If such an object has some non-zero velocity $\mathbf{v}$, then the path traced by it is a straight line and this line is traced with speed $||\mathbf{v}||$.
• If such an object has a zero velocity, i.e. it is at rest, then it remains at rest.

To determine whether a given reference frame is inertial, we rely on empirical methods: If we have analyzed some object and have concluded that it interacts with no other objects but its velocity nevertheless changes, then we conclude that our reference frame is not inertial.

Newton’s Second Law of Motion

The interactions between objects are mediated by entities called forces.

Definition: Force

A force is a vector which models a physical interaction.

NOTATION

Forces are typically denoted using variations of the letter “F”:

$$\mathbf{F}\mathbf{F}\vec{F}$$

One can think of forces as invisible arrows which push or pull stuff around. However, not much thought should be given as to whether forces “truly” exist in the philosophical sense because forces are just a model used by the Newtonian formalism for calculating kinematical quantities. Other, equally correct formalisms, do not make use of forces.

Axiom: Newton's Second Law of Motion

In an inertial reference frame, if ${\mathbf{F}}_1,\dots ,{\mathbf{F}}_n$ are all the forces acting on a point mass $m$, then its instantaneous acceleration $\mathbf{a}$ is given by

$$\mathbf{a}=\frac{1}{m}\sum _{i=1}^n{\mathbf{F}}_i$$

This law tells us two very important things. First, forces obey the superposition principle: the effect of ${\mathbf{F}}_1,\dots ,{\mathbf{F}}_n$ is equivalent to the effect of a single net force ${\mathbf{F}}_{\text{net}}$ which is the sum of ${\mathbf{F}}_1,\dots ,{\mathbf{F}}_n$. Second, it makes apparent how inertia is the ability of an object to resist changes in its motion - the greater the mass $m$, the lesser the effect of the forces ${\mathbf{F}}_1,\dots ,{\mathbf{F}}_n$.

Important: The Second Law in Non-Inertial Frames

Consider a non-inertial reference frame ${\mathcal{R}}_{\text{NI}}$ with origin ${\mathcal{O}}_{\text{NI}}$ and an inertial reference frame ${\mathcal{R}}_{\text{I}}$ with origin ${\mathcal{O}}_{\text{I}}$.

Since ${\mathcal{R}}_{\text{NI}}$ is not inertial, its origin ${\mathcal{O}}_{\text{NI}}$ must be subject to some forces and by applying Newton’s second law, we can determine the acceleration ${}_{{\mathcal{R}}_{\text{I}}}{\mathbf{a}}_{{\mathcal{O}}_{\text{NI}}}$ of ${\mathcal{O}}_{\text{NI}}$ with respect to the inertial reference frame ${\mathcal{R}}_{\text{I}}$. Similarly, if an object $O$ with mass $m$ is subject to forces ${}_{\text{on }O}{\mathbf{F}}_1,\dots ,{}_{\text{on }O}{\mathbf{F}}_n$, we can determine its acceleration ${}_{{\mathcal{R}}_{\text{I}}}{\mathbf{a}}_O$ with respect to ${\mathcal{R}}_{\text{I}}$:

$${}_{{\mathcal{R}}_{\text{I}}}{\mathbf{a}}_O=\frac{1}{m}\sum _{i=1}^n{}_{\text{on }O}{\mathbf{F}}_i$$

Now we can use Galilean relativity to determine the acceleration ${}_{{\mathcal{R}}_{\text{NI}}}{\mathbf{a}}_O$ of $O$ with respect to the non-inertial reference frame ${\mathcal{R}}_{\text{NI}}$. Since ${\mathcal{O}}_{\text{NI}}$ has acceleration ${}_{{\mathcal{R}}_{\text{I}}}{\mathbf{a}}_{{\mathcal{O}}_{\text{NI}}}$ with respect to ${\mathcal{R}}_{\text{I}}$, we know that the acceleration of ${\mathcal{O}}_{\text{I}}$ with respect to ${\mathcal{R}}_{\text{NI}}$ is

$${}_{{\mathcal{R}}_{\text{NI}}}{\mathbf{a}}_{{\mathcal{O}}_{\text{I}}}=-{}_{{\mathcal{R}}_{\text{I}}}{\mathbf{a}}_{{\mathcal{O}}_{\text{NI}}}$$

Using Galilean relativity again, we get that the acceleration ${}_{{\mathcal{R}}_{\text{NI}}}{\mathbf{a}}_O$ of $O$ with respect to ${\mathcal{R}}_{\text{NI}}$ is

$$\begin{array}{rl}{}_{{\mathcal{R}}_{\text{NI}}}{\mathbf{a}}_O& ={}_{{\mathcal{R}}_{\text{NI}}}{\mathbf{a}}_{{\mathcal{O}}_{\text{I}}}+{}_{{\mathcal{R}}_{\text{I}}}{\mathbf{a}}_O\\ & ={}_{{\mathcal{R}}_{\text{NI}}}{\mathbf{a}}_{{\mathcal{O}}_{\text{I}}}+\frac{1}{m}\sum _{i=1}^n{}_{\text{on }O}{\mathbf{F}}_i\end{array}$$

Therefore, in an non-inertial reference frame objects appear to have additional acceleration to the one predicted by Newton’s second law. This acceleration is not due to the forces acting on the object itself, but rather due to the forces acting on the origin. However, we can still treat this acceleration as if it were caused by forces acting on the object by defining ${}_{\text{on }O}{\mathbf{F}}_{n+1}\stackrel{\text{def}}{=}m\cdot {}_{{\mathcal{R}}_{\text{NI}}}{\mathbf{a}}_{{\mathcal{O}}_{\text{I}}}$. Since $\frac{1}{m}{}_{\text{on }O}{\mathbf{F}}_{n+1}={}_{{\mathcal{R}}_{\text{NI}}}{\mathbf{a}}_{{\mathcal{O}}_{\text{I}}}$, we can rewrite the previous equation as

$$\begin{array}{rl}{}_{{\mathcal{R}}_{\text{NI}}}{\mathbf{a}}_O& =\frac{1}{m}{}_{\text{on }O}{\mathbf{F}}_{n+1}+\frac{1}{m}\sum _{i=1}^n{}_{\text{on }O}{\mathbf{F}}_i\\ & =\frac{1}{m}\left({}_{\text{on }O}{\mathbf{F}}_{n+1}+\sum _{i=1}^n{}_{\text{on }O}{\mathbf{F}}_i\right)\\ & =\sum _{i=1}^{n+1}{}_{\text{on }O}{\mathbf{F}}_i\end{array}$$

The vector ${}_{\text{on }O}{\mathbf{F}}_{n+1}$ is not really a force because it is not caused by the physical interactions of the object $O$, despite how we have abused the notation here. It is merely a vector we defined in order to be able to write the equation in a simpler form. As such, we often call ${}_{\text{on }O}{\mathbf{F}}_{n+1}$ a fictitious force or an inertial force and we usually write it as ${\mathbf{F}}_{\text{inertial}}$. Therefore, Newton’s second law in a non-inertial reference frame is

$${}_{{\mathcal{R}}_{\text{NI}}}{\mathbf{a}}_O=\frac{1}{m}\left({\mathbf{F}}_{\text{inertial}}+\sum _{i=1}^n{}_{\text{on }O}{\mathbf{F}}_i\right)$$

Newton’s Third Law of Motion

There is also one last law which tells us how interactions between objects happen.

Axiom: Newton's Third Law of Motion

Physical interactions between two particles $A$ and $B$ always manifest as a pair of forces: the force ${\mathbf{F}}_{A\text{ on }B}$ which $A$ exerts on $B$ and and the force ${\mathbf{F}}_{B\text{ on }A}$ which $B$ exerts on $A$. These forces are equal in magnitude but opposite in direction:

$${\mathbf{F}}_{B\text{ on }A}=-{\mathbf{F}}_{A\text{ on }B}$$

Warning: Inertial Forces

This law does not apply to inertial forces because they are not really forces.

This phenomenon is quite easy to observe. When you push something, you feel it pushing you back in the opposite direction. Similarly, when you pull something, you feel it pulling back. Unfortunately, some people call this pair of forces an “action” and a “reaction”, implying that one force precedes and causes the other. This stems largely from the way we are used to talking about things: I decided to push the box and so the box must be reacting to my decision. However, the reality is quite different. It is not my decision which caused the force I exerted on the box which in turn caused the force which the box exerts on my hand, but rather the physical interaction itself causes both forces simultaneously.