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Newtonian mechanics

Table of Contents

1. Introduction

Newtonian mechanics is a mathematical model of the real world that is good at approximating non-relatavistic macro phenomenon. In other words, it will work for most engineers, and it has a few simple axioms:

2. Definition

First, we assume euclidean 3-dimensional space, and time as the parameterizing variable, or in other words the variable that is constant for everyone and everything. We assume objects can have a definite position, and have some sort of velocity and mass. For these examples we will assume a system with only one particle in a vacuum and then we will try to generalize. First we notice that this particle has a position given by some vector \(\vec{r}\). Then, we say that if this position changes over time:

\begin{align*} \vec{v} = \frac{d\vec{r}}{dt} \end{align*}

where \(\vec{v}\) is velocity, the time-derivative of position. We also define a mass \(m\) for this particle in question. Then, this particle will have a momentum:

\begin{align*} \vec{p} = m\vec{v} \end{align*}

And here we define inertia:

\begin{align*} \vec{p}(t_{1}) = \vec{p}(t_{2}) \end{align*}

Or the momentum of this particle will always stay the same throughout time, at least without other objects. Then we can define force:

\begin{align*} \vec{F} := \frac{d\vec{p}}{dt} \\ = m\frac{d\vec{v}}{dt} \\ \vec{a} := \frac{d\vec{v}}{dt} \\ \vec{F} = m\vec{a} \end{align*}

Now imagine we add another particle. This other particle will have its own position and momentum. These two particles together create a new system. Because we want the laws of physics to work for all systems:

\begin{align*} \vec{p}(t_{1}) = \vec{p}(t_{2}) \end{align*}

Where \(\vec{p}\) becomes the momentum of the whole system. If \(\vec{p}_{1}\) is the momentum of the first particle and \(\vec{p}_{2}\) is the momentum of the second particle, the momentum of the whole system is:

\begin{align*} \vec{p} = \vec{p}_{1} + \vec{p}_{2} \end{align*}

In general, the total momentum is defined to be:

\begin{align*} \vec{p} = \sum_{i=0}^{n}\vec{p}_{i} \end{align*}

And in real life, we observe that things can transfer momentum. That is:

\begin{align*} \vec{p}_{1} = -\vec{p}_{2} + \vec{p}_{i} \end{align*}

Where \(\vec{p}_{i}\) is the initial momentum of the objects. this happens when these two objects have the same position vector \( \vec{r}_{1} = \vec{r}_{2} \) (if they are point masses; don't have volume but have mass which is idealistic but works as an approximation). This operation of momentum transfer is symmetrical:

\begin{align*} \vec{p}_{2} = -\vec{p}_{1} + \vec{p}_{i} \end{align*}

Note that the fact that this operation is symmetrical must be the case to preserve the property:

\begin{align*} \vec{p}(t_{1}) = \vec{p}(t_{2}) \end{align*}

Of the entire system (otherwise you would be adding or subtracting momentum from the system). In other words, the entire system must have inertia, and this statement itself is the conservation of momentum. Conservation of momentum along with transfer of momentum yields a fully functional model of physics.

3. Textbook Formulation

The first law which we discussed is called inertia; the second law is the \( \vec{F} = m\vec{a} \) law as discussed; we can get what is called Newton's Third Law as follows:

\begin{align*} \frac{d\vec{p}_{2}}{dt} = -\frac{d\vec{p}_{1}}{dt} \\ \vec{F}_{2} = -\vec{F}_{1} \\ \vec{F}_{1} = -\vec{F}_{2} \end{align*}

However, the third law follows from conservation of momentum (inertia) and transfer of momentum (not really the second or third law). I do not know why it exists, and I think the formulation of Newtonian physics based off of less (and more descriptive) axioms is far more elegant, so I don't really know how this happened. In any case, if you are using a classic textbook, you will be using this formulation of Newtonian mechanics.

4. Newtonian Gravity

Gravity in Newtonian mechanics is defined via inverse square law. With the stipulation that mass can only be positive, the gravitational force field has the same properties as all inverse square fields by ; therefore:

\begin{align*} \vec{F}(\vec{r}) = \frac{Gm_{1}m_{2}}{r^{2}}\hat{r} \end{align*}

Most of the inverse square results consequently carry over.

Copyright © 2024 Preston Pan