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% 20/01/2022 - 10AM

\section{Introduction}

Books:
\begin{itemize}
    \item Classical Mechanics - Douglas Gregory
        (more examples)
    \item Classical Mechanics - Tom Kibble \& Frank
        Berkshire (more ``chatty'')
\end{itemize}
Lecture Notes - David Tong PDF online. \\
Lecture on Saturday 5th March will be pre-recorded
\& online only.

\subsubsection*{Newtonian Mechanics - Definitions}

\begin{definition*}[Particle]
    A \emph{particle} is an object of
    insignificant size. For now, its only
    attribute is its position.
\end{definition*}

\noindent
For large objects, we take the centre of mass to
define the position and treat them like a
particle. \\
To describe the position, we pick a
\emph{reference frame}:

\begin{definition*}[Reference frame]
    A \emph{reference frame} is a choice of origin
    and 3 coordinate axes.
\end{definition*}

\noindent
With respect to this frame, a particle sweeps out
a trajectory $\bf{x}(t)$. (sometimes, we may write
$\bf{r}(t)$).

\begin{center}
\begin{tsqx}
    (0,0)->>(1,0)
    (0,0)->>(0,1)
    (0,0)->>(-0.4,-0.6)
    label $x$ @ (1.1,0)
    label $z$ @ (0,1.1)
    label $y$ @ 1.1*(-0.4,-0.6)
    (0,-0.8)..(0.3,-0.3)..(0.35,0.3)..(0.8,0.8)
    (0,0)->>(0.35,0.3)
    label $\mathbf{x}(t)$ @ (0.35,0.3)+(0.1,0)
    ! dot((0, 0));
\end{tsqx}
\end{center}

\noindent
The \emph{velocity} of the particle is $\bf{v} =
\dot{\bf{x}} := \dfrac{\bf{x}}{t}$. The
\emph{acceleration} of the particle is $\bf{a} =
\ddot{\bf{x}} := \dfrac[2]{\bf{x}}{t}$.

\begin{note*}
    The derivative of a vector is defined using
    components:
    \[ \dfrac{\bf{x}}{t} =
    \begin{pmatrix}
        \dfrac{x}{t} \\
        \dfrac{y}{t} \\
        \dfrac{z}{t}
    \end{pmatrix}
    .\]
\end{note*}

\noindent
Given two vector functions $\bf{f}(t)$ and
$\bf{g}(t)$,
\[ \dfrac{}{t}(\bf{f}, \bf{g}) = \dfrac{\bf{f}}{t}
\cdot \bf{g} + \bf{f} \cdot \dfrac{\bf{g}}{t} \]
and
\[ \dfrac{}{t} (\bf{f} \times \bf{g}) =
\dfrac{\bf{f}}{t} \times \bf{g} + \bf{f} \times
\dfrac{\bf{g}}{t} .\]
(proof by components)

\subsubsection*{Newton's Laws of Motions}

The framework of Newtonian mechanics rests on
three axioms, known as Newton's Laws:
\begin{enumerate}[N1]
    \item Left alone, a particle moves with
        constant velocity.
    \item The rate of change of momentum is
        proportional to the force.
    \item Every action has an equal and opposite
        reaction.
\end{enumerate}

\subsubsection*{Inertial Frames and The First Law}

For many reference frames, N1 isn't true! It only
holds for frames that are not themselves
accelerating. Such frames are called
\emph{intertial frames}. In an intertial frame,
$\ddot{\bf{x}} = \bf{0}$ when left alone ($\bf{F} =
\bf{0}$).

\bigskip

\noindent
A better framing of the first law is:
\begin{enumerate}[N1']
    \item Inertial frames exist.
\end{enumerate}
Inertial frames provide the setting for much of
what follows in this course. For most purposes,
this room approximates an inertial frame.

\subsubsection*{Galilean Relativity}

Inertial frames are not unique. Given an inertial
frame $S$, in which a particle has coordinate
$\bf{x}$, we can construct another inertial frame
$S'$ in which the coordinates of the particle are
given by $\bf{x}'$.
\begin{itemize}
    \item Translations: $\bf{x}' = \bf{x} +
        \bf{a}$, where $\bf{a}'$ is a constant.
    \item Rotations and reflections: $\bf{x}' =
        R\bf{x}$ where $R$ is a $3 \times 3$
        matrix with $R^\top R = I$ (orthogonal
        matrix).
    \item Boosts: $\bf{x}' = \bf{x} + \bf{v}t$
        where $\bf{v}$ is a constant. 
\end{itemize}
For each of these, if there is no force on a
particle, we have that $\dfrac[2]{\bf{x}}{t} = 0$
since $S$ is an inertial frame, which implies that
$\dfrac[2]{\bf{x}}{t} = 0$ so $S'$ is an inertial
frame. The \emph{Galilean principle of relativity}
tells us that the laws of physics stay the same
\begin{itemize}
    \item At every point in space.
    \item No matter which direction you face.
    \item No matter what constant velocity you
        move at.
    \item At all moments in time.
\end{itemize}
(these are experimentally tested facts).

\bigskip

\noindent
There is no such thing as ``absolutely
stationary'', but notice that acceleration is
absolute - you don't have to accelerate relative
to something.