The text for the video on the moment of inertia tensor

inertia_tensor.txt

% This is the text used in the youtube video
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% https://youtu.be/ccsldiiJH7k

\section{Kinetic energy and moment of inertia tensor of a rigid body}
Fix an (inertial) coordinate system, $O$.

``Clearly'', the configuration space of a rigid body in this system is ${\mathbb R}^3\times {\rm SO}(3).$

\subsection{Preparation: Complex numbers}
$W(\omega):={\rm i}\,\omega, \  \  \  \ U(t):={\rm e}^{W(\omega)\cdot t}$

$\frac{{\rm d}}{{\rm d}t}U(t)=W(\omega)\cdot U(t)$

$\frac{{\rm d}}{{\rm d}t}\left(U(t)\cdot \psi_0\right)=W(\omega)\cdot \left(U(t)\cdot \psi_0\right)$

\subsection{Velocity when working with two frames}
Motion of any particular point $P$ fixed on a rigid body upon motion of that body:

$\frac{{\rm d}}{{\rm d}t} {\bf r}_P^{\mathcal O}(t) = \frac{{\rm d}}{{\rm d}t} {\bf r}_B^{\mathcal O}(t) + {\bf W}({\bf \omega})\cdot{\bf x}_P^{\mathcal B}(t)$

with

${\bf W}({\bf \omega}):=[{\bf \omega}]_{\times}, \ \ \ \ {\bf W}({\bf \omega})\cdot{\bf d}={\bf \omega}\times {\bf d}$

Here ${\bf r}_P^{\mathcal O}$ denotes the coordinates of a point $P$ w.r.t. outer inertial frame $\mathcal O$ 
and ${\bf x}_P^{\mathcal B}$ the coordinate of $P$ w.r.t. a coordinate frame that is fixed to and centered at a point $B$ of the body.
From now on we'll also omit the $\mathcal O$ over any ${\bf r}$ (using ${\bf r}$ will always express using the coordinate system $\mathcal O$).\\
Below we'll write just ${\bf x}^{\mathcal B}$ for the coordinate function of the frame from $B$, taken as function of the coordinate functions ${\bf r}$, 
i.e. with the values of ${\bf x}^{\mathcal B}$ are the coordinate values as seen from $B$, of any point in space given by its coordinates in terms of ${\bf r}$. 
So we could also write ${\bf x}^{\mathcal B}_P$ as ${\bf x}^{\mathcal B}({\bf r}_P)$. 

The formula is at least clear when ${\bf x}_P$ lies on to the roation axis (thought the point $B$), 
which is parallel to ${\bf \omega}$, in which case $P$ moves exactly $B$.
It's also not hard to believe when $B$ is fixed w.r.t. $O$.

The angular velocity ${\bf\omega}$, upon a rotation of the rigid body, is the same vector here for any considered point $P$.

\subsection{Mass density}
Let $\rho:{\mathbb R^3}\to{\mathbb R}$ and define a functional

$\varrho : ({\mathbb R^3}\to{\mathbb R})\to{\mathbb R},\ \ \ \ \varrho[f] := \int f({\bf r})\cdot\rho({\bf r})\,{\rm d}{\bf r}$

Special case:

$\rho({\bf r})=\sum_{Q\in{\rm {Points}}}m_Q\cdot\delta({\bf r}-{\bf r}_Q)\implies \varrho[f]=\sum_{Q\in{\rm {Points}}}m_Q\cdot{}f({\bf r}_Q)$

\subsection{Kinetic energy of the body}

Consider velocities, such as that of $P$ in $O$, as values of a vector field ${\bf v}:{\mathbb R}\times{\mathbb R}^3\to{\mathbb R}^3$
(zero outside of the support of $\varrho$), 
so that e.g. for points $P$,

${\bf v}(t, {\bf r}_{P}(t))=\frac{{\rm d}}{{\rm d}t} {\bf r}_P(t)$.

Also write ${\bf v}_P(t):={\bf v}(t, {\bf r}_{P}(t))$.

In the following, $t$ are omitted for convenience.

$E_{\rm kin}:=\tfrac{1}{2}\,\varrho[{\bf v}^2]=\tfrac{1}{2}\,\varrho[ ({\bf v}_B+{\bf W}({\bf \omega})\cdot {\bf x}^{\mathcal B})^2]
= \tfrac{1}{2} \varrho[1]\, {\bf v}_B^2 
+ \varrho [ {\bf x}^{\mathcal B}] \cdot {\bf W}({\bf \omega})^T\cdot {\bf v}_B 
+ \tfrac{1}{2}\,{\bf \omega}\cdot I \cdot{\bf \omega}$

with

$I_C=\varrho[({\bf x}^{\mathcal B})^2\, {\rm Id}- {\bf x}^{\mathcal B}\otimes {\bf x}^{\mathcal B}]$

In a coordinate system centered at the so called center of mass point $C$, we have $\varrho [ {\bf x}^C]=0$. Thus, there, with $M:=\varrho[1]$,

$E_{\rm kin}=\tfrac{1}{2} M\left(\frac{{\rm d}}{{\rm d}t} {\bf r}_C(t)\right)^2 +\tfrac{1}{2}\,{\bf \omega}\cdot I_C \cdot{\bf \omega}$

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\section{References}
\subsection{Wikipedia links}
$\bullet$ \url{https://en.wikipedia.org/wiki/Moment_of_inertia}

$\bullet$ \url{https://en.wikipedia.org/wiki/Unit_circle#In_the_complex_plane}

$\bullet$ \url{https://en.wikipedia.org/wiki/Cross_product#Alternative_ways_to_compute_the_cross_product}

$\bullet$ \url{https://en.wikipedia.org/wiki/Rigid_body_dynamics} (forces not discussed in the present text)

$\bullet$ \url{https://en.wikipedia.org/wiki/Ehrenfest_paradox} (on rigid bodies in relativity theory, see also \emph{Born rigitity}, etc.)

\section{Books}
$\bullet$ Scheck: Mechanik - \url{https://www.amazon.de/dp/3540713778/ref=sr_1_1?keywords=mechanik+scheck&qid=1575718508&sr=8-1}) (German)

$\bullet$ Arnold: Mathematical Methods of Classical Mechanics - \url{https://www.amazon.com/Mathematical-Classical-Mechanics-Graduate-Mathematics/dp/0387968903} (lots of coordinate free presentations of the content)