The universe of Mechanics: animated mathematical formulas

Question

Many years ago when I saw these videos I has been fascinated by how the formulas were transformed and/or modified. See, for example this video of youtube (Kinetic and Potential Energy) from the minute 8:57:

or this video minutes 9:51 (The evolution of atomic models)

where the formulas are probably constructed with a character that is similar to computer modern(CM). It is interesting the movement and transformation for example of a summation with an integral.

But currently this kind of animated formulas like the videos I mentioned can be made with a specific package of LaTeX or is my question just an unrealizable request?

Answer 1

It is not particularly difficult to create something like this. If you want to see the animation on a pdf, use \usepackage{animate} instead of \usepackage[export]{animate}, and watch the resulting pdf with Acrobat Reader in the fullscreen mode.

\documentclass{standalone} 
%\usepackage{animate}
\usepackage[export]{animate}
\usepackage{tikz}
\tikzset{pics/shadowed/.style={code={
\node[xshift=-0.4ex,yshift=-0.4ex,text=gray!60!black]{#1};
\node[text=white]{#1};}}}
\begin{document}
\begin{animateinline}[autoplay,loop]{2}
\multiframe{42}{i=0+1}{%
 \begin{tikzpicture}[nodes={scale=1.5}]
  \fill[gray] (-5,-4) rectangle (5,4); % background
  \path (0,0) pic{shadowed={$\displaystyle r=\frac{\ifnum\i<36\relax
  L^2\else n^2\,\hbar^2\fi}{D\,M}$}}; %main (?) formula
  \ifnum\i>5 % L formula pops up at i=6
  \ifnum\i<36 % L formula disappears i=36
  \pgfmathsetmacro{\mysp}{max(0.8*(\i-24),0)}% negative space for i>24
  % define a path and move the L formula along it
  \path (1,1) arc[start angle=120,end angle=-200,x radius=1.2,y radius=1.45]
  pic[pos={min((\i-5)/18,1)}]{shadowed={$L_n\hspace{-\mysp pt}=\hspace{-\mysp pt}n\,\hspace{-\mysp pt}\hbar$}};
  \fi
  \fi
\end{tikzpicture}}
\end{animateinline}
\end{document}

Please note that I made no effort to fully reproduce the animation, just indicate some of the methods that might bring you there.

EDIT: Added explanations.

ADDENDUM: An animated version of this post for all who have to prepare online lectures in these crazy days.

\documentclass{standalone}
\usepackage{tikz}
\usepackage[export]{animate}
\begin{document}
\begin{animateinline}[autoplay,loop]{2}
\multiframe{24}{i=2+1}{%
\begin{tikzpicture}[>=stealth]
  \pgfmathsetmacro{\nextx}{4/\i}
  \pgfmathsetmacro{\xmax}{4-\nextx}
  \foreach \x in {0,\nextx,...,\xmax} {%
    \draw[fill=cyan] (\x,0) rectangle (\x+\nextx,{.25*pow(\x+\nextx,2)});
    \draw[fill=orange] (\x,0) rectangle  (\x+\nextx,.25*\x*\x);
  }
  \draw [->] (-.5,0) -- ++(5,0) node[below] {$x$};
  \draw [->] (0,-.5) -- ++(0,5) node[left] {$y$};
  \draw [thick,blue,domain=0:4] plot (\x,{.25*pow(\x,2)});
\end{tikzpicture}}
\end{animateinline}
\end{document}