Basic physics of a (physical) pendulum. The plots of the exact solution with an arbitrarily large amplitude is inspired by this article, and Wikipedia.
For more on the pendulums, please see the “pendulum” tag . For more on the small-angle approximation, please see the “Taylor expansion” tag.
These figures are used in Ben Kilminster’s lecture notes for PHY111.
Simple pendulum:
Force diagram of a pendulum to apply Newton’s second law:
Force diagram of a pendulum, breaking down the force components:
Physical pendulum:
Exact solution
Closed solutions (bounded)
Comparison of exact solution of a pendulum to a sine with the same period, using numerical data produced by these MATLAB scripts. (Note the ideal solution from the small-angle approximation has a fixed period T_0.)
Ratio between real and ideal period of a pendulum, T/T_0, as a function of the angular amplitude (see this Wikipedia section).
Open solutions (unbounded)
Edit and compile if you like:
% Author: Izaak Neutelings (October 2020)
% Instructions: To compile, include data files from
% https://github.com/IzaakWN/CodeSnippets/tree/master/LaTeX/TikZ/physics/dynamics_pendulum
% Other sources:
% https://www.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172007000400024
% https://www.researchgate.net/publication/262746594_Exact_solution_for_the_nonlinear_pendulum
% https://tex.stackexchange.com/questions/545590/phase-portrait-of-van-der-pol-oscillator-with-pgfplots
% http://matlab.cheme.cmu.edu/2011/08/09/phase-portraits-of-a-system-of-odes/
\documentclass[border=3pt,tikz]{standalone}
\usepackage{physics}
\usepackage{siunitx}
\usepackage{tikz,pgfplots}
\usepackage[outline]{contour} % glow around text
\usetikzlibrary{calc}
\usetikzlibrary{angles,quotes} % for pic
\usetikzlibrary{arrows.meta}
\tikzset{>=latex} % for LaTeX arrow head
\contourlength{1.2pt}
\colorlet{xcol}{blue!70!black}
\colorlet{vcol}{green!60!black}
\colorlet{myred}{red!70!black}
\colorlet{myblue}{blue!70!black}
\colorlet{mygreen}{green!70!black}
\colorlet{mydarkred}{myred!70!black}
\colorlet{mydarkblue}{myblue!60!black}
\colorlet{mydarkgreen}{mygreen!60!black}
\colorlet{acol}{red!50!blue!80!black!80}
\tikzstyle{CM}=[red!40!black,fill=red!80!black!80]
\tikzstyle{xline}=[xcol,thick,smooth]
\tikzstyle{mass}=[line width=0.6,red!30!black,fill=red!40!black!10,rounded corners=1,
top color=red!40!black!20,bottom color=red!40!black!10,shading angle=20]
\tikzstyle{faded mass}=[dashed,line width=0.1,red!30!black!40,fill=red!40!black!10,rounded corners=1,
top color=red!40!black!10,bottom color=red!40!black!10,shading angle=20]
\tikzstyle{rope}=[brown!70!black,very thick,line cap=round]
\def\rope#1{ \draw[black,line width=1.4] #1; \draw[rope,line width=1.1] #1; }
\tikzstyle{force}=[->,myred,very thick,line cap=round]
\tikzstyle{velocity}=[->,vcol,very thick,line cap=round]
\tikzstyle{Fproj}=[force,myred!40]
\tikzstyle{myarr}=[-{Latex[length=3,width=2]},thin]
\def\tick#1#2{\draw[thick] (#1)++(#2:0.12) --++ (#2-180:0.24)}
\DeclareMathOperator{\sn}{sn}
\DeclareMathOperator{\cn}{cn}
\DeclareMathOperator{\dn}{dn}
\def\N{80} % number of samples in plots
\begin{document}
% PENDULUM
\def\L{2.8} % string length
\def\ang{28} % angle string
\def\R{0.25} % ball radius
\def\F{1.0} % force magnitude
\begin{tikzpicture}
\message{^^JPendulum}
\coordinate (M) at (\ang-90:\L);
\coordinate (M') at (0,-\L);
\coordinate (O) at (0,0);
\coordinate (B) at (0,-\L-2.2*\R);
\coordinate (FT) at ($(M)+(90+\ang:{\F*cos(\ang)+\R})$);
\coordinate (FG) at ($(M)+(-90:{\F+\R})$);
\coordinate (FGx) at ($(M)+(-90+\ang:{0.55*\F+\R})$);
\coordinate (MA) at ($(M)+(180+\ang:{\F*sin(\ang)+\R})$);
%\draw[faded mass] (M') circle(\R);
\draw[<->] (M)++(\ang-32:0.2*\L) node[below right=-1,scale=0.9] {$x$}
--++ (\ang:0.17*\L) --++ (90+\ang:0.17*\L) node[above right=-1,scale=0.9] {$y$};
\draw[dashed] (O) -- (B);
%\draw[dashed,myred!60!black] (MA) -- (FG);
\draw[dashed,myred!60!black] (M) -- (FGx);
\draw[dashed] (-90+\ang+10:\L) arc(-90+\ang+10:-110:\L) (B);
\rope{(O) -- (M)} \path (O) -- (M) node[midway,above right=-1] {$L$};
\fill[black] (O) circle(0.04);
\draw[force] (M) -- (FT) node[midway,left=0] {$\vb{T}$};
\draw[force] (M) -- (FG) node[right=0] {$m\vb{g}$};
\draw[force,acol] (M) -- (MA) node[right=2,below=0] {$m\vb{a}$}; %{\contour{white}{$m\vb{a}$}};
\draw[mass] (M) circle(\R) node {$m$};
\draw pic[myarr,"$\theta$",xcol,draw=xcol,angle radius=22,angle eccentricity=1.30] {angle=B--O--M}; %_\text{max}
\draw pic[myarr,"$\theta$",xcol,draw=xcol,angle radius=14,angle eccentricity=1.45] {angle=FG--M--FGx};
\end{tikzpicture}
% PENDULUM FORCES - ma
\begin{tikzpicture}
\message{^^JPendulum forces}
\def\F{1.4} % force magnitude
\coordinate (O) at (0,0);
\coordinate (FT) at (90+\ang:{\F*cos(\ang)});
\coordinate (FG) at (-90:\F);
\coordinate (FGx) at (-90+\ang:{0.7*\F});
\coordinate (MA) at (180+\ang:{\F*sin(\ang)});
\draw[dashed,myred!60!black] (O) -- (FGx);
\draw[dashed,myred!60!black] (MA) -- (FG);
\draw[force] (O) -- (FT) node[midway,above right=-2] {$\vb{T}$};
\draw[force] (O) -- (FG) node[right=0] {$m\vb{g}$};
\draw[force,acol] (O) -- (MA) node[left=0] {$m\vb{a}$};
\draw pic[myarr,"$\theta$",xcol,draw=xcol,angle radius=14,angle eccentricity=1.45] {angle=FG--O--FGx};
\end{tikzpicture}
% PENDULUM FORCES
\begin{tikzpicture}
\message{^^JPendulum forces}
\def\F{1.4} % force magnitude
\coordinate (O) at (0,0);
\coordinate (FT) at (90+\ang:{\F*cos(\ang)});
\coordinate (FG) at (-90:\F);
\coordinate (FGx) at (-90+\ang:{\F*cos(\ang)});
\coordinate (FGy) at (180+\ang:{\F*sin(\ang)});
\draw[dashed,myred!60!black] (FGy) -- (FG); %-- (FGx);
\draw[force] (O) -- (FT) node[midway,above right=-2] {$\vb{T}$};
\draw[Fproj] (O) -- (FGy) node[pos=0.4,above left=-3,scale=0.86] {$mg\sin\theta$};
\draw[Fproj] (O) -- (FGx) node[pos=0.5,above right=-3,scale=0.86] {$mg\cos\theta$};
\draw[force] (O) -- (FG) node[right=0] {$m\vb{g}$};
\draw pic[myarr,"$\theta$",xcol,draw=xcol,angle radius=14,angle eccentricity=1.45] {angle=FG--O--FGx};
\end{tikzpicture}
%% PENDULUM
%\begin{tikzpicture}
% \coordinate (M) at (0,-\L);
% \coordinate (O) at (0,0);
% \rope{(O) -- (M)} \path (O) -- (M) node[midway,above right=-2] {$L$};
% \fill[black] (O) circle(0.04);
% \draw[velocity] (M)++(-\R,0) --++ (-2.7*\R,0) node[above=0] {$\vb{v}$};
% \draw[mass] (M) circle(\R) node {$m$};
%\end{tikzpicture}
% PHYSICAL PENDULUM
\begin{tikzpicture}
\message{^^JPhysical pendulum}
\def\L{2.0} % string length
\def\ang{28} % angle string
\def\R{0.25} % ball radius
\def\F{1.3} % force magnitude
\coordinate (M) at (\ang-90:\L);
\coordinate (R) at (\ang-90:1.1*\L);
\coordinate (M') at (0,-\L);
\coordinate (O) at (0,0);
\coordinate (B) at (0,-\L-2.5*\R);
\coordinate (FT) at (90:\F);
\coordinate (FG) at ($(M)+(-90:\F)$);
\coordinate (FGx) at ($(M)+(-90+\ang:{0.55*\F+\R})$);
\coordinate (MA) at ($(M)+(180+\ang:{\F*sin(\ang)+\R})$);
\draw[mass]
(180:0.3*\L) to[out=90,in=180] (80:0.36*\L) to[out=0,in=90] (0:0.3*\L) to[out=-90,in=90]
($(R)+(0:0.7*\L)$) to[out=-90,in=0] ($(R)+(-90:0.4*\L)$) to[out=180,in=-90] ($(R)+(-180:0.6*\L)$) to[out=90,in=-90] cycle; %node {$m$};
\draw[dashed] (O) -- (B);
\draw[dashed] (O) -- (M);
%\draw[dashed,myred!60!black] (MA) -- (FG);
\draw[dashed] (-90+\ang+10:\L) arc(-90+\ang+10:-110:\L) (B);
\fill[black] (O) circle(0.04);
\draw[CM] (M) circle(0.09);
\draw[force] (O) -- (FT) node[below right=0] {$\vb{F}_\mathrm{N}$};
\draw[force] (M) -- (FG) node[right=0] {$M\vb{g}$};
% \draw[force,acol] (M) -- (MA) node[right=2,below=0] {$m\vb{a}$}; %{\contour{white}{$m\vb{a}$}};
% \draw pic[myarr,"$\theta$",xcol,draw=xcol,angle radius=22,angle eccentricity=1.30] {angle=B--O--M}; %_\text{max}
% \draw pic[myarr,"$\theta$",xcol,draw=xcol,angle radius=14,angle eccentricity=1.45] {angle=FG--M--FGx};
\end{tikzpicture}
%% EXACT SOLUTION
%% To compile, include data files from
%% https://github.com/IzaakWN/CodeSnippets/tree/master/LaTeX/TikZ/physics/dynamics_pendulum
%% Other sources:
%% https://www.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172007000400024
%% https://www.researchgate.net/publication/262746594_Exact_solution_for_the_nonlinear_pendulum
%% https://tex.stackexchange.com/questions/545590/phase-portrait-of-van-der-pol-oscillator-with-pgfplots
%% http://matlab.cheme.cmu.edu/2011/08/09/phase-portraits-of-a-system-of-odes/
%\begin{tikzpicture}
% \message{^^JPendulum exact solution}
% \def\xmax{7.0} % max x axis
% \def\ymax{1.4} % max y axis
% \def\A{3.1*\ys} % amplitude, theta0 = 3.1 ~ 178°
% \def\B{2.4*\ys} % amplitude, theta0 = 2.4 ~ 140°
% \def\C{0.6*\ys} % amplitude, theta0 = 0.6 ~ 34.4°
% \pgfmathsetmacro\Tb{\xmax/2.97} % period, theta0 = 2.4 ~ 140°, T/T0 ~ 1.5602
% \pgfmathsetmacro\Ta{\Tb*3.35/1.56} % period, theta0 = 3.1 ~ 178°, T/T0 ~ 3.3485
% \pgfmathsetmacro\Tc{\Tb*1.023/1.56} % period, theta0 = 0.6 ~ 34.4°, T/T0 ~ 1.0230
% \pgfmathsetmacro\xs{\Ta/(2*pi*3.35)} % xscale, note data uses T = 2*pi, omega0 = 1
% \pgfmathsetmacro\ys{\ymax*0.28} % yscale
%
% % AXIS
% \draw[->,thick] (0,-\ymax) -- (0,\ymax+0.2) node[above=-3] {$\theta$ [rad]};
% \draw[->,thick] (-0.2,0) -- (\xmax+0.05,0) node[right=-1] {$t$ [s]};
% \draw[mydarkblue!30,dashed]
% (0, pi*\ys) --++ (0.90*\xmax,0) node[right=1,scale=0.8] {$ \SI{\pi}{rad}= \SI{180}{\degree}$}
% (0,-pi*\ys) --++ (0.84*\xmax,0) node[right=1,scale=0.8] {$-\SI{\pi}{rad}=-\SI{180}{\degree}$};
%
% % PLOT REAL PENDULUM
% \begin{axis}[
% axis lines=none,anchor=origin,x=1cm,y=1cm, % coincide with TikZ coordinates
% xmin=0,xmax=0.94*\xmax/\xs,xscale=\xs,
% y filter/.code={\pgfmathparse{\ys*\pgfmathresult}}
% ]
% \addplot[xline,mygreen] table {dynamics_pendulum/data/pendulum-0p6.txt};
% \addplot[xline,myred] table {dynamics_pendulum/data/pendulum-2p4.txt};
% \addplot[xline,xcol] table {dynamics_pendulum/data/pendulum-3p1.txt};
% \end{axis}
%
% % PLOT APPROXIMATION
% \draw[xline,mydarkgreen,thin,dashed,samples=\N,smooth,variable=\x,domain=0:0.94*\xmax]
% plot(\x,{\C*cos(360/\Tc*\x)});
% \draw[xline,mydarkred,thin,dashed,samples=\N,smooth,variable=\x,domain=0:0.94*\xmax]
% plot(\x,{\B*cos(360/\Tb*\x)});
% \draw[xline,mydarkblue,thin,dashed,samples=\N,smooth,variable=\x,domain=0:0.94*\xmax]
% plot(\x,{\A*cos(360/\Ta*\x)});
%
% % TICKS
% \tick{0, \A}{0} node[above=1,left=-1,scale=0.9,mydarkblue] {3.1}; %\SI{3.1}{\degree}
% \tick{0, \B}{0} node[below=0.5,left=-1,scale=0.9,mydarkred] {2.4};
% \tick{0, \C}{0} node[below=0.5,left=-1,scale=0.9,mydarkgreen] {0.6};
% \tick{0,-\A}{0} node[left=-1,scale=0.9,mydarkblue] {$-3.1$};
% \tick{\Tb,0}{90} node[below,scale=0.9,mydarkred] {$T(2.4)$};
% \tick{2*\Tb,0}{90} node[below,scale=0.9] {$2\color{mydarkred}{T(2.4)}$};
%
%\end{tikzpicture}
%
%
%% EXACT SOLUTION - angular velocity
%\begin{tikzpicture}
% \message{^^JPendulum exact solution - dtheta/dt}
% \def\xmax{7.0} % max x axis
% \def\ymax{1.4} % max y axis
% \def\wmax{2*\ys} % yscale
% \pgfmathsetmacro\Ta{\xmax/2.97} % period, theta0 = 2.4 ~ 140°, T/T0 ~ 1.5602
% \pgfmathsetmacro\Tb{\Ta*3.35/1.56} % period, theta0 = 3.1 ~ 178°, T/T0 ~ 3.3485
% \pgfmathsetmacro\xs{\Tb/(2*pi*3.35)} % xscale, note data uses T = 2*pi, omega0 = 1
% \pgfmathsetmacro\ys{\ymax*0.44} % yscale
% \pgfmathsetmacro\A{2*sin(deg(2.4/2))*\ys} % amplitude, theta0 = 2.4 ~ 140°
% \pgfmathsetmacro\B{2*sin(deg(3.1/2))*\ys} % amplitude, theta0 = 3.1 ~ 178°
%
% % AXIS
% \draw[->,thick] (0,-\ymax) -- (0,\ymax+0.2) node[above=-3] {$\dot{\theta}$ [rad/s]};
% \draw[->,thick] (-0.2,0) -- (\xmax+0.05,0) node[right=-1] {$t$ [s]};
% \draw[mydarkblue!30,dashed]
% (0, \wmax) --++ (0.9*\xmax,0) %node[right=1,scale=0.8] {$ \SI{\pi}{rad}= \SI{180}{\degree}$}
% (0,-\wmax) --++ (0.9*\xmax,0); %node[right=1,scale=0.8] {$-\SI{\pi}{rad}=-\SI{180}{\degree}$};
%
% % PLOT REAL PENDULUM
% \begin{axis}[
% axis lines=none,anchor=origin,x=1cm,y=1cm, % coincide with TikZ coordinates
% xmin=0,xmax=0.94*\xmax/\xs,xscale=\xs,
% y filter/.code={\pgfmathparse{\ys*\pgfmathresult}}
% ]
% \addplot[xline,myred]
% table[y index=2] {dynamics_pendulum/data/pendulum-2p4.txt};
% \addplot[xline,xcol]
% table[y index=2] {dynamics_pendulum/data/pendulum-3p1.txt};
% \end{axis}
%
% % PLOT APPROXIMATION
% \draw[xline,mydarkred,thin,dashed,samples=\N,smooth,variable=\x,domain=0:0.94*\xmax]
% plot(\x,{-\A*sin(360/\Ta*\x)});
% \draw[xline,mydarkblue,thin,dashed,samples=\N,smooth,variable=\x,domain=0:0.94*\xmax]
% plot(\x,{-\B*sin(360/\Tb*\x)});
%
% % TICKS
% \tick{0, \wmax}{0} node[left=-2,scale=0.9] {$2\omega_0$};
% \tick{0,-\wmax}{0} node[left=-2,scale=0.9] {$-2\omega_0$};
% \tick{\Ta,0}{90} node[below,scale=0.9] {\contour{white}{$\color{mydarkred}{T(2.4)}$}};
% \tick{2*\Ta,0}{90} node[below,scale=0.9] {\contour{white}{$2\color{mydarkred}{T(2.4)}$}};
%
%\end{tikzpicture}
%
%
%% EXACT SOLUTION
%\begin{tikzpicture}
% \message{^^JPendulum exact solution (normalized)}
% \def\xmax{7.0} % max x axis
% \def\ymax{1.4} % max y axis
% \def\A{3.1*\ys} % amplitude, theta0 = 3.1 ~ 178°
% \def\B{2.4*\ys} % amplitude, theta0 = 2.4 ~ 140°
% \def\C{1.6*\ys} % amplitude, theta0 = 1.6 ~ 91.7
% \pgfmathsetmacro\T{0.95*\xmax/2} % period
% \pgfmathsetmacro\ys{\ymax*0.28} % yscale
%
% % AXIS
% \draw[->,thick] (0,-\ymax) -- (0,\ymax+0.2) node[above=-3] {$\theta$ [rad]};
% \draw[->,thick] (-0.2,0) -- (\xmax+0.05,0) node[right=-1] {$t$ [s]};
% \draw[mydarkblue!30,dashed]
% (0, pi*\ys) --++ (0.98*\xmax,0) node[right=1,scale=0.8] {$ \SI{\pi}{rad}= \SI{180}{\degree}$}
% (0,-pi*\ys) --++ (0.90*\xmax,0) node[right=1,scale=0.8] {$-\SI{\pi}{rad}=-\SI{180}{\degree}$};
%
% % PLOT REAL PENDULUM
% \begin{axis}[
% axis lines=none,anchor=origin,x=1cm,y=1cm, % coincide with TikZ coordinates
% xmin=0,xscale=\T, %0.94*7/3, %xmax=0.94*\xmax/\xs,
% y filter/.code={\pgfmathparse{\ys*\pgfmathresult}}
% ]
% \addplot[xline,mygreen] table {dynamics_pendulum/data/pendulum-1p6_norm.txt};
% \addplot[xline,myred] table {dynamics_pendulum/data/pendulum-2p4_norm.txt};
% \addplot[xline,xcol] table {dynamics_pendulum/data/pendulum-3p1_norm.txt};
% \end{axis}
%
% % PLOT APPROXIMATION
% \draw[xline,mydarkblue,thin,dashed,samples=\N,smooth,variable=\x,domain=0:0.94*\xmax]
% plot(\x,{\A*cos(360/\T*\x)});
% \draw[xline,mydarkred,thin,dashed,samples=\N,smooth,variable=\x,domain=0:0.94*\xmax]
% plot(\x,{\B*cos(360/\T*\x)});
% \draw[xline,mydarkgreen,thin,dashed,samples=\N,smooth,variable=\x,domain=0:0.94*\xmax]
% plot(\x,{\C*cos(360/\T*\x)});
%
% % TICKS
% \tick{0, \A}{0} node[above=1,left=-1,scale=0.9] {\color{mydarkblue}{3.1}};
% \tick{0, \B}{0} node[below=0.5,left=-1,scale=0.9] {\color{mydarkred}{2.4}};
% \tick{0, \C}{0} node[below=0.5,left=-1,scale=0.9] {\color{mydarkgreen}{1.6}};
% \tick{0,-\A}{0} node[left=-1,scale=0.9] {\color{mydarkblue}{$-3.1$}};
% \tick{\T,0}{90} node[below,scale=0.9] {$T$};
% \tick{2*\T,0}{90} node[below,scale=0.9] {$2T$};
%
%\end{tikzpicture}
%
%
%% EXACT SOLUTION - angular velocity
%\begin{tikzpicture}
% \message{^^JPendulum exact solution - dtheta/dt (normalized)}
% \def\xmax{7.0} % max x axis
% \def\ymax{1.4} % max y axis
% \def\wmax{2*\ys} % yscale
% \pgfmathsetmacro\T{0.95*\xmax/2} % period
% \pgfmathsetmacro\ys{\ymax*0.44} % yscale
% \pgfmathsetmacro\A{2*sin(deg(3.1/2))*\ys} % amplitude, theta0 = 3.1 ~ 178°
% \pgfmathsetmacro\B{2*sin(deg(2.4/2))*\ys} % amplitude, theta0 = 2.4 ~ 140°
% \pgfmathsetmacro\C{2*sin(deg(1.6/2))*\ys} % amplitude, theta0 = 1.6 ~ 91.7°
%
% % AXIS
% \draw[->,thick] (0,-\ymax) -- (0,\ymax+0.2) node[above=-3] {$\dot{\theta}$ [rad/s]};
% \draw[->,thick] (-0.2,0) -- (\xmax+0.05,0) node[right=-1] {$t$ [s]};
% \draw[mydarkblue!30,dashed]
% (0, \wmax) --++ (0.9*\xmax,0) %node[right=1,scale=0.8] {$ \SI{\pi}{rad}= \SI{180}{\degree}$}
% (0,-\wmax) --++ (0.9*\xmax,0); %node[right=1,scale=0.8] {$-\SI{\pi}{rad}=-\SI{180}{\degree}$};
%
% % PLOT REAL PENDULUM
% \begin{axis}[
% axis lines=none,anchor=origin,x=1cm,y=1cm, % coincide with TikZ coordinates
% xmin=0,xscale=\T,
% y filter/.code={\pgfmathparse{\ys*\pgfmathresult}}
% ]
% \addplot[xline,xcol]
% table[y index=2] {dynamics_pendulum/data/pendulum-3p1_norm.txt};
% \addplot[xline,myred]
% table[y index=2] {dynamics_pendulum/data/pendulum-2p4_norm.txt};
% \addplot[xline,mygreen]
% table[y index=2] {dynamics_pendulum/data/pendulum-1p6_norm.txt};
% \end{axis}
%
% % PLOT APPROXIMATION
% \draw[xline,mydarkblue,thin,dashed,samples=\N,smooth,variable=\x,domain=0:0.94*\xmax]
% plot(\x,{-\A*sin(360/\T*\x)});
% \draw[xline,mydarkred,thin,dashed,samples=\N,smooth,variable=\x,domain=0:0.94*\xmax]
% plot(\x,{-\B*sin(360/\T*\x)});
% \draw[xline,mydarkgreen,thin,dashed,samples=\N,smooth,variable=\x,domain=0:0.94*\xmax]
% plot(\x,{-\C*sin(360/\T*\x)});
%
% % LABELS
% \node[above=0,scale=0.8,mydarkblue] at (0.78*\T,0.99*\A) {$\theta_0=3.1$};
% \node[right,scale=0.8,mydarkred] at (0.81*\T,0.88*\B) {\contour{white}{$\theta_0=2.4$}};
% \node[right,scale=0.8,mydarkgreen] at (0.96*\T,0.4*\C) {$\theta_0=1.6$};
%
% % TICKS
% \tick{0, \wmax}{0} node[left=-2,scale=0.9] {$2\omega_0$};
% \tick{0,-\wmax}{0} node[left=-2,scale=0.9] {$-2\omega_0$};
% \tick{\T,0}{90} node[left=2,below,scale=0.9] {$T$};
% \tick{2*\T,0}{90} node[below,scale=0.9] {$2\color{mydarkred}{T}$};
%
%\end{tikzpicture}
%
%
%% EXACT SOLUTION - T/T0
%% https://en.wikipedia.org/wiki/Pendulum_(mathematics)
%\begin{tikzpicture}[xscale=1.1]
% \message{^^JPendulum exact solution period (closed/bounded)}
% \def\xmax{4.1} % max x axis
% \def\ymax{3.2} % max y axis
% \def\ys{0.82} % yscale
% \def\xs{1.02} % xscale
% \def\yz{\ys*0.4} % y zero
% \def\xtick#1{\draw[thick] (#1)++(0,0.11) --++ (0,-0.22)}
% \def\ytick#1{\draw[thick] (#1)++(0.1,0) --++ (-0.2,0)}
%
% % AXIS
% \draw[->,thick]
% (0,\yz-0.1*\ymax) -- (0,\ymax+0.1) node[below=3,left=1] {$\dfrac{T}{T_0}$};
% \draw[->,thick]
% (-0.1*\ymax,\yz) -- (\xmax,\yz) node[right=3,below=0] {$\theta_0$ [rad]};
% \draw[dashed] (\xs*pi,\yz) --++ (0,\ymax-\yz);
%
% % REAL PENDULUM
% \begin{axis}[
% axis lines=none,anchor=origin,
% x=1cm,y=1cm,ymin=0,ymax=\ymax,xscale=\xs,
% y filter/.code={\pgfmathparse{\ys*\pgfmathresult}}
% ]
% \addplot[xline]
% table[x index=0,y index=1] {dynamics_pendulum/data/pendulum_period.txt};
% \end{axis}
%
% % LINES
% \draw[dashed,mydarkgreen]
% (0,\ys) --++ (0.82*\xmax,0)
% node[right=1,scale=0.8,align=left] {small-angle\\[-1mm]approximation};
% \draw[dashed,mydarkred]
% (0,\ys*1.18) -| (0.50*pi,\yz);
%
% % TICKS
% \xtick{0.25*\xs*pi,\yz} node[below=1,scale=0.9] {$\dfrac{\pi}{4}$};
% \xtick{0.50*\xs*pi,\yz} node[below=1,scale=0.9] {$\dfrac{\pi}{2}$};
% \xtick{0.75*\xs*pi,\yz} node[below=-0.9,scale=0.9] {$\dfrac{3\pi}{4}$};
% \xtick{ \xs*pi,\yz} node[below=1,scale=0.9] {$\pi$};
% \ytick{0,\ys} node[below=1.5,left=0,scale=0.9] {1};
% \ytick{0,\ys*1.18} node[above=3,left=0,scale=0.8,mydarkred] {1.18};
% \ytick{0,\ys*2} node[left=0,scale=0.9] {2};
% \ytick{0,\ys*3} node[left=0,scale=0.9] {3};
%
%\end{tikzpicture}
%
%
%% EXACT SOLUTION - OPEN TRAJECTORY
%\begin{tikzpicture}
% \message{^^JPendulum exact solution (open/unbounded, normalized)}
% \def\xmax{7.0} % max x axis
% \def\ymax{2.8} % max y axis
% \pgfmathsetmacro\T{0.98*\xmax/2.5} % period
% \pgfmathsetmacro\ys{\ymax*0.06} % yscale
%
% % AXIS
% \draw[->,thick] (0,-0.2) -- (0,\ymax+0.2) node[above=-3] {$\theta$ [rad]};
% \draw[->,thick] (-0.2,0) -- (\xmax+0.25,0) node[right=-1] {$t$ [s]};
% \draw[mydarkblue!30,dashed,yscale=\ys]
% (0,pi) --++ (\xmax,0)
% (0,3*pi) --++ (\xmax,0)
% (0,5*pi) --++ (\xmax,0);
%
% % PLOT REAL PENDULUM
% \begin{axis}[
% axis lines=none,anchor=origin,x=1cm,y=1cm, % coincide with TikZ coordinates
% xmin=0,xscale=\T, %0.94*7/3, %xmax=0.94*\xmax/\xs,
% y filter/.code={\pgfmathparse{\ys*\pgfmathresult}}
% ]s
% \addplot[xline,xcol] table {dynamics_pendulum/data/pendulum_open-2p00000002_norm.txt};
% \addplot[xline,myred] table {dynamics_pendulum/data/pendulum_open-2p01_norm.txt};
% \addplot[xline,mygreen] table {dynamics_pendulum/data/pendulum_open-2p6_norm.txt};
% \end{axis}
%
% % LABELS
% \node[above right=-1,scale=0.8,mydarkblue] at (0.97*\T,\ys*pi) {$\Omega_0/2\omega_0=1.00000001$};
% %\node[right=2,scale=0.8,mydarkred] at (\T,1.96*\ys*pi) {\contour{white}{$\Omega_0/2\omega_0=1.005$}};
% \node[below right=0,scale=0.8,mydarkred] at (1.27*\T,3*\ys*pi) {\contour{white}{$\Omega_0/2\omega_0=1.005$}};
% %\node[below right=-1,scale=0.8,mydarkgreen] at (1.4*\T,3*\ys*pi) {$\Omega_0/2\omega_0=1.3$};
% \node[below right=0,scale=0.8,mydarkgreen] at (2.01*\T,4.24*\ys*pi) {$\Omega_0/2\omega_0=1.3$};
%
% % TICKS
% \tick{0,pi*\ys}{0} node[left=-1,scale=0.9] {$\pi$};
% \tick{0,2*pi*\ys}{0} node[left=-1,scale=0.9] {$2\pi$};
% \tick{0,3*pi*\ys}{0} node[left=-1,scale=0.9] {$3\pi$};
% \tick{0,4*pi*\ys}{0} node[left=-1,scale=0.9] {$4\pi$};
% \tick{0,5*pi*\ys}{0} node[left=-1,scale=0.9] {$5\pi$};
% \tick{\T,0}{90} node[below,scale=0.9] {$T$};
% \tick{2*\T,0}{90} node[below,scale=0.9] {$2T$};
%
%\end{tikzpicture}
%
%
%% EXACT SOLUTION - OPEN TRAJECTORIES - T/T0
%% https://en.wikipedia.org/wiki/Pendulum_(mathematics)
%\begin{tikzpicture}
% \message{^^JPendulum exact solution period (open/unbounded)}
% \def\xmax{6.1} % max x axis
% \def\ymax{2.9} % max y axis
% \def\ys{2.2} % yscale
% \def\xs{0.79} % xscale
% \def\xtick#1{\draw[thick] (#1)++(0,0.11) --++ (0,-0.22)}
% \def\ytick#1{\draw[thick] (#1)++(0.12,0) --++ (-0.24,0)}
%
% % AXIS
% \draw[->,thick] (0,-0.1*\ymax) -- (0,\ymax+0.2) node[below=3,left=1] {$\dfrac{T}{T_0}$};
% \draw[->,thick] (-0.1*\xs*\ymax,0) -- (\xs*\xmax,0) node[right=3,below=0] {$\Omega_0/2\omega_0$};
% %\draw[dashed] (pi,0) --++ (0,\ymax-0);
%
% % UNIFORM MOTION
% \draw[xline,mydarkgreen,thin,dashed,samples=\N,smooth,variable=\x,domain=\ys/(2.05*\ymax):5]
% plot(\xs*\x,{\ys/(2*\x)});
% \node[mydarkgreen,scale=0.75,above right=-4]
% at (0.12,\ymax) {uniform circular motion, $\dfrac{T}{T_0}=\dfrac{\omega_0}{\Omega_0}$};
%
% % REAL PENDULUM
% \begin{axis}[
% axis lines=none,anchor=origin,
% x=1cm,y=1cm,ymin=0,ymax=\ymax,xmin=0,xmax=5,xscale=\xs,
% y filter/.code={\pgfmathparse{\ys*\pgfmathresult}}
% ]
% \addplot[xline]
% table[x index=1,y index=2] {dynamics_pendulum/data/pendulum_period_open.txt};
% \end{axis}
%
% % LINES
% \draw[dashed,mydarkblue] (\xs,0) --++ (0,0.95*\ymax);
%
% % TICKS
% \xtick{\xs*1,0} node[below=1,scale=0.9] {$1$};
% \xtick{\xs*2,0} node[below=1,scale=0.9] {$2$};
% \xtick{\xs*3,0} node[below=1,scale=0.9] {$3$};
% \xtick{\xs*4,0} node[below=1,scale=0.9] {$4$};
% \xtick{\xs*5,0} node[below=1,scale=0.9] {$5$};
% \ytick{0,\ys*0.5} node[below=1.5,left=0,scale=0.9] {$0.5$};
% \ytick{0,\ys} node[below=1.5,left=0,scale=0.9] {$1$};
%
%\end{tikzpicture}
% PENDULUM EQUATIONS - CLOSED TRAJECTORIES
\begin{tikzpicture}[scale=1]
\message{^^JPendulum equations (closed/bounded)}
\def\myarg{\left(\alpha,k^2\right)}
\node[align=left] at (0,0) {
\begin{minipage}{9.8cm}
The differential equation for a simple pendulum is
\begin{equation*}
\dv{^2\theta}{t^2} + \omega_0^2\sin\theta = 0.
\end{equation*}
For the initial conditions
\begin{equation*}
\left\{\begin{aligned}
\theta(0) &= \theta_0\\
\left.\dv{\theta}{t}\right|_{0} &= 0
\end{aligned}\right.
\end{equation*}
with $0<\theta_0<\pi$,
the bounded solution is given by
\begin{equation*}
\theta(t)
= 2\arcsin\!\left[k\sn\!\left(\frac{\omega_0T}{4}-\omega_0t,k^2\right)\right]
\leq \theta_0,
\end{equation*}
where
\begin{equation*}
%0<\theta_0<\pi,\quad
T = \frac{4K(k^2)}{\omega_0},\quad
k = \sin{\frac{\theta_0}{2}}
\end{equation*}
and where $\sn$ is the Jacobi elliptic functions,
and $K$ is complete elliptic integral of the first kind.
The first derivative is
\begin{equation*}
\dv{\theta}{t}
= -\frac{2 k \omega_0}{\sqrt{1-(k \sn\!\myarg)^2}} \cn\!\myarg\dn\!\myarg
\end{equation*}
with $\alpha = \omega_0T/4-\omega_0t$.
%\begin{equation*}
% \alpha = \frac{\omega_0T}{4}-\omega_0t.
%\end{equation*}
The maximum angular velocity is
\begin{equation*}
\Omega_\mathrm{max} = \left.\dv{\theta}{t}\right|_{T/2} = 2k\omega_0.
\end{equation*}
Therefore, the total energy is
\begin{equation*}
E = \frac{mg^2\Omega_\mathrm{max}^2}{2\omega_0^4}
= 2\frac{mg^2}{\omega_0^2}\sin^2\frac{\theta_0}{2}.
\end{equation*}
with kinetic and potential energy
\begin{align*}
K &= \frac{mg^2}{2\omega_0^4}\left(\dv{\theta}{t}\right)^2
= 2\frac{mg^2}{\omega_0^2}\left( \sin^2\frac{\theta_0}{2} - \sin^2\frac{\theta}{2} \right) \\
U &= 2\frac{mg^2}{\omega_0^2}\sin^2\frac{\theta}{2}.
\end{align*}
\end{minipage}
};
\end{tikzpicture}
% PENDULUM EQUATIONS - OPEN TRAJECTORIES
\begin{tikzpicture}[scale=1]
\message{^^JPendulum equations (open/unbounded)}
\def\myarg{\left(\frac{\Omega_0}{2}t,\frac{1}{k^2}\right)}
\node[align=left] at (0,0) {
\begin{minipage}{9.8cm}
The differential equation for a simple pendulum is
\begin{equation*}
\dv{^2\theta}{t^2} + \omega_0^2\sin\theta = 0.
\end{equation*}
For the initial conditions
\begin{equation*}
\left\{\begin{aligned}
\theta(0) &= 0 \\
\left.\dv{\theta}{t}\right|_{0} &= \Omega_0 %> 2\omega_0
\end{aligned}\right.
\end{equation*}
with $\Omega_0 > 2\omega_0$,
the unbounded solution is given by
\begin{equation*}
\theta(t)
= 2\arcsin\!\left[\sn\!\myarg\right]
\geq 0,
\end{equation*}
where
\begin{equation*}
%\Omega_0 > 2\omega_0,\quad
T = \frac{4K(k^2)}{\Omega_0},\quad
k = \frac{\Omega_0}{2\omega_0} > 2
\end{equation*}
and where sn is the Jacobi elliptic functions,
and $K$ is complete elliptic integral of the first kind.
Note that the solution $\theta(t)$ is continuous and increase monotonically,
so if one assumes $\abs{\arcsin x}\leq\pi/2$ for each real $x$,
\begin{equation*}
\theta(t)
= 2\pi n + (-1)^n 2\arcsin\!\left[\sn\!\myarg\right],
\end{equation*}
where $n=\lfloor(t+T/2)/T\rfloor=\lfloor t/T+1/2\rfloor$.\\
The first derivative is
\begin{equation*}
\dv{\theta}{t} = 2\Omega_0\dn\!\myarg,
\end{equation*}
The maximum angular velocity $\Omega_\mathrm{max}=\Omega_0$,
and the minimum is
\begin{equation*}
\Omega_\mathrm{min} = 2\omega_0\sqrt{k^2-1}.
\end{equation*}
\end{minipage}
};
\end{tikzpicture}
\end{document}
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MATLAB figures for the exact solution of a pendulum:
Closed solutions (bounded):
Open solutions (unbounded):
