Euler-Lagrange modeling

.gitignore

@@ -23,3 +23,4 @@
 *.synctex.gz
 *.bak
 *.mp4
+*.ps

modeling.tex

+\documentclass[12pt,a4paper]{report}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{tikz}
+\usetikzlibrary{patterns,angles,calc,quotes}
+
+\begin{document}
+
+\section*{Dynamics from Euler-Lagrange equations}
+
+\begin{figure}
+\begin{center}
+\begin{tikzpicture}[thick,>=latex]
+    \begin{scope}
+    %\draw[step=1cm,gray,very thin] (-1,-1) grid (4,5);
+    \draw[->,thick] (-0.1,0) -- (4,0) node[anchor=west]{x}; 
+    \draw[->,thick] (0,-0.1) -- (0,4.5) node[anchor=south]{y};
+    \coordinate (cart) at (2,0);
+    \coordinate (mass1) at ($(cart) + (104:4)$);
+    \coordinate (mass2) at ($(cart) + (127:2.5)$);
+    \coordinate (upp) at ($(cart) + (0,5)$);
+    
+    \draw[draw=black,fill=black] (cart) circle (.1cm) node (cart){};
+    \draw[dashed] (cart.center) -- (upp) {};
+    \node at (cart.south) [anchor=north] {$(x,0)$}; 
+    
+    \draw (cart.center) -- (mass1)  node[midway,xshift=2mm,yshift=5mm] (mid1) {};
+    \draw[draw=black,fill=white] (mass1.center) circle (.15cm);
+    \node at (mass1.north) [anchor=south,xshift=1mm,yshift=1mm] { $(x_1,y_1)$};
+    \pic["$\theta_1$", draw=black, ->, angle eccentricity=1.15, angle radius=2.2cm]
+    {angle = upp--cart--mass1};
+    \draw (cart.center) -- (mass2) node[midway,xshift=1mm,yshift=3mm] (mid2) {};
+    \draw[draw=black,fill=white] (mass2.center) circle (.15cm);
+    \node at (mass2.north) [anchor=south,xshift=1mm,yshift=1mm] { $(x_2,y_2)$};
+    %\pic [draw, ->,  angle eccentricity=1] {angle = upp--cart--mass2};
+    \pic["$\theta_2\;$", draw=black, ->, angle eccentricity=1.25, angle radius=1.2cm]
+    {angle = upp--cart--mass2};
+    \end{scope}
+\end{tikzpicture}
+\label{fig01}
+\caption{Two pendulums on a moving cart}
+\end{center}
+\end{figure}
+Two pendulums with lengths $l_1$ and $l_2$ and masses $m_1$ and $m_2$ are mounted on a moving cart with mass $M$, see Figure~\ref{fig01}. Introduce $L = T - V$ where kinetick and potential energies are given by
+\begin{align}
+T &= \frac{1}{2}M\dot x^2 + \frac{1}{2}m_1(\dot x_1^2 + \dot y_1^2) + \frac{1}{2}m_2(\dot x_2^2 + \dot y_2^2)\\
+V &= m_1gl_1 c_1 + m_2gl_2c_2
+\end{align}
+where $c_i$ and $s_i$ and short for $\cos(\theta_i)$ and $\sin(\theta_i)$, $i=1,2$ respectively.
+Note that $x_i = x-l_is_i$ and $y_i = l_ic_i$. 
+The Euler-Lagrange equations
+\begin{align}
+\begin{cases}
+0=\dfrac{\partial L}{\partial \theta_i} - \dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{\theta}_i} \right), & i=1,2 \\
+F=\dfrac{\partial L}{\partial x} - \dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{x}} \right).
+\end{cases}
+\end{align}
+give after some calculations that
+\begin{align}
+l_i \ddot \theta_i &= gs_i + c_i \ddot x, \quad i=1,2\\
+F&=M\ddot x + m_1(\ddot x - l_1c_1\ddot \theta_1 + l_1s_1\dot{ \theta}_1^2) + m_2(\ddot x - l_2c_2\ddot \theta_2+ l_2s_2\dot{ \theta}_2^2).
+\end{align}
+For $m_i\approx 0$ and setting $u = \frac{F}{M}$, this simplifies to
+\begin{align}
+l_1 \ddot \theta_1 &= g\sin(\theta_1) + \cos(\theta_1) u\\
+l_2 \ddot \theta_2 &= g\sin(\theta_2) + \cos(\theta_2) u\\
+\ddot x &= u
+\end{align}
+
+\end{document}
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