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\author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz, Danny van Dyk} (UZH)}
\institute{UZH}
\title[Linear equation systems: exact methods]{Linear equation systems:exact methods}
\date{21 September 2016}


\begin{document}
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{
\setbeamertemplate{sidebar right}{\llap{\includegraphics[width=\paperwidth,height=\paperheight]{bubble2}}}
\begin{frame}[c]%{\phantom{title page}} 
\begin{center}
\begin{center}
	\begin{columns}
		\begin{column}{0.9\textwidth}
			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Chaos in ODEs}
		\end{column}
		\begin{column}{0.2\textwidth}
		  %\includegraphics[width=\textwidth]{SHiP-2}
		\end{column}
	\end{columns}
\end{center}
	\quad
	\vspace{3em}
\begin{columns}
\begin{column}{0.6\textwidth}
\flushright \vspace{-1.8em} {\fontspec{Trebuchet MS} \large Marcin Chrząszcz, Danny van Dyk\\\vspace{-0.1em}\small \href{mailto:mchrzasz@cern.ch}{mchrzasz@cern.ch}, \href{mailto:dany.van.dyk@gmail.com}{danny.van.dyk@gmail.com}}

\end{column}
\begin{column}{0.4\textwidth}
\includegraphics[height=1.3cm]{uzh-transp}
\end{column}
\end{columns}

\vspace{1em}
%		\footnotesize\textcolor{gray}{With N. Serra, B. Storaci\\Thanks to the theory support from M. Shaposhnikov, D. Gorbunov}\normalsize\\
\vspace{0.5em}
	\textcolor{normal text.fg!50!Comment}{Numerical Methods, \\ 16 November, 2016}
\end{center}
\end{frame}
}


\begin{frame}\frametitle{Classical mechanics formulation}

\ARROW This you should know by heart:\\{~}\\
\begin{columns}
\column{0.33\textwidth}
\ARROW Newton:
\begin{align*}
\overrightarrow{F} = \frac{d \overrightarrow{p}}{dt}
\end{align*}
\column{0.33\textwidth}
\ARROW Lagrange:
\begin{align*}
\mathcal{L}=T-V\\
\frac{\partial \mathcal{L} }{ \partial x} - \frac{d}{dt} \frac{\partial \mathcal{L}}{\partial \dot{x}}=0
\end{align*}
\column{0.33\textwidth}
\ARROW Hamilton:
\begin{align*}
\mathcal{H}=T+ V\\
\frac{d p}{d t}=-\frac{\partial \mathcal{H}}{\partial x}\\
\frac{d x }{d t}= \frac{\partial \mathcal{H}}{\partial p}
\end{align*}

\end{columns}

\ARROW Now what are advantages of each one?

\end{frame}




\backupbegin   

\begin{frame}\frametitle{Backup}


\end{frame}







\backupend			
\end{document}