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authorJODBaer <JODBaer@github.com>2021-04-20 12:38:29 +0200
committerJODBaer <JODBaer@github.com>2021-04-20 12:38:29 +0200
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+%
+% so2.tex -- Illustration of so(2) -> SO(2)
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{Von der Lie-Gruppe zur -Algebra}
+ \vspace{-20pt}
+ \begin{columns}[t,onlytextwidth]
+ \begin{column}{0.48\textwidth}
+ \uncover<1->{
+ \begin{block}{Lie-Gruppe}
+ Darstellung von \gSO2:
+ \begin{align*}
+ \mathbb R
+ &\to
+ \gSO2
+ \\
+ t
+ &\mapsto
+ \begin{pmatrix}
+ \cos t & -\sin t \\
+ \sin t & \phantom-\cos t
+ \end{pmatrix}
+ \end{align*}
+ \end{block}
+ }
+ \uncover<2->{
+ \begin{block}{Ableitung am neutralen Element}
+ \begin{align*}
+ \frac{d}{d t}
+ &
+ \left.
+ \begin{pmatrix}
+ \cos t & -\sin t \\
+ \sin t & \phantom-\cos t
+ \end{pmatrix}
+ \right|_{ t = 0}
+ \\
+ =
+ &
+ \begin{pmatrix} -\sin0 & -\cos0 \\ \phantom-\cos0 & -\sin0 \end{pmatrix}
+ =
+ \begin{pmatrix} 0 & -1 \\ 1 & \phantom-0 \end{pmatrix}
+ \end{align*}
+ \end{block}
+ }
+ \end{column}
+ \begin{column}{0.48\textwidth}
+ \uncover<3->{
+ \begin{block}{Lie-Algebra}
+ Darstellung von \aso2:
+ \begin{align*}
+ \mathbb R
+ &\to
+ \aso2
+ \\
+ t
+ &\mapsto
+ \begin{pmatrix}
+ 0 & -t \\
+ t & \phantom-0
+ \end{pmatrix}
+ \end{align*}
+ \end{block}
+ }
+ \end{column}
+ \end{columns}
+\end{frame}
+
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{Von der Lie-Algebra zur -Gruppe}
+ \vspace{-20pt}
+ \begin{columns}[t,onlytextwidth]
+ \begin{column}{0.48\textwidth}
+ \uncover<1->{
+ \begin{block}{Differentialgleichung}
+ Gegeben:
+ \[
+ J
+ =
+ \dot\gamma(0) = \begin{pmatrix} 0 & -1 \\ 1 & \phantom-0 \end{pmatrix}
+ \]
+ Gesucht:
+ \[ \dot \gamma (t) = J \gamma(t) \qquad \gamma \in \gSO2 \]
+ \[ \Rightarrow \gamma(t) = \exp(Jt) \gamma(0) = \exp(Jt) \]
+ \end{block}
+ }
+ \end{column}
+ \begin{column}{0.48\textwidth}
+ \uncover<2->{
+ \begin{block}{Lie-Algebra}
+ Potenzen von $J$:
+ \begin{align*}
+ J^2 &= -I &
+ J^3 &= -J &
+ J^4 &= I &
+ \ldots
+ \end{align*}
+ \end{block}
+ }
+ \end{column}
+ \end{columns}
+\uncover<3->{
+ Folglich:
+ \begin{align*}
+ \exp(Jt)
+ &= I + Jt
+ + J^2\frac{t^2}{2!}
+ + J^3\frac{t^3}{3!}
+ + J^4\frac{t^4}{4!}
+ + J^5\frac{t^5}{5!}
+ + \ldots \\
+ &= \begin{pmatrix}
+ \vspace*{3pt}
+ 1 - \frac{t^2}{2} + \frac{t^4}{4!} - \ldots
+ &
+ -t + \frac{t^3}{3!} - \frac{t^5}{5!} + \ldots
+ \\
+ t - \frac{t^3}{3!} + \frac{t^5}{5!} - \ldots
+ &
+ 1 - \frac{t^2}{2!} + \frac{t^4}{4!} - \ldots
+ \end{pmatrix}
+ =
+ \begin{pmatrix}
+ \cos t & -\sin t \\
+ \sin t & \phantom-\cos t
+ \end{pmatrix}
+ \end{align*}
+ }
+\end{frame}
+\egroup