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Diffstat (limited to 'vorlesungen/slides/10/so2.tex')
| -rw-r--r-- | vorlesungen/slides/10/so2.tex | 237 |
1 files changed, 124 insertions, 113 deletions
diff --git a/vorlesungen/slides/10/so2.tex b/vorlesungen/slides/10/so2.tex index b63a67e..dcbcdc8 100644 --- a/vorlesungen/slides/10/so2.tex +++ b/vorlesungen/slides/10/so2.tex @@ -8,123 +8,134 @@ \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} - \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} - \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} - \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} + \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} - \begin{block}{Differentialgleichung} - Gegeben: - \[ - A + \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} = - \dot\gamma(0) = \begin{pmatrix} 0 & -1 \\ 1 & \phantom-0 \end{pmatrix} - \] - Gesucht: - \[ \dot \gamma (t) = \gamma(t) A \qquad \gamma \in \gSO2 \] - \[ \Rightarrow \gamma(t) = \exp(At) \gamma(0) = \exp(At) \] - \end{block} -\end{column} -\begin{column}{0.48\textwidth} - \begin{block}{Lie-Algebra} - Potenzen von A: - \begin{align*} - A^2 &= -I & - A^3 &= -A & - A^4 &= I & - \ldots - \end{align*} - \end{block} -\end{column} -\end{columns} -Folglich: -\begin{align*} - \exp(At) - &= I + At - + A^2\frac{t^2}{2!} - + A^3\frac{t^3}{3!} - + A^4\frac{t^4}{4!} - + A^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*} - + \begin{pmatrix} + \cos t & -\sin t \\ + \sin t & \phantom-\cos t + \end{pmatrix} + \end{align*} + } \end{frame} \egroup |