From 4313f2c207d5d60171898ccfd4c3b3d0d2fb4a75 Mon Sep 17 00:00:00 2001
From: Roy Seitz <roy.seitz@ost.ch>
Date: Sun, 18 Apr 2021 17:49:56 +0200
Subject: =?UTF-8?q?Pr=C3=A4sentation=20feritg.?=
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---
 vorlesungen/slides/10/so2.tex | 237 ++++++++++++++++++++++--------------------
 1 file changed, 124 insertions(+), 113 deletions(-)

(limited to 'vorlesungen/slides/10/so2.tex')

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
-- 
cgit v1.2.1

