update theo and num
This commit is contained in:
+1
-1
@@ -213,7 +213,7 @@
|
||||
}
|
||||
% replace all relations with align characters (&) and add the needed padding
|
||||
\regex_replace_all:nnN
|
||||
{ (\c{implies}&|&\c{implies}|\c{approx}&|&\c{approx}|\c{equiv}&|&\c{equiv}|=&|&=|\c{le}&|&\c{le}|\c{ge}&|&\c{ge}|&\c{stackrel}{.*?}{.*?}|\c{stackrel}{.*?}{.*?}&|&\c{neq}|\c{neq}&) }
|
||||
{ (\c{impliedby}&|&\c{impliedby}|\c{implies}&|&\c{implies}|\c{approx}&|&\c{approx}|\c{equiv}&|&\c{equiv}|=&|&=|\c{le}&|&\c{le}|\c{ge}&|&\c{ge}|&\c{stackrel}{.*?}{.*?}|\c{stackrel}{.*?}{.*?}&|&\c{neq}|\c{neq}&) }
|
||||
{ \c{kern} \u{l_tmp_dim_needed} \1 \c{kern} \u{l_tmp_dim_needed} }
|
||||
\l__lec_text_tl
|
||||
\l__lec_text_tl
|
||||
|
||||
+1
-1
Submodule sose2020/ana/lectures updated: f4c36cb129...0368910241
Binary file not shown.
@@ -98,7 +98,7 @@
|
||||
\quad \text{also insbes.} \quad m \Vert u_{k} \Vert_{\infty} \le \Vert u_k \Vert_1
|
||||
\quad \forall k \in \N
|
||||
.\]
|
||||
Wegen $\Vert u_k \Vert = 1$ $\forall k \in \N$, folgt also
|
||||
Wegen $\Vert u_k \Vert_{\infty} = 1$ $\forall k \in \N$, folgt also
|
||||
\[
|
||||
m \le \Vert u_k\Vert_1 \quad \forall k \in \N \quad \contr \text{ zu } \Vert u_k \Vert
|
||||
\xrightarrow{k \to \infty} 0
|
||||
|
||||
Binary file not shown.
@@ -33,8 +33,10 @@
|
||||
\frac{\partial L}{\partial \dot{\vartheta}}
|
||||
&= M R^2 \dot{\vartheta} =: p_{\vartheta} \\
|
||||
\intertext{Damit folgt die Hamilton Funktion}
|
||||
H &= \frac{p_{\vartheta}^2}{mR^2} - L = \frac{p_{\vartheta}^2}{mR^2}
|
||||
- \frac{1}{2} mR^2 (\dot{\vartheta}^2 + \omega^2\sin^2\vartheta) + mg R \cos\vartheta
|
||||
H &= \frac{p_{\vartheta}^2}{mR^2} - L \\
|
||||
&= \frac{p_{\vartheta}^2}{mR^2}
|
||||
- \frac{1}{2} mR^2 (\dot{\vartheta}^2 + \omega^2\sin^2\vartheta) + mg R \cos\vartheta \\
|
||||
&= \frac{p_{\vartheta}^2}{2mR^2} - \frac{1}{2} \omega^2 \sin^2\vartheta + mgR\cos\vartheta
|
||||
.\end{align*}
|
||||
\item Es liegt hier zwar ein Potential vor, allerdings ist
|
||||
die kinetische Energie des Systems gegeben als
|
||||
@@ -51,23 +53,23 @@
|
||||
\begin{align*}
|
||||
\frac{\partial H}{\partial p_{\vartheta}} &= \frac{2 p_{\vartheta}}{mR^2} - \frac{p_{\vartheta}}{mR^2}
|
||||
= \frac{p_{\vartheta}}{mR^2} = \dot{\vartheta} \\
|
||||
\frac{\partial H}{\partial \vartheta} &= mR^2 \omega^2 \sin\vartheta \cos \vartheta - mg R \sin\vartheta
|
||||
= mR\sin\vartheta(R\omega^2 \cos\vartheta - g) = - \dot{p}_{\vartheta}
|
||||
\frac{\partial H}{\partial \vartheta} &= mR^2 \omega^2 \sin\vartheta \cos \vartheta + mg R \sin\vartheta
|
||||
= mR\sin\vartheta(R\omega^2 \cos\vartheta + g) = \dot{p}_{\vartheta}
|
||||
.\end{align*}
|
||||
\item
|
||||
Für stationäre Lösungen gilt $\dot{\vartheta} = 0$, also $p_{\vartheta} = 0$, damit folgt
|
||||
\begin{align*}
|
||||
mR \sin\vartheta (R \omega^2 \cos \vartheta -g ) &= 0
|
||||
mR \sin\vartheta (R \omega^2 \cos \vartheta +g ) &= 0
|
||||
.\end{align*}
|
||||
Als stationäre Lösungen folgen damit $\vartheta_1 = 0$ und $\vartheta_2 = \pi$, denn dann ist
|
||||
$\sin\vartheta = 0$. Für $\vartheta_3 = \frac{\pi}{2}$, folgt $- mRg = 0$, dies ist also
|
||||
$\sin\vartheta = 0$. Für $\vartheta_3 = \frac{\pi}{2}$, folgt $mRg = 0$, dies ist also
|
||||
nur möglich, falls Masse oder Radius 0 sind.
|
||||
|
||||
Für $R \neq 0$, $m \neq 0$, $\sin\vartheta \neq 0$ und $\omega \neq 0$ folgt
|
||||
\begin{align*}
|
||||
\underbrace{\sin\vartheta}_{\neq 0}(R \omega^2 \cos\vartheta - g) &= 0 \\
|
||||
\implies R \omega^2 \cos\vartheta - g &= 0 \\
|
||||
\implies \vartheta &= \arccos \left( \frac{g}{R\omega^2} \right)
|
||||
\underbrace{\sin\vartheta}_{\neq 0}(R \omega^2 \cos\vartheta + g) &= 0 \\
|
||||
\implies R \omega^2 \cos\vartheta + g &= 0 \\
|
||||
\implies \vartheta &= \arccos \left( -\frac{g}{R\omega^2} \right)
|
||||
.\end{align*}
|
||||
Wie zu erwarten ist $\vartheta \xrightarrow{\omega \to \infty} \frac{\pi}{2}$.
|
||||
\end{enumerate}
|
||||
@@ -134,20 +136,21 @@
|
||||
\frac{1}{r^2} \frac{\partial}{\partial r}\left( r^2 \frac{\partial R}{\partial r} \right)
|
||||
+ \frac{c}{v^2}R = 0
|
||||
.\end{align*}
|
||||
\item Sei $c > 0$. Ansatz: $R(r) = \frac{\tilde{R}(r)}{r}$.
|
||||
\item Sei $c > 0$ und $\omega^2 = c$. Ansatz: $R(r) = \frac{\tilde{R}(r)}{r}$.
|
||||
Aus der DGL für $R$ folgt
|
||||
\begin{align*}
|
||||
&\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\tilde{R} - r \frac{\partial \tilde R}{\partial r^2}}{r^2} \right) + \frac{c}{v^2} \frac{\tilde R }{r} = 0 \\
|
||||
\implies &\frac{1}{r} \left( - \frac{\partial^2 \tilde R}{\partial r}
|
||||
+ \frac{c}{v^2} \tilde R \right) = 0 \\
|
||||
\implies & - \frac{\partial^2 \tilde R}{\partial r^2} + \frac{c}{v^2} \tilde R = 0 \\
|
||||
\implies & \tilde{R} = A\exp\left(\sqrt{\frac{c}{v^2}}r\right) + B\exp\left(-\sqrt{\frac{c}{v^2}}r\right) \\
|
||||
\implies & R = \frac{A\exp\left(\sqrt{\frac{c}{v^2}}r\right) + B\exp\left(-\sqrt{\frac{c}{v^2}}r\right)}{r}
|
||||
&\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{r \frac{\partial \tilde R}{\partial r} - \tilde{R}}{r^2} \right) + \frac{\omega^2}{v^2} \frac{\tilde R }{r} = 0 \\
|
||||
\implies &\frac{1}{r} \left( \frac{\partial^2 \tilde R}{\partial r}
|
||||
+ \frac{\omega^2}{v^2} \tilde R \right) = 0 \\
|
||||
\implies & \frac{\partial^2 \tilde R}{\partial r^2} + \frac{\omega^2}{v^2} \tilde R = 0 \\
|
||||
\implies & \tilde{R} = A \sin\left( \frac{\omega}{v}r \right) + B \cos\left( \frac{\omega}{v}r \right) \\
|
||||
\implies & R = \frac{A \sin\left( \frac{\omega}{v}r \right) + B \cos\left( \frac{\omega}{v}r \right)}{r}
|
||||
\intertext{Mit der Vorraussetzung $R$ bei $r=0$ stetig folgt $B=0$:}
|
||||
&R = \frac{A}{r} \sin \left( \frac{\omega}{v}r \right)
|
||||
.\end{align*}
|
||||
Damit folgt
|
||||
\[
|
||||
q = T \cdot R = \frac{A_0}{r} \cos(\sqrt{c} t - \delta)
|
||||
\left( A\exp\left(\sqrt{\frac{c}{v^2}}r\right) + B\exp\left(-\sqrt{\frac{c}{v^2}}r\right)\right)
|
||||
q = T \cdot R = \frac{A_0}{r} \cos(\omega t - \delta) \sin\left(\frac{\omega}{v} r\right)
|
||||
.\]
|
||||
\end{enumerate}
|
||||
\end{aufgabe}
|
||||
|
||||
Reference in New Issue
Block a user