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-rw-r--r--buch/papers/lambertw/teil4.log1620
-rw-r--r--buch/papers/lambertw/teil4.tex81
2 files changed, 1701 insertions, 0 deletions
diff --git a/buch/papers/lambertw/teil4.log b/buch/papers/lambertw/teil4.log
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+++ b/buch/papers/lambertw/teil4.log
@@ -0,0 +1,1620 @@
+This is pdfTeX, Version 3.141592653-2.6-1.40.23 (TeX Live 2021/W32TeX) (preloaded format=pdflatex 2021.11.16) 4 APR 2022 20:41
+entering extended mode
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+**teil4.tex
+(./teil4.tex
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+! Undefined control sequence.
+l.6 \section
+ {Beispiel Verfolgungskurve
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+! LaTeX Error: Missing \begin{document}.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type H <return> for immediate help.
+ ...
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+l.6 \section{B
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+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no U in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no q in font nullfont!
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+Missing character: There is no i in font nullfont!
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+Missing character: There is no c in font nullfont!
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+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no u in font nullfont!
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+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
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+Missing character: There is no c in font nullfont!
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+Missing character: There is no o in font nullfont!
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+Missing character: There is no s in font nullfont!
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+Missing character: There is no s in font nullfont!
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+Missing character: There is no p in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
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+Missing character: There is no i in font nullfont!
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+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 6--23
+[]
+ []
+
+! Undefined control sequence.
+l.24 \subsection
+ {De finibus bonorum et malorum
+The control sequence at the end of the top line
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+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type H <return> for immediate help.
+ ...
+
+l.24 \subsection{D
+ e finibus bonorum et malorum
+You're in trouble here. Try typing <return> to proceed.
+If that doesn't work, type X <return> to quit.
+
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+
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+[]
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+
+)
+! Emergency stop.
+<*> teil4.tex
+
+*** (job aborted, no legal \end found)
+
+
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diff --git a/buch/papers/lambertw/teil4.tex b/buch/papers/lambertw/teil4.tex
new file mode 100644
index 0000000..74b6b02
--- /dev/null
+++ b/buch/papers/lambertw/teil4.tex
@@ -0,0 +1,81 @@
+%
+% teil3.tex -- Beispiel-File für Teil 3
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Beispiel Verfolgungskurve
+\label{lambertw:section:teil4}}
+\rhead{Beispiel Verfolgungskurve}
+In diesem Abschnitt wird rechnerisch das Beispiel einer Verfolgungskurve beschreiben.
+
+\subsection{Ziel bewegt sich auf einer Gerade
+\label{lambertw:subsection:malorum}}
+Das zu verfolgende Ziel \(Z\) wandert auf einer Gerade, wobei diese Gerade der \(y\)-Achse entspricht. Der Verfolger \(V\) startet auf einem beliebigen Punkt auf dem ersten Quadrant. Diese Anfangspunkte oder Anfangsbedingungen können wie folgt formuliert werden:
+\begin{equation}
+ Z
+ =
+ \left( \begin{array}{c} 0 \\ t \end{array} \right)
+ ;
+ V
+ =
+ \left( \begin{array}{c} x \\ y \end{array} \right)
+ \label{lambertw:equation2}
+\end{equation}
+Wenn man diese Startpunkte in die Gleichung der Verfolgungskurve einfügt ergibt sich folgender Ausdruck:
+\begin{equation}
+ \frac{\left( \begin{array}{c} 0-x \\ t-y \end{array} \right)}{\sqrt{x^2 + (t-y)^2}}
+ \circ
+ \left(\begin{array}{c} \dot{x} \\ \dot{y} \end{array}\right)
+ =
+ 1
+ \label{lambertw:equation3}
+\end{equation}
+Macht man den linken Term Bruchfrei und löst das Skalarprodukt auf, dann ergibt sich folgende DGL:
+\[
+ \left( \begin{array}{c} 0-x \\ t-y \end{array} \right)
+ \circ
+ \left(\begin{array}{c} \dot{x} \\ \dot{y} \end{array}\right)
+ = \sqrt{x^2 + (t-y)^2}\\
+\]
+\begin{equation}
+ -x \cdot \dot{x} + (t-y) \cdot \dot{y}
+ = \sqrt{x^2 + (t-y)^2}
+ \label{lambertw:equation4}
+\end{equation}
+Im nächsten Schritt quadriert man beide Seiten, erweitert den neu entstandenen quadratischen Term, bringt alles auf die linke Seite und klammert gemeinsames aus.
+\begin{align*}
+ ((t-y) \dot{y} - x \dot{x})^2
+ &= x^2 + (t-y)^2 \\
+ x^2 \dot{x}^2 - 2x(t-y) \dot{x} \dot{y} + (t-y)^2 \dot{y}
+ &= x^2 + (t-y)^2 \\
+ \dot{x}^2 x^2 - x^2 - 2x(t-y) \dot{x} \dot{y} + \dot{y}^2 (t-y)^2 - (t-y)^2
+ &= 0 \\
+ (\dot{x}^2 - 1) \cdot x^2 - 2x(t-y) \dot{x} \dot{y} + (\dot{y}^2 - 1) \cdot (t-y)^2
+ &= 0
+\end{align*}
+Der letzte Ausdruck kann mittels folgender Beziehung \(\dot{x}^2 + \dot{y}^2 = 1\) vereinfacht werden und anschliessend mit \(-1\) multiplizieren:
+\[
+ \underbrace{(\dot{x}^2 - 1)}_{\mathclap{-\dot{y}^2}} \cdot x^2 - 2x(t-y) \dot{x} \dot{y} + \underbrace{(\dot{y}^2 - 1)}_{\mathclap{-\dot{x}^2}} \cdot (t-y)^2
+ = 0
+\]
+\begin{align*}
+ - \dot{y}^2 \cdot x^2 - 2x(t-y) \dot{x} \dot{y} - \dot{x}^2 \cdot (t-y)^2
+ &= 0 \\
+ \dot{y}^2 \cdot x^2 + 2x(t-y) \dot{x} \dot{y} + \dot{x}^2 \cdot (t-y)^2
+ &= 0
+\end{align*}
+Im letzten Ausdruck erkennt man das Muster einer binomischen Formel, was den Ausdruck wesentlich vereinfacht:
+\begin{align*}
+ x^2 \dot{y}^2 + 2 \cdot x \dot{y} \cdot (t-y) \dot{x} + (t-y)^2 \dot{x}^2
+ &= 0 \\
+ (x \dot{y} + (t-y) \dot{x})^2
+ &= 0
+\end{align*}
+Wenn man nun beidseitig die Quadratwurzel zieht, dann ergibt sich im Vergleich zu \eqref{lambertw:equation4} eine wesentlich einfachere DGL:
+\begin{equation}
+ x \dot{y} + (t-y) \dot{x}
+ = 0
+ \label{lambertw:equation5}
+\end{equation}
+
+