Commit c0621536 authored by Holger Homann's avatar Holger Homann

added unziped altland files

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\documentclass[12pt,a4paper]{book}
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\includeonly{newton,
lagrange,
starrer_koerper,
hamilton,
chaos}
\begin{document}
\parindent0cm
\title{\huge \bf Theoretische Mechanik}
\date{Alexander Altland\\[1.0cm]
{\it Bitte im Skript vorhandene Fehler -- wahrscheinlich noch viele !
-- an mich zur{\"u}ckmelden.}}
%\author{{\bf Alexander Altland}\\
\maketitle
\tableofcontents
%\include{newton}
%\include{lagrange}
%\include{starrer_koerper}
\include{hamilton}
%\include{chaos}
\begin{thebibliography}{99}
\bibitem{arnold} V.I. Arnold, {\it Mathematical Methods of Classical
Mechanics}, second edition, Springer Verlag, 1989.
\bibitem{goldstein} H. Goldstein, {\it Classical Mechanics}, second
edition, Addison Wesley Publishing, 1980 ("Ubers.: {\it Klassische
Mechanik},
Aula Verlag, 1985.)
\bibitem{grossmann} S. Grossmann, {\it Mathematischer Einf"uhrungskurs
f"ur die Physik}, Teubner 1984.
\bibitem{jelitto} R.J. Jellito, {\it Theoretische Physik 1: Mechanik 1
\& 2}, Akademische Verlagsgesellschaft Wiesbaden, 1982.
\bibitem{ll} L.D. Landau and E.M. Lifshitz, {\it Course of Theoretical
Physics, Vol.1}, Butterworth Heinemann, 1999.
\bibitem{scheck} F.Scheck, {\it Mechanik}, 5. Auflage, Springer
Verlag, 1996.
\bibitem{schuster} H.G. Schuster, {\it Deterministic Chaos}, Physik
Verlag, 1984.
\bibitem[tabor] {tabor} M. Tabor, {\it Chaos and Integrability in Nonlinear
Dynamics}, Wiley, 1989
\end{thebibliography}
\end{document}
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\chapter{Newton Mechanik}
\label{cha:newton}
\section{Formale Grundlagen}
\label{sec:new:grundlagen}
Wir beginnen dieses Kapitels mit der formalen Entwicklung der
Newton'schen Mechanik. Der erste Schritt besteht in der Definition der
'Arena', in der sich die Mechanik abspielt, der Raum-Zeit und ihrer
Gallilei'schen Struktur
\subsection{Raum-Zeit}
\label{sec:new:galilei}
Das Konzept der Galilei'schen Raum-Zeit basiert wesentlich auf der
mathematischen Struktur des affinen Raumes ($\to$
Lin.Alg.):
\Def Ein {\bf $n$-dimensionaler affiner Raum} $A^4$ ist ein Tripel
$(M,{\cal R}^n,+)$ bestehend aus einer Punktmenge $M$, einem $n$-dimensionalen rellen
Vektorraum ${\cal R}^n$ und einer Addition $+:M \times {\cal R}^n \to
M$ mit den Eigenschaften
\begin{enumerate}
\item $a\in M,v,w\in {\cal R}^n:\qquad a+(v+w)=(a+v)+w$,
\item $a\in M,v\in {\cal R}^n:\qquad a+v=a \Leftrightarrow v=0$,
\item Zu je zwei Punkten $a,b\in M$ existiert ein $v\in {\cal R}^n$,
mit $a=b+v$.
\end{enumerate}
Einige Bemerkungen hierzu:
\begin{Titemize}
\item Ein affiner Raum hat viel mit einem Vektorraum gemein. Der
zentrale Unterschied ist, da"s, anders als bei Vektorr"aumen, kein
Ursprung ausgezeichnet ist; solange kein Referenzpunkt festgelegt
ist, sind alle Punkte gleichberechtigt.
\item Es macht keinen Sinn, von der 'Summe' $a+b$ zweier Punkte zu
sprechen! Die Differenz zweier Punkte, $a-b \in {\cal R}^n$
bezeichnet man als die entsprechende (parallele) Verschiebung.
\item Notation: Falls nicht anders festgelegt, verwenden wir fr"uhe
lateinische Buchstaben, $a,b,\dots$ f"ur Punkte eines affinen
Raumes, sp"ate, $v,w,\dots$ f"ur Elemente eines Vektorraumes.
\end{Titemize}
In der Mechanik spielt der r"aumliche Abstandsbegriff eine zentrale
Rolle. Wir wollen daher von vornherein vereinbaren, da"s alle
Vektorr"aume {\bf euklidisch}, also mit einem Skalarprodukt
ausgestattet sind. Für alle praktischen Zwecke verwenden wir das
nat"urliche Skalarprodukt $\langle \cdot ,\cdot \rangle$ des ${\cal R}^n$,
\[
v,w \in {\cal R}^n:\qquad \langle v,w \rangle \equiv
\sum_{i=1}^n v_i w_i,
\]
wobei $v_i$ die $i$-te Komponente des Vektors $i$ bezeichnet. Der {\it
Abstand} $\rho(a,b)$ zwischen zwei Punkten eines affinen Raumes ist
definiert durch,
$$
\rho(a,b) = \sqrt{\langle a-b,a-b\rangle}.
$$
Nach dieser Vorbereitung sind wir in der Lage, den Raum-Zeit
Begriff der klassischen Mechanik einzuf"uhren
\Def Die {\bf Galilei'sche Raum-Zeit} besteht aus drei Elementen:
\begin{enumerate}
\item Dem {\bf Universum}, einem vierdimensionalen affinen raum,
$A^4$. Die Elemente dieses Raumes bezeichnet man als 'Ereignisse',
manchmal auch als 'Welt-Punkte'
\item Der {\bf Zeit}, einer linearen Abbildung,
$t:{\cal R}^4\to {\cal R}$, von dem dem Universum zugeordneten
Vektorraum der Verschiebungen in die reellen Zahlen. Das
Zeitintervall zwischen den Ereignissen $a$ und $b$ is die Zahl
$t(b-a)$. Die Ereignisse hei"sen gleichzeitig, falls $t(b-a)=0$.
Die Menge der Ereignisse, die zu einem vorgegebenen Ereignis $a$
gleichzeitig sind, bildet in nat"urlicher Weise einen
dreidimensionalen affinen Unterraum des $A^4$. Man bezeichnet ihn
als den Raum gleichzeitiger Ereignisse, $A^3$.
\item Der {\bf Abstand zwischen zwei gleichzeitigen Ereignissen}
$a,b\in A^3$, ist $\rho(a,b)$ ist durch das Skalarprodukt auf dem
Verschiebungsraum ${\cal R}^3$ definiert. Vorsicht: Es macht
keinen Sinn, vom 'Abstand' zwischen allgemeinen,
nicht-gleichzeitigen Weltpunkten zu sprechen.
\end{enumerate}
Ein affiner Raum, der mit einer Galilei'schen Raum-Zeit Struktur
ausgestattet ist, hei"st {\it Galilei'scher Raum}.
Ein das Arbeiten mit affinen bzw. speziellen Galilei'schen R"aumen
erleichterndes Konzept sind {\bf Koordinatensysteme}. Ein
Koordinatensystem l"aßt sich folgenderma"sen einf"uhren:
\begin{enumerate}
\item Wir legen einen Punkt $o\in A^4$ als Ursprungspunkt fest.
\item Nach Konstruktion l"aßt sich jeder Punkt $a$ des Raumes als
$a=o+v$ mit einem $v\in {\cal R}^4$ darstellen.
\item Den ${\cal R}^4$ seinerseits k"onnen wir mittels einer
Orthonormalbasis $\{{\bf e}_0,{\bf e}_1,{\bf e}_2,{\bf e}_3\}$
aufspannen. Wir w"ahlen diese Vektoren zweckm"a"sigerweise so, da"s
${\bf e}_1,{\bf e}_2,{\bf e}_3$ den Raum der zu $o$ gleichzeitigen
Ereignisse erzeugen. Der Vektor ${\bf e}_0$ zeigt in die
Zeitrichtung.
Jeder Weltpunkt $a$ ist nun eindeutig als
$$
a=o + t {\bf e}_0 + \sum_{i=1}^3 x_i {\bf e}_i
$$
Darstellbar, d.h wir k"onnen jedes Ereignis "uber seine Koordinaten
$$
a \leftrightarrow \left(
\matrix{t\cr x_1\cr x_2\cr x_3}\right)
$$
identifizieren. Die Koordinate $t$ ist die Zeit\footnote{Beachten
Sie, da"s die zu der das Ereignis stattfindet, $(x_1,x_2,x_3)$ geben
seinen Ort an.
\end{enumerate}
\subsection{Galilei Transformationen}
\label{sec:new:galilei_trafo}
Die Galilei'sche Struktur spielt eine immens wichtige Rolle bei der
Entwicklung der klassischen Mechanik. Die bald einzuf"uhrenden
Newton'schen Gesetze werden ohne R"uckgriff auf diese Struktur nicht
vern"unftig erkl"aren lassen. Eine Frage, mit der wir immer wieder
konfrontiert werden sein ist folgende: Gegeben einen Galilei'schen
Raum $A^4$; was ist die maximale Menge von Transformationen, d.h.
affinen Abbildungen $g:A^4 \to A^4$, die seine Struktur invariant lassen?
'Invariant' hei"st hierbei, da"s (i) sich der zeitliche Abstand zwischen zwei
Ereignissen unter der Transformation nicht "andert,
$t(a-b)=t(g(a)-g(b)$, und (ii) der Abstand zwischen gleichzeitigen
Ereignissen invariant bleibt: $\rho(a,b)=\rho(g(a),g(b)$.
Um Abbildungen zwischen Punkten eines affinen Raumes effizient
studieren zu k"onnen, f"uhren wir ein weiteres Konzept ein,
Einige Beispiele von Transformationen, die diese Bedingung
offensichtlich erf"ullen, sind leicht formuliert:
\begin{enumerate}
\item g(
\end{enumerate}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "mechanik"
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\relax
\@input{newton.aux}
\@input{lagrange.aux}
\@input{starrer_koerper.aux}
\@input{hamilton.aux}
\@input{chaos.aux}
\bibcite{arnold}{1}
\bibcite{goldstein}{2}
\bibcite{grossmann}{3}
\bibcite{jelitto}{4}
\bibcite{ll}{5}
\bibcite{scheck}{6}
\bibcite{schuster}{7}
\bibcite{tabor}{tabor}
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