From a0fcb885c26d00a7b85f9ad65ecb6f46609ff915 Mon Sep 17 00:00:00 2001
From: Roger Wolf <roger.wolf@kit.edu>
Date: Thu, 19 Oct 2023 11:31:39 +0200
Subject: [PATCH] upload intermediate result to allow for a reasonable branch
 off

---
 Kreisel/doc/Hinweise-Aufgabe-1.md    |  81 ++++++++++-----------------
 Kreisel/figures/Traegheitsmoment.odg | Bin 19712 -> 24727 bytes
 Kreisel/figures/Traegheitsmoment.png | Bin 38003 -> 47405 bytes
 3 files changed, 30 insertions(+), 51 deletions(-)

diff --git a/Kreisel/doc/Hinweise-Aufgabe-1.md b/Kreisel/doc/Hinweise-Aufgabe-1.md
index c076a09..417eec9 100644
--- a/Kreisel/doc/Hinweise-Aufgabe-1.md
+++ b/Kreisel/doc/Hinweise-Aufgabe-1.md
@@ -2,15 +2,15 @@
 
 ## Aufgabe 1: Physik starrer Körper [1/2]
 
-### Von Winkelgeschwindigkeit bis Trägheitstensor
+### Winkelgeschwindigkeit und Trägheitstensor
 
-Zur Beschreibung der Dynamik des Massepunkts in der Mechanik verwenden wir die physikalischen Größen Geschwindigkeit $\vec{v}$, Impuls $\vec{p}$ und Kraft $\vec{F}$. Mit dem Kreisel betrachten wir einen ausgedehnten, starren Körper, in Rotation, zu dessen Beschreibung wir zu $\vec{v}$, $\vec{p}$ und $\vec{F}$ äquivalente Größen heranziehen:  
+Zur Beschreibung der Dynamik des Massepunkts in der Mechanik verwenden wir die physikalischen Größen Geschwindigkeit ($\vec{v}$), Impuls ($\vec{p}$) und Kraft ($\vec{F}$). Mit dem Kreisel betrachten wir einen ausgedehnten, starren Körper, in Rotation, zu dessen Beschreibung wir zu $\vec{v}$, $\vec{p}$ und $\vec{F}$ äquivalente Größen heranziehen:  
 
 - Das Äquivalent zu $\vec{v}$ ist die **Winkelgeschwindigkeit** $\vec{\omega}$; 
 - das Äquivalent zu $\vec{p}$ ist der **Drehimpuls** $\vec{L}$; und 
 - das Äquivalent zu $\vec{F}$ ist das **Drehmoment** $\vec{M}$.  
 
-Für ein gegebenes Massenelement $\delta m$ wird der Zusammenhang dieser Größen zueinander durch das äußere Produkt ([Kreuzprodukt](https://de.wikipedia.org/wiki/Kreuzprodukt)) "$\times$" mit dem Ortsvektor $\vec{r}$ des Massenelements hergestellt: 
+Für ein gegebenes Massenelement $\mathrm{d}m$ wird der Zusammenhang dieser Größen zueinander durch das äußere Produkt ([Kreuzprodukt](https://de.wikipedia.org/wiki/Kreuzprodukt)) "$\times$" mit dem Ortsvektor $\vec{r}$ des Massenelements hergestellt: 
 $$
 \begin{equation*}
 \vec{v} = \vec{r}\times \vec{\omega}; \qquad
@@ -18,28 +18,28 @@ $$
 \vec{M} = \vec{r}\times \vec{F}. \\
 \end{equation*}
 $$
-Ein Zusammenhang zwischen $\vec{L}$ und $\vec{\omega}$ ergibt sich aus
+Ein Zusammenhang zwischen $\vec{L}$ und $\vec{\omega}$ ergibt sich daraus zu 
 $$
 \begin{equation}
-\vec{L} = \vec{r}\times\vec{p} = \delta m\left(\vec{r}\times\vec{v}\right) = \delta m\left(\vec{r}\times\left(\vec{r}\times\vec{\omega}\right)\right).
+\vec{L} = \vec{r}\times\vec{p} = \mathrm{d}m \left(\vec{r}\times\vec{v}\right) = \mathrm{d}m \left(\vec{r}\times\left(\vec{r}\times\vec{\omega}\right)\right).
 \end{equation}
 $$
-Um das doppelte Kreuzprodukt in Gleichung **(1)** weiter zu verstehen, können wir auf eine Regel aus der [analytischen Geometrie](https://de.wikipedia.org/wiki/Analytische_Geometrie) (die sog. "bac-cab"-Regel) zurückgreifen:
+Um das doppelte Kreuzprodukt in Gleichung **(1)** weiter zu aufzulösen, können wir auf eine Regel aus der [analytischen Geometrie](https://de.wikipedia.org/wiki/Analytische_Geometrie) (die sog. "bac-cab"-Regel) zurückgreifen:
 $$
 \begin{equation*}
 \vec{a}\times\vec{b}\times\vec{c} = \big(\vec{b}\cdot\vec{a}\big)\,\vec{c} - \big(\vec{c}\cdot\vec{a}\big)\,\vec{b}
 \end{equation*}
 $$
-Das Kreuzprodukt $\vec{b}\times\vec{c}$ kann durch einen Vektor $\vec{\kappa}$ beschrieben werden, der senkrecht auf die aus $\vec{b}$ und $\vec{c}$ aufgespannte Ebene steht. Das erneute Kreuzprodukt $\vec{a}\times\vec{\kappa}$ führt auf einen Vektor, der wiederum in diese Ebene fällt. Als Konsequenz kann dieser Vektor als eine Linearkombination aus $\vec{b}$ und $\vec{c}$ geschrieben werden. Die Linearkombination ergibt sich aus den vorangestellten [Skalarprodukten](https://de.wikipedia.org/wiki/Skalarprodukt). 
+Das Kreuzprodukt $\vec{b}\times\vec{c}$ kann durch einen Vektor $\vec{\kappa}$ beschrieben werden, der senkrecht auf die aus $\vec{b}$ und $\vec{c}$ aufgespannte Ebene steht. Das erneute Kreuzprodukt $\vec{a}\times\vec{\kappa}$ führt auf einen Vektor, der wiederum in diese Ebene fällt. Als Konsequenz kann dieser Vektor als eine Linearkombination aus $\vec{b}$ und $\vec{c}$ geschrieben werden. Diese Linearkombination ergibt sich aus den vorangestellten [Skalarprodukten](https://de.wikipedia.org/wiki/Skalarprodukt). 
 
 Anwendung auf Gleichung **(1)** führt auf: 
 $$
 \begin{equation}
 \begin{split}
-&\vec{L} = \delta m\left(r^{2}\vec{\omega} - \big(\vec{\omega}\cdot\vec{r}\big)\vec{r}\right)\\
+&\vec{L} = \mathrm{d}m \left(r^{2}\vec{\omega} - \big(\vec{\omega}\cdot\vec{r}\big)\vec{r}\right)\\
 &\\
-&L_{i} = \underbrace{\delta m\left(r^{2}\delta_{ij} - r_{i}r_{j}\right)}\omega_{j}.\\
-&\hphantom{L_{i} = \delta m\,r^{2}\,\,}\equiv \Theta_{ij}\\
+&L_{i} = \underbrace{\mathrm{d}m \left(r^{2}\delta_{ij} - r_{i}r_{j}\right)}\omega_{j}.\\
+&\hphantom{L_{i} = \mathrm{d}m \,r^{2}\,}\equiv \Theta_{ij}\\
 \end{split}
 \end{equation}
 $$
@@ -58,20 +58,21 @@ $$
 bezeichnen wir als [Trägheitstensor](https://de.wikipedia.org/wiki/Tr%C3%A4gheitstensor). Aufgrund seines Transformationsverhaltens unter Drehungen im Raum handelt es sich um einen Tensor 2. Stufe. Konkret als Matrix ausgeschrieben hat $\boldsymbol{\Theta}$ die Form: 
 $$
 \begin{equation}
-\boldsymbol{\Theta} = \delta m
+\boldsymbol{\Theta} = \mathrm{d}m
 \left(
 \begin{array}{ccc}
 r^{2}-x^{2} & x\,y & x\,z \\
 y\,x & r^{2}-y^{2} & y\,z \\
 z\,x & z\,y & r^{2}-z^{2} \\
 \end{array}
-\right)
+\right).
 \end{equation}
 $$
+Der Trägheitstensor ist durch seine Konstruktion symmetrisch ($\Theta_{ij}=\Theta_{ji}$). Die drei Diagonalelemente von $\boldsymbol{\Theta}$ (die wir im Folgenden auch mit $(\theta_{i};\hspace{0.1cm}i=x,y,z)$ bezeichnen werden) heissen [Trähgeitsmomente](https://de.wikipedia.org/wiki/Tr%C3%A4gheitsmoment). Die drei nicht-diagonalen Elemente heissen [Deviationsmomente](https://de.wikipedia.org/wiki/Deviationsmoment).
 
-### Bestimmung von $\vec{L}$ aus $\vec{\omega}$ als Eigenwertproblem  
+### Eigenwertproblem  
 
-Bei Gleichung **(2)** handelt es sich um ein gekoppeltes [lineares Gleichungssystem](https://de.wikipedia.org/wiki/Lineares_Gleichungssystem) der Form
+Schreibt man Gleichung **(2)** aus erhält man ein gekoppeltes [lineares Gleichungssystem](https://de.wikipedia.org/wiki/Lineares_Gleichungssystem) der Form
 $$
 \begin{equation}
 \left(
@@ -96,38 +97,40 @@ L_{z} \\
 \omega_{y} \\
 \omega_{z} \\
 \end{array}
-\right)
+\right),
 \end{equation}
 $$
-für das $\boldsymbol{\Theta}$ einer linearen Abbildung von $\mathbb{R}^{3}\to\mathbb{R}^{3}$ in der Darstellung einer $3\times3$-Matrix entspricht. In bestimmten Fällen kann ein solches Gleichungssystem, durch geeignete Transformationen 
+wofür wir $\boldsymbol{\Theta}$ als $3\times3$-Matrix verwendet haben. In bestimmten Fällen kann ein solches Gleichungssystem, durch eine Transformation $\bold{U}$ auf eine geeignete Basis 
 $$
 \begin{equation*}
-\boldsymbol{\widetilde{\Theta}} = \bold{U}^{-1}\cdot\boldsymbol{\Theta}\cdot\bold{U}
+\boldsymbol{\widetilde{\Theta}} = \bold{U}\cdot\boldsymbol{\Theta}\cdot\bold{U}^{-1}
 \end{equation*}
 $$
-entkoppelt werden, so dass $\boldsymbol{\widetilde{\Theta}}$ die Form einer [Diagonalmatrix](https://de.wikipedia.org/wiki/Diagonalmatrix) annimmt. Die Suche nach solchen Abbildungen $\bold{U}$ bezeichnet man als [Eigenwertproblem](https://de.wikipedia.org/wiki/Eigenwerte_und_Eigenvektoren). Die Matrix $\bold{U}$ führt die Basisvektoren des Vektors 
+entkoppelt werden, so dass $\boldsymbol{\widetilde{\Theta}}$ die Form einer [Diagonalmatrix](https://de.wikipedia.org/wiki/Diagonalmatrix) annimmt. Die Suche nach solchen Abbildungen $\bold{U}$ bezeichnet man als [Eigenwertproblem](https://de.wikipedia.org/wiki/Eigenwerte_und_Eigenvektoren). Die Matrix $\boldsymbol{\widetilde{\Theta}}$ führt die Basisvektoren des Vektors 
 $$
 \begin{equation*}
 \vec{\tilde{\omega}} = \bold{U}\cdot\vec{\omega}
 \end{equation*}
 $$
-bis auf die Multiplikation mit dem Eigenwert $\theta_{i}$ in sich selbst über. Die Transformation $\bold{U}$ entspricht also einem Wechsel in eine Einheitsbasis, die durch $\boldsymbol{\widetilde{\Theta}}$ in sich selbst abgebildet wird. 
+bis auf die Multiplikation mit dem Eigenwert $\theta_{i}$ in sich selbst über. Die Transformation $\bold{U}$ entspricht also einem Wechsel von einer beliebigen Orthonormalbasis in eine Orthonormalbasis, die durch $\boldsymbol{\widetilde{\Theta}}$ in sich selbst abgebildet wird. 
 
-Wie Sie sich selbst aus Gleichung **(3)** überzeugen können besitzt der Trägheitstensor die Eigenschaft **symmetrisch** zu sein, was für die Lösung des Eigenwertproblems weitreichende Konsequenzen hat: 
+### Hauptachsentransformation
 
-- Für symmetrische Abbildung ist das Eigenwertproblem **immer lösbar**. Unter den Eigenwerten kann jedoch [Entartung](https://de.wikipedia.org/wiki/Eigenwerte_und_Eigenvektoren#Berechnung_der_Eigenwerte) vorliegen. 
+Die Eigenschaft, dass der Trägheitstensor aus Gleichung **(3)** **symmetrisch** ist, hat für die Lösung des Eigenwertproblems die folgenden weitreichenden Konsequenzen: 
+
+- Für symmetrische Abbildung ist das Eigenwertproblem **immer lösbar**. Die Eigenwerte können allerdings n-fach [entartet](https://de.wikipedia.org/wiki/Eigenwerte_und_Eigenvektoren#Berechnung_der_Eigenwerte) vorliegen. 
 - Die Eigenwerte sind **immer reell**.
 - Die durch die zugehörigen Eigenvektoren aufgespannten Unterräume stehen **immer senkrecht aufeinander**.
 
-Die Matrizen $\bold{U}$ zur Diagonalisierung von $\boldsymbol{\Theta}$ sind Rotationsmatrizen $\bold{R}$ mit der definierenden Eigenschaft: 
+Die Matrizen $\bold{U}$ zur Diagonalisierung von $\boldsymbol{\Theta}$ sind in diesem Fall Rotationsmatrizen $\bold{R}$ mit der (definierenden) Eigenschaft: 
 $$
 \begin{equation*}
 \bold{R}^{-1} = \bold{R}^{\intercal}.
 \end{equation*}
 $$
-In diesem Fall hat die Lösung des Eigenwertproblems also eine **anschauliche geometrische Bedeutung**: 
+Die Lösung des Eigenwertproblems gewinnt dadurch eine **anschauliche geometrische Bedeutung**: 
 
-Es ist die Drehung von einem allgemeinen in ein spezielles Bezugssystem, in dem die Eigenvektoren des Problems parallel zu den [Hauptträgheitsachsen](https://de.wikipedia.org/wiki/Haupttr%C3%A4gheitsachse) des starren Körpers verlaufen. Diese Achsen sind durch die Form und Massenbelegung des Körpers vorgegeben. 
+Es handelt sich um die Drehung von einem allgemeinen in ein spezielles Bezugssystem, in dem die Eigenvektoren des Problems aus Gleichung **(2)** parallel zu den [Hauptträgheitsachsen](https://de.wikipedia.org/wiki/Haupttr%C3%A4gheitsachse) des starren Körpers verlaufen. Diese Achsen sind durch die Form und Massenbelegung des Körpers vorgegeben. 
 
 Das Eigenwertproblem erhält aufgrund dieser Besonderheiten den eigenen Namen [Hauptachsentransformation](https://de.wikipedia.org/wiki/Hauptachsentransformation). Nach Hauptachsentransformation hat Gleichung **(4)** die einfache Form: 
 $$
@@ -161,33 +164,9 @@ $$
 L_{i} = \theta_{i}\,\delta_{ij}\,\omega_{j}.
 \end{equation*}
 $$
-
-### Bestimmung eines beliebigen Trägheitsmoments aus dem Trägheitstensor
-
-Es macht nur Sinn von einem Trägheitsmoment $\theta_{\hat{n}}$ eines starren Körpers bezüglich einer Achse $\hat{n}$ zu sprechen, um die der Körper rotiert. Beim Trägheitstensor $\boldsymbol{\Theta}$ handelt es sich um eine lineare Abbildung von $\vec{\omega}$ auf $\vec{L}$, die i.a. als $3\times3$-Matrix dargestellt wird. Beim Trägheitsmoment $\theta_{\hat{n}}$ handelt es sich um eine Zahl.  Aus $\boldsymbol{\Theta}$ erhät man $\theta_{\hat{n}}$ aus der beidseitigen Multiplikation von $\boldsymbol{\Theta}$ mit $\hat{n}$:
-$$
-\begin{equation*}
-\begin{split}
-\theta &= \delta m \, n_{i}\left(r^{2}\delta_{ij}-r_{i}r_{j}\right)n_{j} \\
-&\\
-&=\delta m\,\left(r^{2}n_{i}^{2} - \big(n_{i}\, r_{i}\big)\big(n_{j}\,r_{j}\big)\right) \\
-&\\
-&=\delta m\,\left(r^{2}\hat{n}^{2} - \big(\hat{n}\cdot\vec{r}\big)^{2}\right) \\
-&\\
-&=\delta m\,\left(r^{2} - \big(\hat{n}\cdot\vec{r}\big)^{2}\right) \\
-\end{split}
-\end{equation*}
-$$
-Auch dieses Ergebnis hat eine anschauliche Bedeutung, wie in **Skizze 1** dargestellt:
-
-<img src="../figures/Trägheitsmoment.png" width="900" style="zoom:100%;" />
-
-**Skizze 1** (Anschauliche Bedeutung des Trägheitsmoments $\theta_{\hat{n}}$)
-
----
-
-Das Trägheitsmoment $\theta_{\hat{n}}$ jedes Massenelements $\delta m$ eines starren Körpers berechnet sich aus dessen Abstand $\vec{r}_{\perp}$ senkrecht zu $\hat{n}$.
+Die $\{\theta_{i}\}$ heisst in diesem Fall Hauptträgheitsmomente.
 
 # Navigation
 
-[Main](https://git.scc.kit.edu/etp-lehre/p1-for-students/-/tree/main/Geometrische_Optik) | [Weiter](https://git.scc.kit.edu/etp-lehre/p1-for-students/-/blob/main/Geometrische_Optik/doc/Hinweise-Aufgabe-1-a.md)
+[Main](https://gitlab.kit.edu/kit/etp-lehre/p1-praktikum/students/-/tree/main/Kreisel) | [Weiter](https://gitlab.kit.edu/kit/etp-lehre/p1-praktikum/students/-/blob/main/Kreisel/doc/Hinweise-Aufgabe-1.md)
+
diff --git a/Kreisel/figures/Traegheitsmoment.odg b/Kreisel/figures/Traegheitsmoment.odg
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