Commit 2e1de3ed authored by Harry Fuchs's avatar Harry Fuchs
Browse files

2019-07-10

parent 8b60d458
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...@@ -4472,7 +4472,8 @@ $$ ...@@ -4472,7 +4472,8 @@ $$
{} & H^0(T^2) \arrow[rr] & {} & H^0(U)\oplus H^0(V) \arrow[rr] & {} & H^0(U\cap V) \arrow[r] & {} \\ {} & H^0(T^2) \arrow[rr] & {} & H^0(U)\oplus H^0(V) \arrow[rr] & {} & H^0(U\cap V) \arrow[r] & {} \\
\arrow[r] & H^1(T^2) \arrow[rr] & {} & H^1(U)\oplus H^1(V) \arrow[rr] & {} & H^1(U\cap V) \arrow[r] & {} \\ \arrow[r] & H^1(T^2) \arrow[rr] & {} & H^1(U)\oplus H^1(V) \arrow[rr] & {} & H^1(U\cap V) \arrow[r] & {} \\
\arrow[r] & H^2(T^2) \arrow[rr] & {} & H^2(U)\oplus H^2(V) \arrow[rr] & {} & H^2(U\cap V) \arrow[r] & {} \\ \arrow[r] & H^2(T^2) \arrow[rr] & {} & H^2(U)\oplus H^2(V) \arrow[rr] & {} & H^2(U\cap V) \arrow[r] & {} \\
\arrow[r] & \ldots & {} & {} & {} & {} & {} \arrow[r] & \ldots & {} & {} & {} & {} &
{}
\end{tikzcd} \end{tikzcd}
\end{center} \end{center}
ist exakt. ist exakt.
...@@ -4482,7 +4483,8 @@ ist exakt. ...@@ -4482,7 +4483,8 @@ ist exakt.
\begin{tikzcd} \begin{tikzcd}
{} & \mathbb R \arrow[r] & \mathbb R^2 \arrow[r] & \mathbb R^2 \arrow[r, "m"] & {} \\ {} & \mathbb R \arrow[r] & \mathbb R^2 \arrow[r] & \mathbb R^2 \arrow[r, "m"] & {} \\
\arrow[r] & H^1(T^2) \arrow[r, "h"] & \mathbb R^2 \arrow[r] & \mathbb R^2 \arrow[r] & {} \\ \arrow[r] & H^1(T^2) \arrow[r, "h"] & \mathbb R^2 \arrow[r] & \mathbb R^2 \arrow[r] & {} \\
\arrow[r] & \underbrace{ H^2(T^2) }_{\mathbb R} \arrow[r] & 0 \arrow[r] & 0 & {} \arrow[r] & \underbrace{ H^2(T^2) }_{\mathbb R} \arrow[r] & 0 \arrow[r] & 0 &
{}
\end{tikzcd} \end{tikzcd}
\end{center} \end{center}
...@@ -4585,3 +4587,201 @@ $$ ...@@ -4585,3 +4587,201 @@ $$
$$ $$
weil $H^n(S^{n-1})\cong H^n(S^{n-1}\times (0,1)) = 0$ weil $H^n(S^{n-1})\cong H^n(S^{n-1}\times (0,1)) = 0$
%2019-07-09
TODO missing 2019-07-09
%2019-07-10
Gestern: Variationsproblem
$$
\text{Wirkung} \rightarrow S(\underbrace{x}_{x\in C^\infty[a,b]} ) = \int_a^b 2(x, \cdot x, t)\diffd t \rightarrow \operatorname{min}
$$
** Proposition
Wenn $x_0$ die Wirkung minnimiert/maximiert
$\Rightarrow$ $x_0$ erfüllt Euler-Lagrange-Gleichung:
$$
\frac{\diffd}{\diffd t} \frac{\partial 2 (x_0, \cdot x_0, 1)}{\partial \cdot x} = \frac{\partial 2 (x_0, \cdot x_0, t)}{\partial x}
$$
** Lemma
Wenn $2 = 2(x,\cdot x)$ [hängt nicht von $t$ ab]
$\Rightarrow$ jede Lösung $x_0$ der Euler-Lagrange-Gleichung erfüllt:
$$
\cdot x_0 \frac{\partial 2}{\partial \cdot x} - L(x_0, \cdot x_0) = \operatorname{const}
$$
** Beispiel: Brachistochrone
$$
2 (y, y') = \sqrt{\frac{1+(y')^2}{-y}}
$$
$$
\frac{\partial L}{\partial y'} = \cancel{2} y' \cdot \frac{1}{2}(1+(y')^2)^{-\frac{1}{2}} (-y)^{-\frac{1}{2}}
$$
$$
y' \cdot \frac{\partial L}{\partial y'} = (y')^2 (1+(y')^2)^{-\frac{1}{2}}(-y)^{-\frac{1}{2}}
$$
$$
L = (1+(y')^2)(-y)^{-\frac{1}{2}}
$$
%TODO missing
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$$
C(-y)^{\frac{1}{2}}(1+(y')^2)^{\frac{1}{2}} = \cancel{(y')^2} - (1+\cancel{ (y')^2 } ) = -1
$$
$$
\Rightarrow y(1+(y')^2) = D
$$
$$
y\left(1 + \left(\frac{}{}\right)^2 \right) TODO missing
$$
%TODO missing
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$$
\int \frac{\diffd y}{\sqrt{\frac{}{}}} TODO missing
$$
%TODO missing
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** Geodäten
Sei $(M, g)$, $A$, $B\in M$.
%TODO Bildchen
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$$
\gamma\colon [a, b] \to M,\quad \gamma(a) = A,\quad \gamma(b) = B
$$
$$
L(\gamma) = \int_a^b g(x(t)) (\dot \gamma (t), \dot\gamma(t))^{\frac{1}{2}} \diffd t \rightarrow \operatorname{min}
$$
In Koordinaten ist $g(\gamma(t)) (\dot \gamma(t), \dot \gamma(t)) = \sum_{i=1}^{n} g_{ij}(\gamma(t)) \dot\gamma^i (t) \dot\gamma^j(t)$
$\rightarrow$ haben ein Variationsproblem mit $2 = 2(\gamma^1, \ldots, \gamma^n, \dot\gamma^1,\ldots, \dot \gamma^n)$
Indem man $\gamma^1, \ldots, \gamma^n$ einzeln variert bekommt man $n$ Euler-Lagrange-Gleichungen
$$
\frac{\diffd}{\diffd t} \frac{\partial 2}{\partial \dot \gamma^i} = \frac{\partial 2}{\partial \gamma^i}, \quad i = 1, \ldots, n
$$
Vorbereitung: Wie lösen anderes (!) Variationsproblem mit
$$
2(\gamma, \dot\gamma) &=& \frac{1}{2}g(\gamma(t))(\dot\gamma(t), \dot\gamma(t))
\\ &=& \frac{1}{2}\sum_{i,j = 1}^n g_{ij}(\gamma(t)) \dot\gamma^i(t)\dot\gamma^j (t)
$$
$$
\frac{\partial 2}{\partial \dot\gamma^k} &=& \sum_{j=1}^n g_{kj} \dot\gamma^j
$$
$$
\frac{\diffd}{\diffd t} \frac{\partial 2}{\partial \dot\gamma^k} = \sum_{j=1}^n g_{kj}\ddot\gamma^j + \sum_{j=1}^n \sum_{i=1}^n \frac{g_{kj}}{\partial x^i} \dot\gamma^i \dot\gamma^j
$$
$$
\frac{\partial 2}{\partial \gamma^k} = \frac{1}{2} \sum_{i,j = 1}^n \frac{\partial g_{ij}}{\partial x^k} (\gamma(t)) \dot\gamma^i\dot\gamma^j
$$
Euler-Lagrange-Gleichungen:
$$
\sum_{j=1}^n g_{kj} \ddot\gamma^j + \sum_{i,j = 1}^n \left( \frac{\partial g_{kj}}{\partial x^i} - \frac12 \frac{\partial g_{ij} }{\partial x^k} \right)\dot \gamma^i \dot\gamma^j = 0
$$
$$
\text{irgendwas} = \frac{1}{2} \sum_{i,j = 1}^n \left( \frac{\partial g_{kj}}{\partial x^i} + \frac{\partial g_{ki}}{\partial x^j} -\frac{\partial g_{ij}}{\partial x^k} \right) \dot\gamma^i \dot\gamma^j
$$
Dies ist äquivalent zu
$$
\ddot\gamma^m + \sum_{k=1}^n g^{mk}\cdot\frac{1}{2} \sum_{i,j = 1}^n \left( \frac{\partial g_{kj}}{\partial x^i} + \frac{\partial g_{kj} }{\partial x^j} -\frac{\partial g_{ij}}{\partial x^k} \right) \cdot \dot\gamma^i \dot\gamma^j
$$
$$
\Leftrightarrow \ddot\gamma^m &=& \Gamma^m_{ij} \dot\gamma^i\dot\gamma^j = 0
\\ \Leftrightarrow \nabla_{\dot\gamma} &=& 0
$$
%TODO Bildchen
TODO Bildchen
** Letzter Abschnitt: Physik
$$
A \in \Omega^1(\mathbb R^4) = \Omega^1(M, \mathbb R)
\\ F &=& \diffd A \in \Omega^2(M, \underbrace{ \mathbb R}_{u(1)} )
\\ \diffd F = 0 \leftarrow \text{die ersten 2 Maxwell-Gleichungen}
$$
$$
S = \int_M F\wedge \ast F = \int_M \langle F, F \rangle \operatorname{vol}
$$
$4 = 2\dot 2$, $2 = 4 - 2$
$$
2 = \langle F, F \rangle = \langle \diffd A, \diffd A \rangle
$$
$$
A \rightsquigarrow A + \epsilon H
$$
$$
S(A + \epsilon H) &=& \int_M \langle \diffd A + \epsilon \diffd H, \diffd A + \epsilon \diffd H \rangle \operatorname{vol}
\\ &=& \int_M\langle \diffd A, \diffd A \rangle \operatorname{vol} + 2 \cdot \epsilon \underbrace{\int_M\langle \diffd A, \diffd H \rangle}_{} + \epsilon^2\cdots
$$
$$
(*)0 =\int_M\langle \diffd A, \diffd H \rangle \operatorname{vol} &=& \int_M \diffd H \wedge * \diffd A \underbrace{=}_{M \text{ Kompakt geragen }/ H \text{ Kompakt getragen} }
\\&=& -\int H\wedge \diffd \ast \diffd A
\\&=& \int H\wedge \diffd \ast F
$$
$(*)$ für jedes $H$ $\Rightarrow \diffd \ast F = 0 \Leftrightarrow \ast \diffd \ast F = \diffd^\ast F$
$F$ erfüllt also:
1. $\diffd F = 0 \leftarrow$ 1. Paar
2. $\diffd^\ast F = 0 \leftarrow$ 2. Paar
$M$ Mannigfaltigkeit, $\rightsquigarrow$ studieren Riemansche Matriken $g$ auf $M$
$$
g \rightsquigarrow R \in \Omega^2(M, \underline{so} (u)) \rightarrow \underbrace{s}_{\text{Skalarkrümmung}}(R) \in C^\infty(M)
$$
Einstein-Hilbert-Wirkung:
$$
S(g) = \int_M s\cdot \operatorname{vol} \left( + \Lambda \int \operatorname{vol} \right)
$$
$\rightsquigarrow (M, g)$, s.d. $g$ Extrempunkt für $S$ ist, heißen Einstein-Mannigfaltigkeiten
Das Gleiche für nicht-Riemansche, sondern Pseudo-Riemansche Mannigfaltigkeiten $\rightarrow$ Allgemeine Relativitätstheorie
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