From 40393753d33d6edf7e92186ad0b1100aa58dc316 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Vely=C4=8Dko?= Date: Thu, 15 May 2025 07:23:57 +0200 Subject: [PATCH] prednaska 2.5.2025 --- prvni-integral.tex | 67 +++++++++++++++++++++++++++++++++++++++++++++ skripta.pdf | Bin 180910 -> 194710 bytes skripta.tex | 1 + stabilita.tex | 6 ++++ 4 files changed, 74 insertions(+) create mode 100644 prvni-integral.tex diff --git a/prvni-integral.tex b/prvni-integral.tex new file mode 100644 index 0000000..8b66abc --- /dev/null +++ b/prvni-integral.tex @@ -0,0 +1,67 @@ +\section{První integrál} + +V celé této kapitole budeme uvažovat autonomní rovnici +\begin{equation} + \label{eq-auto} + x' = f(x) +\end{equation} +pro $f$ spojitou a lokálně lipschitzovskou. + +\begin{definition} + Funkci $U: \Omega \to \R$ nazveme \textit{prvním integrálem} rovnice \eqref{eq-auto}, jestliže $U \in C^1(\Omega)$ a je nekonstantní a zároveň $t \to U(x(t))$ je konstantní pro každé řešení $x$ dané rovnice v $\Omega$. +\end{definition} + +Například, máme-li rovnici $x'' + kx = 0$ (lze pomocí ní popsat kmitání pružiny s hybností $k > 0$), funkce $V(x', x) = \frac{1}{2}x'^2 + \frac{k}{2} x^2$ je jejím prvním integrálem, neboť tato funkce je zřejmě hladká a nekonstantní a +$$ \odv*{V(x'(t), x(t))}{t} = x'\cdot x'' + kxx' = x'(x'' + kx) = 0. $$ + +\begin{theorem}[Charakterizace prvních integrálů pomocí orbitálních derivací] + \label{thm-first-integral} + Buď $\Omega \subset \R^n$ otevřená, $f: \Omega \to \R^n$ spojitá a $U: \Omega \to \R$ je třídy $C^1$. Potom následující tvrzení jsou ekvivalentní: + \begin{enumerate}[(i)] + \item Pro všechna řešení $x$ rovnice \eqref{eq-auto} je $t \mapsto U(x(t))$ konstantní; + \item $\nabla U(\xi)f(\xi) = 0$ pro všechna $\xi \in \Omega$. + \end{enumerate} +\end{theorem} + +\begin{proof} + (ii) $\implies$ (i): Přímým výpočtem dostaneme $\odv*{U(x, t)}{t} = \nabla U(x(t)) \cdot x'(t) = \nabla U(x(t))f(x(t)) = 0$. + + (i) $\implies$ (ii): Mějme bod $\xi \in \Omega$. Dle Peanovy věty (Věta \ref{thm-peano}) bodem $\xi$ prochází nějaké řešení $x$ takové, že $x(0) = \xi$. Z toho již dostáváme, že $\nabla U(\xi) \cdot f(\xi) = \odv*{U(x(t))}{t}_{t=0} = 0$. +\end{proof} + +\begin{definition} + První integrály $U_1, \dots, U_k$ jsou \textit{lineárně nezávislé} v bodě $x_0$, jestliže matice $\pdv{U_i}{x_j}_{i=1,\dots,k}^{j=1,\dots,n}$ má v tomto bodě hodnost $k$. +\end{definition} + +Je dobré si uvědomit, že prvních integrálů, které jsou lineárně nezávislé v bodě $x_0$, může být nejvýše $n - 1$, což plyne z Věty \ref{thm-first-integral}, a to tak, že pro každý první integrál platí $\nabla U_i(x_0) \perp f(x_0) \neq 0$ a v prostoru dimenze $n$ existuje přesně $n - 1$ lineárně nezávislých vektorů kolmých na daný vektor. + +Metodu prvních integrálů budeme používat k redukci počtu rovnic v soustavách ODR. Skutečně, mějme soustavu +$$ \begin{brace}x' = f(x, y)\\y' = g(x, y)\end{brace},$$ +a nechť $V(x, y) = K$ je její první integrál. Ve většině případů z toho můžeme vyjádřit $x$ jakožto funkci $x = h(y, K)$, což po dosazení do druhé rovnice nám dává +$$ y' = g(h(y, K), y), $$ +čímž jsme zredukovali počet rovnic na jednu. K přesné formulaci právě popsaného postupu použijeme následující větu. + +\begin{theorem}[O snížení řádu] + \label{thm-first-int-solution} + Nechť $U_1, \dots, U_k$ jsou první integrály \eqref{eq-auto} lineárně nezávislé v bodě $x_0$. Potom řešení procházející bodem $x_0$ lze \textit{lokálně} popsat pomocí $(n - k)$ rovnic $z' = g(z)$, $z \in \R^{n-k}$, $g: \R^{n-k} \to \R^{n-k}$. +\end{theorem} + +\begin{proof} + Označme $K_i = U_i(x_0)$ a $\Gamma = \{x \in \R^n: U_i(x) = K_i \text{ pro } i = 1,\dots,k\}$. Víme, že $\nabla U_1(x_0), \dots \nabla U_k(x_0)$ jsou lineárně nezávislé vektory, to znamená, že matice $\pdv{U_i}{x_j}_{i=1,\dots,k}^{j=1,\dots,n}$ má $k$ lineárně nezávislých sloupců. Bez újmy na obecnosti nechť toto jsou sloupce $1, \dots, k$. Označme $x = (x_1, \dots, x_k, y)$, kde $x \in \R^n$ a pro $y \in \R^{n-k}$ platí $y = (x_{k+1}, \dots, x_n)$. + + Jelikož matice $\pdv{U_i}{x_j}_{i=1,\dots,k}^{j=1,\dots,k}$ je regulární, díky větě o implicitních funkcích existují konstanty $\delta, \Delta > 0$ takové, že pro každé $y$ z $\delta$-okolí bodu $x_0$ existuje právě jedna $k$-tice $(x_1, \dots, x_k)$ z $\Delta$-okolí $x_0$ (uvažujeme restrikci na prvních $k$ souřadnic) taková, že platí $U_i(x_1, \dots, x_k, y) = K_i$ pro všechna $i = 1, \dots, k$. Pokud příslušnou $k$-tici označíme jako $(\varphi_1(y), \dots, \varphi_k(y))$, potom funkce $\varphi$ je třídy $C^1$. + + Z výše uvedeného dostáváme, že pokud $(x_1(t), \dots, x_n(t))$ je řešení rovnice \eqref{eq-auto}, potom pro $i \geq k + 1$ platí + $$ x_i'(t) = f_i(x_1(t), \dots, x_n(t)) = f_i(\varphi_1(y(t)), \dots, \varphi_k(y(t)), y(t)) =: \tilde f_i(y(t)). $$ + + Tímto jsme získali požadovanou soustavu rovnic + $$ (x_{k+1}', \dots, x_n') = y' = \tilde f(y(t)), \tilde f: \R^{n-k} \to \R^{n-k}.$$ +\end{proof} + +\begin{theorem}[Existence lineárně nezávislých prvních integrálů] + Nechť $f \in C^1(\Omega)$, $f(x_0) \neq 0$. Potom má rovnice $x' = f(x)$ na okolí $x_0$ $(n - 1)$ prvních integrálů, které jsou lineárně nezávislé v bodě $x_0$. +\end{theorem} + +Právě vybudovanou teorii můžeme využít k aplikaci takzvané metody charakteristik, což je matematický aparát, který nám umožňuje přecházet mezi autonomními ODR a jistou třídou lineárních parciálních diferenciálních rovnic (podrobnosti viz přednáška Úvod do parciálních diferenciálních rovnic\footnote{pokud ji zrovna učí někdo příčetný}). + +\hfill \textit{konec 10. přednášky (2.5.2025)} diff --git a/skripta.pdf b/skripta.pdf index ba2e5da188a074d1ef3855aefc916a29242b2b7d..0364fb789256b2b03f41637886e849cdcce1b5eb 100644 GIT binary patch delta 82595 zcmZtMLwJ}C(>CBVw%IhcZQDs>+qRqBu^QX9ZQHgR+vfKy{>8iaXEU>&)y#QZ)xmJd zw{U)qKa9)(R#r|XetuYI7bjCgTUd|vOC9YPGOjq^9eu~-l+)!CY>A+n-@1FmZbgSM zC%Qy)ys}HD!YquExp8L6P`*4UZrlsgjWc8AXJ=(rkgVKGY4ZC0`kMp%AD19dQ{SP1 zBY~l%`5)H~?`#6^Xy4B}q57SCA1^&1h#%v5ETJucJJfX|6e;&FJnHMnf`N?%g6hU$ 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--- a/stabilita.tex +++ b/stabilita.tex @@ -144,3 +144,9 @@ Matice $A$ splňující $\Re \lambda < 0$ pro všechna $\lambda \in \sigma(A)$ s \end{proof} \hfill \textit{konec 9. přednášky (25.4.2025)} + +Na závěr této kapitoly si uvedeme jednu větu bez důkazu, která se hodí k vyšetřování stability na okolí stacionárních bodů. + +\begin{theorem}[Hartman-Grobman] + Uvažujme autonomní rovnici $x' = f(x)$. Nechť $x_0$ je hyperbolický stacionární bod této rovnice (tedy $\sigma(\nabla f(x_0)) \cap i\R = \emptyset$). Pak existuje okolí $U$ bodu $x_0$ a okolí $V$ bodu $0 \in \R^n$ a homeomorfismus $\phi: U \to V$ při kterém se řešení rovnice $x' = f(x)$ zobrazí na řešení rovnice $x' = Ax$, kde $A = \nabla f(x_0)$. +\end{theorem}