From 4307eef4d5d16367fb774155208e7ba8ec1959ff Mon Sep 17 00:00:00 2001 From: Jonas Hahn Date: Thu, 24 Apr 2025 11:16:55 +0200 Subject: [PATCH] test pipe --- S2/AnaMech/VL/AnMeVL2.typ | 111 +++++++++++++++++++++++++++++++++++++ S2/DiffII/VL/DiIIVL3.typ | 66 +++++++++++++++++++++- S2/DiffII/pdfs/DiIIVL3.pdf | Bin 0 -> 43580 bytes book.typ | 35 ++++++++---- 4 files changed, 197 insertions(+), 15 deletions(-) create mode 100644 S2/AnaMech/VL/AnMeVL2.typ create mode 100644 S2/DiffII/pdfs/DiIIVL3.pdf diff --git a/S2/AnaMech/VL/AnMeVL2.typ b/S2/AnaMech/VL/AnMeVL2.typ new file mode 100644 index 0000000..9894c46 --- /dev/null +++ b/S2/AnaMech/VL/AnMeVL2.typ @@ -0,0 +1,111 @@ +#import "../preamble.typ": * + +#show: conf.with(num: 1) += Einleitung + +Newton < Lagrange < Hamilton + +== Ziel der Analytischen Mechanik + +Dynamik von mechanischen Systemen (N Massepunkte) + +Aufstellen von BWGL mit der Forminvarianz + +Loesen von BWGLn. Hier gibt es nur wenig loesbare Beispiele: Zentralpotential und gekoppelte Oszillatoren. + +Erhaltungsgroessen & Symetrien beim Loesen ausnutzen und neue Erhaltungsgroessen auffinden. + +Newton <= Lagrange = Hamilton < Hamilton'sches Prinzip + +Abstraktionsweg: + +$"Kraefte" -> "Potentiale" -> "Lagrange-/ Hamiltonfunktion" -> "Wirkung"$ + +== Vorlesung + +24VL + +Newton'sche Mechanik (6VL) + +1D Probleme, Numerik, Reduktion auf DGL 1. Ordnung, Zentralpotential, Streuprobleme + +#example[ + Ich meine das PDF gibts auch online +] + += Notes + +Es wird ein Skript geben, welches vom letzten Jahr etwas umgebaut werden wird. + +Hoersaaluebung sei das wichtigste (Fr 16-18). Vielleicht sogar wichtiger und ersetzend zu den kleinen Uebungen. + +Donnerstag nach Ostern Vorstellung der TeamCaptains. Das sind irgendwelche Studierende, welche freiwillig Ansprechpartner fuer Fragen sein wollen und einen Kommunikationskanal zum Dozenten herstellen. + += Tafelanschriebe + +- Alles vom Standpunkt Newton mit Erhaltungsgroessen (6-7VL) +- Loesen vom Zentralkraftproblem + +== Newton Mechanik + +=== 1D Systeme + +Grundlage fuer die ersten Wochen + +Newton II. BWGL + +$ +dot(arrow(p)) = dif / (dif t) arrow(p) = m dot.double(r) = sum_(i=0)^(n) arrow(F_i) (arrow(r) dot(arrow(r))) +$ + +Dabei ist $arrow(r)$ ein Vektor in Kartesichen Koordinaten. + +Ziel: $arrow(r) = arrow(r)(t)$ bzw $x(t), y(t) z(t)$ also eine Lsg von @bwg durch das Anfangswertproblem + +$ + arrow(r)(t) = arrow(r_0) \ + dot(arrow(r))(t_0 ) = dot(arrow(r_0)) +$ + +#example[ + 1D Oszillator im Gravitationsfeld. + + Das KS wird so gewaehlt, dass $z$ zu einem kraeftefreien Punkt wird. $arrow(F_g) + arrow(F_r)(z) = arrow(0)$ + + $ + z = z(t) \ + arrow(F_g) = -m g \ + arrow(F_k) = -k z + m g = -k Delta l + $ + + BWGL: + + $ + m dot.double(z) = arrow(F_g) + arrow(F_k) = - m g - k z + m g\ + m dot.double(z) + k z = 0 + $ + + Gewoehnliche DGL 2. Ordnung in Zeit $t$ fuer $z=z(t)$ BWGL fuer $z$. + + - linear homogen, Koeffizienten konstant + + Standardloesungsansatz: + + $ + z(t) = z_0 e^(lambda t), z_0 in CC, lambda in CC + $ + + Ableiten und Einsetzen + $ + dot.double(z(t)) = z_0 lambda^2 e^(lambda t) ==> z_0 e^(lambda t) [lambda^2 + k/m] = 0, forall t + $ + + Lsg. $z_0 = 0 or lambda = p_m i omega_0, omega_0 = sqrt(k/m)$ +] + +Allgemeine Loesung des Beispiels: + +// TODO: mit dem Skript weiterschreiben zuende machen + + +// okayt diff --git a/S2/DiffII/VL/DiIIVL3.typ b/S2/DiffII/VL/DiIIVL3.typ index 1342ba8..fd26033 100644 --- a/S2/DiffII/VL/DiIIVL3.typ +++ b/S2/DiffII/VL/DiIIVL3.typ @@ -1,6 +1,66 @@ -#import "./preamble.typ": * +#import "../preamble.typ": * -#show: conf.with(num: 1) +#show: conf.with(num: 3) -= Uebersicht += Klassen von Raeumen +In metrischen Raeumen mit Metrik $d$ gibt es den Begriff von offenen Mengen, wodurch dann auch eine Topologie $tau_(d) : "alle Mengen, die unter" d "offen sind"$ gegeben ist. +Diese Topologie ist eine Familie von Mengen. + += Konvergenz mittels Topologien +Fuer Konvergenzbegriffe ist dann einzig $tau$ notwendig. +#lemma[ + Sei $(M,d)$ ein metrischer Raum, und $(a_n )_(n in NN)$ eine Folge in $M$ und $a in M$. Dann gilt $lim_(n -> oo) a_n = a$ genau dann wenn golgendes gilt: + $ + forall U in tau_(d): a in U: exists n in NN: forall n >= n: a_n in U + $ +] + +#definition[ + Sei $M$ ein metrischer Raum mit Metriken $d_1 "und" d_2 $. + Wir nennen $d_1 $und $d_2 $ aequivalent falls $tau_(d_1 ) = tau_(d_2 ) $. + Ist $V$ ein K-VR mit $K in {RR,CC}$ mit zwei Normen $norm(dot)$ und $norm(dot)^(star) $ so nennen wir diese aequivalent, falls die erzeugten Metriken aequivalent sind. +] + +#lemma[ + Sei $V$ ein K-VR, $K in {RR,CC}$ und $norm(dot),norm(dot)^star, V -> RR^(+) $ Normen auf $V$ . Dann sind $norm(dot) "und" norm(dot)^(star) $ aequivalent genau dann wenn es Konstanten $c_1, c_2 >0$ gibt sodass + $ + C_1 norm(x) <= norm(x)^(star) <= c_2 norm(x), forall x in V. + $ +] + +#proof[ + Seien $tau_(1) "und" tau_(2) $ die von den Normen $(1) "und" (2)$ erzeugten Topologien. Sei $U in tau_(2) $. Nun gilt es zu zeigen, dass $U in tau_(1) $. Sei $x in U$. Dann gibt es ein $r >0$ sodass $K_(r) ^((2)) (x) subset.eq U $. + Wir erhalten, dass gilt + + $ + B_(c_(2) ^((1)) ) (x)={y in V | norm(x-y)0$ sodass $B_(r) ^((2)) <= B_(1) ^((1)) (0)$. + Sei $x in V \\ {0}$. Dann ist $1+1$ + +] + +#example[ + Betrachte $norm(dot)_(1) "und" norm(dot)_(oo) $ auf $RR^n$. Fuer $x in RR^n$ gilt $norm(x)_(oo) = max_(1 <= k <= n) abs(x_(k)) <= sum_(k=1)^(n) abs(x_k )= norm(x)_(1) $ und $norm(x)_(1)= sum_(k=1)^(n) abs(x_k )<= n max_(1 <= k <= n) abs(x_k ) = n norm(x)_(oo) $, also sind $norm(dot)_(1) "und" norm(dot)_(oo) $ aequivalent. +] + +#theorem[ + Je zwei Normen auf $RR^n$ sind aequivalent. +] + +Hier bezeichnet $e_(n) $ den Einheitsvektor in Richtung $n$. + +#proof[ + Sei die Euklidische Norm als $(1)$ fix. Ferner sei $(2)$ eine beliebige weitere Norm auf $RR^n$. + Sei $x in RR^n$. Dann ist + $ + norm(x)= norm(sum_(i=0)^(n) x_i e_i ) <= sum_(i=0)^(n) abs(1+1) norm(e_i ) <= sqrt(sum_(i=0)^(n) abs(x_i )) sqrt(sum_(i=0)^(n) norm(e_i )^2 ) + $ + Sei $S^(n-1) := {x in RR^(n) | norm(x)_(2) =1}$ und $c := inf {norm(x) | x in S^(n-1) }$. Behauptung: es gilt $c>0$. + + Angenommen $c=0$. Waehle $x_l in S^(n-1) , l in NN $ mit $lim_(l -> oo) norm(x_l )=0$. +] diff --git a/S2/DiffII/pdfs/DiIIVL3.pdf b/S2/DiffII/pdfs/DiIIVL3.pdf new file mode 100644 index 0000000000000000000000000000000000000000..01430b754115fec18555e28c0c0bed3b1ce40c57 GIT binary patch literal 43580 zcmce-1yCH(wk{lkOMu`S2Dji07Tn$4-Ccr1@B}BgdkF3j9D)RQceh{(ZsG4C=gN8S z)V=lJTlHqIs-ahJS$p;BX6F0mqf`_XXJBFEM56ovypWJMK+GTqV{0T{UL+8RN!8QQ z48$bvVDAb!QgJhOeRe?H%n*;Tk&7AREQm=~SyWz8h+fvp*v#40%HGXI#mw2toI%;l z!p+vmnL*0b%-+??)ssOIGEB_g#KF|c-U7t*&wf!e7ZWpkQzLs<$fPbH9Znz`WxeM} z9MtTsfKg^3Zio+wpC1Y635gU5(&@hqSwu<{GUxyA$RTS3V^us|T+QsH?9ClO&zS+d zz}Bo>T%A2ZG(tceX0$+0z^mYF3TzzM8qL3L?H?yp+#DTk&Fp|QfFP*@;s<&|LV|4C z*~-z?!5PH$yfwfts`1>3Ndu(A4EVsz%pew^aWb=jSRsv*1;hq4Fc&L`9cZjzpbyYk zxj8``kOpQ2aRLp@%nr=S1!Ci51=7Y1V&z~5u>eysgV@00+Q6n8A$Pz>5>a2Dl(u0y4l3F-RP2V2E)7`)A{T7&{l> z01kk~AP%4_up5XESOUly7?>GI7vvhSF){-$AacM3SrTvq8v>Sui~!thAP#PBAhQtT zWCl(FNdxx80rUfw0p^D6AJ`CZWq`=pAJZ%D!aM}5nh 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zB1hLsOhNuU^K2*hV`U?Zt; z%yScp1Oi~|$ONjDjjsX(d#W4k#g1saBNiWqCXz5{5^!au#TZeNR|3AxiHiG=N=iy2 zV^Z4iicta|0#{>h=r-=gU=$Rj2sr7No&jI_?;I5b2oO)<7$Kh^rDO#<2a1J6ATmcm za1sH?J|L5U9vuHUB5uPZer1;spEf7|ewE z17DAU-`-T}f>Q)wx>S>4Fn~5vU4}!e_lw8k;AEW9_9P+(hYN}H<^GNe<-|j5_ufJgH4Imx3TdK^7$W2 CvqY-^ literal 0 HcmV?d00001 diff --git a/book.typ b/book.typ index 373eca1..9d3f397 100644 --- a/book.typ +++ b/book.typ @@ -15,22 +15,33 @@ language: "de", summary: [ // this field works like summary.md of mdbook + = Semester I - = ExPhy II - - #chapter("S2/ExPhyII/VL/ExIIVL1.typ")[VL1] - - #chapter("S2/ExPhyII/VL/ExIIVL2.typ")[VL2] + = Semester II - = CWR - - #chapter("S2/CWR/VL/CwrVL1.typ")[VL1] - - #chapter("S2/CWR/VL/CwrVL2.typ")[VL2] + == ExPhy II + - #chapter("S2/ExPhyII/VL/ExIIVL1.typ")[VL1] + - #chapter("S2/ExPhyII/VL/ExIIVL2.typ")[VL2] - = AnaMech - - #chapter("S2/AnaMech/VL/AnMeVL1.typ")[VL1] - - #chapter("S2/AnaMech/VL/AnMeVL3.typ")[VL3] + == CWR + - #chapter("S2/CWR/VL/CwrVL1.typ")[VL1] + - #chapter("S2/CWR/VL/CwrVL2.typ")[VL2] - = Diff II - - #chapter("S2/DiffII/VL/DiIIVL1.typ")[VL1] - - #chapter("S2/DiffII/VL/DiIIVL2.typ")[VL2] + == AnaMech + - #chapter("S2/AnaMech/VL/AnMeVL1.typ")[VL1] + - #chapter("S2/AnaMech/VL/AnMeVL2.typ")[VL2] + - #chapter("S2/AnaMech/VL/AnMeVL3.typ")[VL3] + + == Diff II + - #chapter("S2/DiffII/VL/DiIIVL1.typ")[VL1] + - #chapter("S2/DiffII/VL/DiIIVL2.typ")[VL2] + - #chapter("S2/DiffII/VL/DiIIVL3.typ")[VL3] + + == Computational Neuroscience + - #chapter("S2/Neuro/VL/NeuroVL1.typ")[VL1] + + = Semester III + ] )