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226
S2/AnaMech/VL/AnMeVL10.typ
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226
S2/AnaMech/VL/AnMeVL10.typ
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// Main VL template
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#import "../preamble.typ": *
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// Fix theorems to be shown the right way in this document
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#import "@preview/ctheorems:1.1.3": *
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#show: thmrules
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// Main settings call
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#show: conf.with(
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// May add more flags here in the future
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num: 10,
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type: 0, // 0 normal, 1 exercise
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date: datetime.today().display(),
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//date: datetime(
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// year: 2025,
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// month: 5,
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// day: 1,
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//).display(),
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)
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= Uebersicht
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Bei der Lagrange II fuer einen MP mit $arrow(r) in RR^(3) $ gilt die Trafo: $x_(i) = x_(i) (q_1, q_2, q_3 )$, wobei $q_i $ beliebige Koordinaten sind.
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Und die $x_(i) $ die kartesischen Koordinaten sind.
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Mit Newton gilt
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$
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m dot.double(x)_(i) = f_i , space arrow(f) = vec(f_1, f_2, f_3 ) \
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<=> dif / (dif t) ((partial T) / (partial dot(q)_(i) ) )- (partial T) / (partial q_i) = arrow(g)_(i) * arrow(f) , space i = 1,2,3.
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$
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= Konservative Systeme
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Es gilt
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$
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arrow(f) = - arrow(nabla) V (arrow(r)) <=> f_i = - partial_(i) V , space partial_(i) = partial / (partial x_(i) ).
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$
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Lagrangefunktion
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$
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L (q_1, q_2, q_3, dot(q)_(1) , dot(q)_(2) , dot(q)_(3) ) = T (q_1, q_2, q_3, dot(q)_(1) , dot(q)_(2) , dot(q)_(3) ) - V (q_1, q_2, q_3 ).
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$
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Lagrange BWGL II
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$
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p_(i) = (partial L) / (partial dot(q)_(i) ) \
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dif / (dif t) (partial L) / (partial dot(q)_(i) ) - (partial L) / (partial q_(i) ) = 0 , space i = 1,2,3
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$
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ist forminvariant.
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Die skalare Funktion der Lagrangefunktion ist eine Hilfsgroesse, wobei sie beliebigen $q_i "und" dot(q)_(i) $, welche unabhaengige Variablen sind
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einen skalaren Wert zuordnet.
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Wobei gilt
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$
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dot(q) = (dif q) / (dif t).
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$
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= Vorgehen
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1. Transformation in allgemeine Koordinaten auf deren Bewegung keine Kraefte wirken. Diese muss man Raten oder sie sind vorgegeben
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2. Die kinetische und potentielle Energie als Funktion von kartesischen Korrdinaten aufstellen
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3. Diese Energien in allgemeine Koordinaten transformieren
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Dabei gilt
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$
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T = m/2 dot(x)_(i) , space dot(x) = m/2 g_(i j) dot(q)_(i) dot(q)_(j) , space g_(i j) = g_(i j) (q).
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$
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#example[
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Zentralpotential mit konstantem Drehimpuls
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$
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dif / (dif t) arrow(L) = 0 => 2"D".
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$
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Waehle die generalisierten Koordinaten
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$
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q_(1) = r, \
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q_(2) = phi.
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$
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Berechne Transformationen
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$
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x_1 = r cos phi \
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x_2 = r sin phi \
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dot(x)_(1) = dot(r) cos phi - r sin phi dot(phi) \
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dot(x)_(2) = dot(r) sin phi + r cos phi dot(phi) \
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$
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Berechne die kinetische Energie
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$
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T &= m/2 (dot(x)_(1) ^2 + dot(x)_(2) ^2 ) = m/2 vec(dot(q)_(1), dot(q)_(2) )^(T) mat(
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1, 0;
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0, r^2 ;
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) vec(dot(q)_(1) , dot(q)_(2) ) \
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&= m/2 (dot(r)^2 + r^2 dot(phi)^2 ).
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$
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Dann folgt fuer die Lagrangefunktion
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$
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L = T - V, \
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V = -alpha/r.
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$
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Berechne die Partiellen Ableitungen
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$
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(partial L) / (partial dot(r) ) = m dot(r) \
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(partial L) / (partial r) = m r dot(phi)^2 - V' (r) \
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(partial L) / (partial dot(phi)) = m r^2 dot(phi) , space (partial L) / (partial phi) = 0
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=> m dot.double(r) - m r dot(phi)^2 + V' (r) = 0 \
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dif / (dif t) (m r^2 dot(phi)) = 0 \
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=> m r^2 dot(phi) = "const." = L_(z) = abs(arrow(L)) = p_(phi).
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$
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]
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#example[
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Das Mathematische Pendel.
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Hier gilt die Transformation
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$
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vec(x,y)= l vec(sin phi, - cos phi)
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$
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wobei $l$ die Laenge des Pendels ist.
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Hier gibt es zwei Zwangsbedingungen, denn es muss immer gelten
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$
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g_(2) (x,y,z,t) = x^2 + y^2 - l^2 = 0 \
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g_(1) (x,y,z,t) = z = 0.
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$
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Die Zahl der unabhaengigen Koordinaten ist gegeben durch
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$
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f = N - R = 1.
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$
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Die Lagrange Funktion in kartesischen Koordinaten
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$
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L = T - V = m/2 (dot(x)^2 + dot(y)^2 ) - m g y \
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L = L (phi, dot(phi)) = m/2 dot(l)^2 dot(phi)^2 + m g l cos phi \
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(partial L) / (partial dot(phi)) = l^2 m dot(phi) , space (partial L) / (partial phi) = - m g l sin phi \
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=> l^2 m dot.double(phi) + m g l sin phi = 0.
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$
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Fuer die Zwangskraefte gilt dann
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$
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arrow(Z)_(1) prop arrow(e)_(z) \
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arrow(Z)_(2) = - arrow(f)_(perp).
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$
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]
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Von den generalisierten Koordinaten wird erwartet, dass sie die Zwangsbedingungen immer erfuellen.
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Das Ziel ist nun die Forminvarianz der BWGL fuer N Massepunkte.
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#definition[
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Zwangsbedingungen koennen holonom und skeleronom sein.
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Dabei gilt dann fuer die skalare Funktion
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$
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g_(alpha) (arrow(x), t) = 0 , space alpha = 1, ..., R
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$
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wobei $R$ die Anzahl der Zwangsbedingungen ist.
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Es gilt
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$
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arrow(x), arrow(F), m in RR^(3 N).
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$
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]
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Im allgeinen ist die Anzahl der unabhaengigen Koordinaten gegeben durch
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$
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f = 3 N - R.
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$
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#definition[
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Fuer jede Zwangsbedingung $g_(alpha) $ gibt es eine Zwangskraft $arrow(Z)_(alpha) $, welche diese Zwangsbedingung physikalisch realisiert.
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Diese sind im allgemeinen nicht zeitlich konstant.
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]
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Als Loesungsansatz gilt dann
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$
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arrow(Z)_(alpha) = lambda_(alpha) arrow(nabla) g_(alpha) => "skalare Groesse" lambda_(alpha).
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$
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= Lagrangegleichung I fuer N MP
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Zunaechst sei angenommen $R <= 2$ und
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$
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m dot.double(arrow(r)) = arrow(f) + sum_(alpha = 1)^(R) lambda_(alpha) arrow(nabla) g_(alpha) \
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g_(alpha) (arrow(r), t) = 0.
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$
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Im Allgemeinen gilt dann
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$
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m_(n) dot.double(x)_(n) = F_(n) + sum_(alpha = 1)^(R) lambda_(alpha) (partial g_(alpha) ) / (partial x_(n) ) , space n = 1, ..., 3 N \
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g_(alpha) (arrow(x), t) = 0 , space alpha 1, ..., R \
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arrow(nabla) _(n) g_(alpha) = (partial g_(alpha) ) / (partial x_(n) ).
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$
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=== Allgemeines Loesungsverfahren
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1. Zwangsbedingungen aufstellen
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2. Lagrangegleichung I
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3. Man muss die $lambda_(alpha) $ aus den BWGL eleminieren $==>$ $lambda_(alpha) = lambda_(alpha) (arrow(x), dot(arrow(x)), t) $: funktional
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4. Loese die 3N BWGL mit 6N Integrationskonstanten
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5. Die 2R Integrationskonstanten sind schon durch die Zwangsbedingungen fixiert
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6. Die konrete Loesung $==>$ $arrow(x), dot(arrow(x)) -> lambda_(alpha) (arrow(x), dot(arrow(x)), t) $ $==>$ $Z_(n) = lambda_(alpha) partial_(x_(n) ) g_(alpha) $
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#example[
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Schiefe Ebene.
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Skizze der Schiefen Ebene mit relevanten Groessen.
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Zuerst die elementare Loesung
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$
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m dot.double(s) = - m g sin alpha => s (t) = - g/2 r^2 + v_0 t + s_0 \
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arrow(r) = vec(s cos alpha, 0 , s sin alpha).
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$
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Lagrange I liefert
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$
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g_(2) (x,y,z,t) = y = 0 \
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g_(1) (x,y,z,t) = x sin alpha - z cos alpha = 0 \
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m dot.double(arrow(r)) = - m g arrow(e)_(z) + lambda_(1) arrow(nabla) g_(1) + lambda_(2) arrow(nabla) g_(2).
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$
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Dann muessen wir die (zweimal) Zwangsbedingungen ableiten
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$
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dot.double(y) = 0 , space dot.double(x) sin alpha - dot.double(z) cos alpha = 0 \
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m dot.double(x) = lambda_1 sin alpha \
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m dot.double(y) = lambda_2 => lambda_2 = 0 => dot.double(z) = arrow(0) \
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m dot.double(z) = - m g - lambda_1 cos alpha => lambda_1 = - cos alpha m g
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$
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]
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Technisch kompliziert kann die richtige Beruecksichtigung der Kettenregel sein.
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169
S2/AnaMech/VL/AnMeVL9.typ
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169
S2/AnaMech/VL/AnMeVL9.typ
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@@ -0,0 +1,169 @@
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// Main VL template
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#import "../preamble.typ": *
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// Fix theorems to be shown the right way in this document
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#import "@preview/ctheorems:1.1.3": *
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#show: thmrules
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// Main settings call
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#show: conf.with(
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// May add more flags here in the future
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||||
num: 9,
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||||
type: 0, // 0 normal, 1 exercise
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||||
date: datetime.today().display(),
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//date: datetime(
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// year: 2025,
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// month: 5,
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// day: 1,
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//).display(),
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)
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= Exkurs in die Geometrie
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Zunaechst betrachten wir einen Massepunkt $m$ in $arrow(r) in RR^(3) $.
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Wir kennen die kartesischen Raumkoordinaten.
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Der Ursprung bleibt bei der Transformation gleich.
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Es gilt fuer die Basisvektoren
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$
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arrow(e)_(i) * arrow(e)_(j) = delta_(i j).
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$
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Die Koordinatentransformation muss umkehrbar sein in fast jedem Punkt
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$
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x_(i) = x_(i) (q_1, q_2, ..., q_n ).
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$
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Theoretische Physik geht los wenn alle griechischen Buchstabe fuer Indizes verbraucht sind.
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$
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arrow(r) = underbrace(x_(i) arrow(e)_(i), forall P) = x_(i) (q_1, q_2, q_3 ) arrow(e)_(i) =^(!) q_j arrow(q)_(j) <- "haengen von" P "ab".
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$
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Die $q_i $ Kurven koennen krummlinieg verlaufen. Die Basisvektoren im Punkt $P$ sind gegeben durch
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$
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arrow(q)_(i) = (partial arrow(r) (p)) / (partial q_i) , arrow(r) = x_(i) (q_1, q_2, q_3) arrow(e)_(i), \
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arrow(q)_(i) = (partial x_(j) ) / (partial q_i ) arrow(e)_(arrow(j)) , arrow(e)_(i) = (partial q_j ) / (partial x_(i) ) arrow(q)_(j).
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$
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In fast jedem Punkt sind diese linear unabhaengig. Dreibein ${arrow(q)_(1) , arrow(q)_(2) , arrow(q)_(3) }$.
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#example[
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Kugelkoordinaten.
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$
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x_1 = r cos phi sin theta \
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x_2 = r sin phi sin theta \
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x_3 = r cos theta \
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r >0 , space theta in [0, pi] , space phi in [0, 2 pi)
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$
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Durch Ableiten kann so das Dreibein gebildet werden. Dieses erfuellt die gefordeten Eigenschaften von linearer Unabhaengigkeit.
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]
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Es gilt
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$
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d arrow(r) &= (partial arrow(r)) / (partial r) d r + (partial arrow(r)) / (partial theta) d theta + (partial arrow(r)) / (partial phi) d phi ==> d arrow(r) * d arrow(r) = d^2 r + r^2 d^2 theta + r^2 sin^2 theta d^2 phi \
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&= d x_1 arrow(e)_(2) + d x_2 arrow(e)_(2) + d x_3 arrow(e)_(3).
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$
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= Metrischer Tensor
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Wird auch metrisches Dings genannt.
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Es gilt
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$
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g_(i j) &= arrow(g)_(i) * arrow(g)_(j), \
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g_(i j) &= mat(
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1, 0, 0;
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0, r^2 , 0;
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0, 0, r^2 sin^2 theta;
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), \
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g_(i j) &= (partial x_m ) / (partial q_i ) arrow(e)_(m) * (partial x_k ) / (partial q_(j) ) arrow(e)_(k) = (partial x_(m) ) / (partial q_(i) ) (partial x_(k) ) / (partial q_(j) ) underbrace(arrow(e)_(m) * arrow(e)_(k), = delta_(m k) ) = (partial x_k ) / (partial q_i ) (partial x_k ) / (partial q_j ).
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$
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= Bewegungsgleichung fuer $q_i $
|
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Hier sind $dot(q)_(j)$ die verallgemeinerten Geschwindigkeiten.
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Berechne
|
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$
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dif / (dif t) T = m dot(x)_(i) dot.double(x)_(i) \
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T = m/2 dot(arrow(r)) * dot(arrow(r)) = m/2 dot(x)_(i) dot(x)_(i) \
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dot(arrow(r)) (t) = (partial arrow(r)) / (partial q_(j) ) dot(q)_(j) = dot(q)_(j) arrow(g)_(j) , space dot(q)_(j) "verallgemeinerte Geschwindigkeiten" \
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(partial dot(arrow(r))) / (partial dot(q)_(j) ) = arrow(g)_(i); quad arrow(g)_(i) = (partial arrow(r)) / (partial q_(i) ) \
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arrow(r) = x_(i) arrow(e)_(i) \
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(dif arrow(r)) / (dif t) = dot(x)_(i) arrow(e)_(i) = (partial x_(i) ) / (partial q_j ) dot(q)_(j) arrow(e)_(i) = (partial arrow(r)) / (partial q_(j) ) dot(q)_(j).
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||||
$
|
||||
|
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Wir starten von Newton II
|
||||
$
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m dot.double(arrow(r)) = arrow(f) \
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<==> m arrow(e)_(i) * dot.double(arrow(r)) = arrow(e)_(i) * arrow(f) \
|
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m dot.double(x)_(i) = f_(i) , i = 1,2.3.
|
||||
$
|
||||
|
||||
Jetzt werden beliebige Koordinaten gewaehlt
|
||||
$
|
||||
m arrow(g)_(i) * dot.double(arrow(r)) = arrow(g)_(i) * arrow(f) \
|
||||
<==> m (arrow(g)_(i) * dot.double(arrow(r)) + dot(arrow(g))_(i) * dot(arrow(r))) = arrow(g)_(i) * arrow(f) + m dot(arrow(g))_(i) * dot(arrow(r)) \
|
||||
<==> m dif / (dif t) (arrow(g)_(i) * dot(arrow(r))_(i) ) = arrow(g)_(i) * arrow(f) + m dot(arrow(r)) * (partial dot(arrow(r))) / (partial q_(i) ) \
|
||||
<==> m partial / (partial t) ((partial dot(arrow(r))_(i) ) / (partial q_(i) ) * dot(arrow(r))_(i) ) = arrow(g)_(i) arrow(f) + m dot(arrow(r))* (partial dot(arrow(r))) / (partial q_(i) )
|
||||
$
|
||||
|
||||
Nebenrechung
|
||||
|
||||
$
|
||||
dot(arrow(g))_(i) = dif / (dif t) (partial arrow(r)) / (partial q_(i) ) = partial / (partial q_(j) ) ((partial arrow(r)) / (partial q_(i) ) )dot(q)_(j) \
|
||||
= (partial ^2 arrow(r)) / (partial q_(j) q_(i) ) dot(q)_(j) partial / (partial q_(i) ) ((partial arrow(r)) / (partial q_(j) ) dot(q)_(j) ) = partial / (partial q_i ) dot(arrow(r)) "und" arrow(q)_(i) = (partial dot(arrow(r))) / (partial dot(q)_(i) ) = (partial arrow(r)) / (partial q_(i) )
|
||||
$
|
||||
|
||||
Betrachtung der kinetischen Energie
|
||||
$
|
||||
T = T (dot(x)_(1) , dot(x)_(2) , dot(x)_(3) ) = T (q_1, q_2, q_3, dot(q)_(1) , dot(q)_(2) , dot(q)_(3) ) \
|
||||
==> dif / (dif t) ((partial T) / (partial dot(q)_(i) ) ) = (partial T) / (partial q_(i) ) + arrow(g)_(i) * arrow(f).
|
||||
$
|
||||
|
||||
|
||||
Allgemein gilt fuer die Produktregel
|
||||
$
|
||||
(partial T) / (partial dot(q)_(j) ) = m/2 ((partial dot(x)_(i) ) / (partial dot(q)_(j) ) dot(x)_(i) + dot(x)_(i) (partial dot(x)_(i) ) / (partial dot(q)_(j) ) ).
|
||||
$
|
||||
|
||||
Skalarprodukt ist eine Projektion.
|
||||
|
||||
Verallgemeinere die kinetische Energie
|
||||
|
||||
$
|
||||
T &= m/2 dot(arrow(r))^2 = m/2 (dot(x)_(i) * dot(x)_(i) ) = m/2 (dot(q)_(i) arrow(g)_(i) ) * (dot(q)_(j) arrow(g)_(j)) = m/2 dot(q)_(i) dot(q)_(j) space arrow(g)_(i) * arrow(g)_(j) \
|
||||
&= m/2 sum_(i,j) g_(i j) dot(q)_(i) dot(q)_(j), \
|
||||
&g_(i j) = g _(i j) (q_1, q_2, q_3 ).
|
||||
$
|
||||
|
||||
= Konservative Kraftfelder
|
||||
|
||||
Betrachte die Kraft mit $V = V (x_1, x_2, x_3 ) = V (x_1 (q_1, q_2, q_3), ...) = V (q_1, q_2, q_3 )$ und der Lagrangefunktion als $L (q_1, q_2, q_3, dot(q)_(1) , dot(q)_(2) , dot(q)_(3) ) := T (q_(i) , dot(q)_(i) ) - V (q_(i) )$
|
||||
$
|
||||
arrow(f) (arrow(r)) = - arrow(nabla) V (arrow(r)) = - (partial V) / (partial arrow(r)) \
|
||||
arrow(f) * arrow(g)_(i) = - (partial V) / (partial arrow(r)) * (partial arrow(r)) / (partial q_(i) ) = - (partial V) / (partial x_(j) ) (partial x_(j) ) / (partial q_(i) ) = - (partial V) / (partial q_(i) ) \
|
||||
==> dif / (dif t) ((partial T) / (partial dot(q)_(i) ) ) - (partial T) / (partial q_(i) ) + (partial V) / (partial dot(q)_(i) ) =^(!) 0 \
|
||||
V = V (q_1, q_2, q_3 ) ==> (partial V) / (partial dot(q)_(i) ) = 0 \
|
||||
==> dif / (dif t) (partial L) / (partial dot(q)_(i) ) - (partial L) / (partial q_(i) ) = 0
|
||||
$
|
||||
|
||||
Diese Lagrangegleichung der II Art ist
|
||||
- Forminvariant
|
||||
- Nicht messbar
|
||||
- Nicht eindeutig
|
||||
|
||||
Das meschanische System ist so definiert durch $q_(i) "und" L$.
|
||||
Der *verallgemeinerte Impuls* ist gegeben durch
|
||||
$
|
||||
p_(i) := (partial L) / (partial dot(q)_(i) ).
|
||||
$
|
||||
Eine *zyklische verallgemeinerte Koordinate* $q_(i) $ erfuellt
|
||||
$
|
||||
(partial L) / (partial q_(i) ) = 0 ==> dif / (dif t) (partial L) / (partial dot(q)_(i) ) = 0 <==> (dif p_(i) ) / (dif t) = 0 <==> p_(i) "erhalten"
|
||||
$
|
||||
|
||||
19
S2/AnaMech/other/Hahn_Blatt5A5.py
Normal file
19
S2/AnaMech/other/Hahn_Blatt5A5.py
Normal file
@@ -0,0 +1,19 @@
|
||||
# Hilfreiche Pakete
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
theta = np.linspace(-np.pi, np.pi, 1000)
|
||||
cs = 1 / np.sin(theta/2)**4
|
||||
|
||||
plt.semilogy(theta*180/np.pi, cs)
|
||||
plt.xlabel(r"Streuwinkel $\theta$ [deg]")
|
||||
plt.ylabel(r"$\csc^4(\theta/2)$")
|
||||
plt.title("Differentieller Wirkungsquerschnitt für $U(r) = \\alpha/r^2$")
|
||||
plt.show()
|
||||
|
||||
# Task
|
||||
# Calculate the differential Wirkungsquerschnitt ds/dOm fuer das repulsive
|
||||
# Potential V(r) = a/r^2, a > 0
|
||||
|
||||
# Plotte den Wirkungsquerschnitt als funktion des Raumwinkels theta
|
||||
|
||||
@@ -4,7 +4,7 @@
|
||||
#let rot = math.op("rot")
|
||||
#let grad = math.op("grad")
|
||||
|
||||
#let conf(num: none, date: "", type: none, body, ueb: false) = {
|
||||
#let conf(num: none, date: "", type: none, ueb: false, body) = {
|
||||
// Global settings
|
||||
show: default
|
||||
|
||||
|
||||
Reference in New Issue
Block a user