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243
S2/AnaMech/VL/AnMeVL11.typ
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243
S2/AnaMech/VL/AnMeVL11.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: 11,
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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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E: 26.05.2025
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= Wiederholung
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Die Hoersaaluebung am 30.5. findet statt.
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Naechste Vorlesung VL12 werden die Erhaltungssaetze in Lagrange I und II diskutiert.
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Die letzten HA beinhalten nicht alle Themen der Klausur.
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Es wird mehr Bonuspunkt geben.
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In Lagrange I wird mehr geloest als benoetigt wird.
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= Mechanische Systeme mit Zwangsbedingungen
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Einmal mit N MP in 3D. Hier gibt es dann R ZB der Form
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$
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g_(alpha) = (arrow(x), t) = 0 , space alpha = 1, ..., R , space arrow(x) in RR^(3 N) .
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$
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Die BWGL in Lagrange I lauten dann
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$
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m_(n) dot.double(x)_(n) = F_(n) + sum_(alpha = 1)^(R) lambda_(alpha) arrow(nabla) g_(alpha) (arrow(x), t).
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$
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Hier sind dann noch 3N Gleichungen zu bestimmen und die R $lambda$ mit dem $arrow(x)$.
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Die Zahl der unabhaengigen Koordinaten ist
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$
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f = 3 N - R.
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$
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Dieses Problem mit Zwangsbedingungen wird als ein Problem mit Nebenbedingungen bezeichnet.
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= Generalisierte Koordinaten
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Die Generaliserten Koordinaten haben die Form und Eigenschaft
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$
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q_(k), space k = 1,...f , space x_(n) = x_(n) (q, t), \
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q = {q_1, q_2, ..., q_(f) }.
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$
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Es gilt dann fuer alle Werte von $q_(k) $ und fuer alle Zeiten
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$
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g_(alpha) (x(q_1, ...,q_(f) ), t) = 0.
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$
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#example[
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Fuer einen MP auf einer Kugeloberflaeche gilt
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$
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x_(1) ^2 + y_(1) ^2 + z_(1) ^2 - R^2 = 0.
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$
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Hier gibt es dann die generaliserten Koordinaten
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$
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theta "und" phi.
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$
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]
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#example[
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Das Doppelpendel.
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Hier ist eine Skizze des Pendels in kartesischen Koordinaten.
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Wir sind in der Ebene $==>$ Es gibt 4 kartesischen Koordinaten..
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Die Zwangsbedingungen sind dann
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$
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x_(1) ^2 + y_(1) ^2- l^2 = 0 , space abs(arrow(r_(1) ))= l_(1) \
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(x_(1) - x_(2) )^2 + (y_(1) - y_(2) )^2 - l_(2) ^2 = 0 , space abs(arrow(r)_(1) - arrow(r)_(2) ) = l_(2).
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$
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Die Generalisierten Koordinaten sind dann die Winkel
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$
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phi_(1) "und" phi_(2)
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$
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mit den Trafos
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$
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x_(1) = l_(1) sin phi_(1) \
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y_(1) = - l_(1) cos phi_(1) \
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x_(2) = x_(1) + l_(2) sin phi_(2) \
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y_(2) = - l_(1) cos phi_(1) - l_(2) cos phi_(2).
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$
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]
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= Elimieren der Zwangskraefte
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Es gibt hier $f$ BWGL fuer $q_(k) $.
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Lagrange II folgt
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$
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dif / (dif t) (partial L) / (partial dot(q)_(k) ) - (partial L) / (partial q_(k) ) = 0 , space L = T - V.
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$
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Als Ansatz gilt dass die $g_(alpha) $ nicht varrieren bei Varriation von $q_k $!
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Es gilt so
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$
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forall q_(k): (partial g_(alpha) ) / (partial q_(k) ) = 0 =>^("Kettenregel") sum_(n = 1)^(3 N) (partial g_(alpha) ) / (partial x_(n) ) (partial x_(n) ) / (partial q_(k) ) = 0 space forall k = 1, ... ,f.
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$
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Fixiere $q_(k) $ dann multipliziere alle 3N Gleichungen mit $(partial x_(n) ) / (partial q_(k) ) $ und dann bilde die Summe
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ueber alle 3N Lagrange I Gleichungen.
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Es gibt hier $f$ Moeglichkeiten $==>$ $f$ Gleichungen.
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Dadurch folgt
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$
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sum_(n) m_(n) dot.double(x)_(n) (partial x_(n) ) / (partial q_(k) ) = sum _(n) F_(n) (partial x_(n) ) / (partial q_(k) ) + underbrace( sum _(alpha) lambda_(alpha) sum _(n) (partial g_(alpha) ) / (partial x_(n) ) (partial x_(n) ) / (partial q_(k) ), = 0).
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$
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So entstehen $f$ Gleichungen.
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Beispiel fuer die Notation
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$
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h (q) &= g (r (q)) \
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&= g (f (q_1 ), ..., f_(R) (q_(1) ) ).
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$
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= Generaliserte Geschwindigkeiten
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Es gilt fuer die Geschwindigkeiten
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$
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dot(q)_(k) = (dif q_(k) ) / (dif t) , space k = 1, ..., f \
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x_(n) = x_(n) (q,t) => ^(?) dot(x)_(n) = dot(x)_(n) (q, dot(q), t) \
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dot(x)_(n) = dif / (dif t) x_(n) (q,t) = sum_(k = 1)^(f) (partial x_(n) ) / (partial q_(k) ) dot(q)_(k) + (partial x_(n) ) / (partial t)
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=> (partial dot(x)_(n) ) / (partial dot(q)_(k) ) = (partial x_(n) ) / (partial q_(k) ).
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$
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Erinnerung 1MP
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$
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T = sum_(i=1)^(3) m/2 dot(x)_(i) ^2 = sum_(j, k = 1)^(3) m/2 g_(i k) dot(q)_(j) dot(q)_(k) , space g_(j k) = arrow(g)_(j) * arrow(g)_(k).
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$
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Hier folgt dann
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$
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T &= sum_(i=1)^(3 N) m_(n) /2 dot(x)_(n) ^2 = sum_(i=1)^(3 N) m_(n) /2 (sum_(i=1)^(f) (partial x_(n) ) / (partial q_(k) ) + (partial x_(n) ) / (partial t) ) (sum_(i=1)^(f) (partial x_(n) ) / (partial q_(j) ) dot(q)_(j) + (partial x_(n) ) / (partial t) ) \
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&= sum_(k, j = 1)^(f) m_(k j) dot(q)_(k) dot(q)_(j) + underbrace(sum_(k)^(f) b_(k) (q,t) dot(q)_(k) + c (q,t), "nur wenn" (partial x_(n) ) / (partial t) != 0).
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$
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Hier steht dann insgesamt
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$
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sum_(k, j) sum_(n) (m_(n) /2 (partial x_(n) ) / (partial q_(k) ) (partial x_(n) ) / (partial q_(j) ) )dot(q)_(k) dot(q)_(j).
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$
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Ferner gilt
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$
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g_(alpha) (arrow(x), t)= 0 \
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=> x_(n) = x_(n) (q,t).
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$
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= Partielle Ableitungen von der kinetischen Energie
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Schreibe
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$
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T = sum _(n) m_(n) /2 dot(x)_(n) ^2 = T (q, dot(q), t) \
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(partial T) / (partial q_(k) ) = sum _(n) m_(n) dot(x)_(n) (partial dot(x)_(n) ) / (partial q_(k) ).
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$
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Betrachte nun
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$
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(partial T) / (partial dot(q)_(k) ) = sum _(n) m_(n) dot(x)_(n) (partial dot(x)_(n) ) / (partial dot(q)_(k) ) = sum _(n) m_(n) dot(x)_(n) (partial x_(n) ) / (partial q_(k) ) \
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$
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Nun die totale Zeitableitung
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$
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dif / (dif t) (partial T) / (partial dot(q)_(k) ) = sum_n m_(n) dot.double(x)_(n) (partial x_(n) ) / (partial q_(k) ) + sum _(n) m_(n) dot(x)_(n) underbrace(dif / (dif t)(partial x_(n) ) / (partial q_(k) ), = (partial dot(x)_(n) ) / (partial q_(k) ) )
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$
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Der Faktor $1/2$ verschwindet hier durch die Kettenregel.
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Zusammen ergibt das dann
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$
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dif / (dif t) (partial T) / (partial dot(q)_(k) ) - (partial T) / (partial q_(k) ) = sum _(n) m_(n) dot.double(x)_(n) + underbrace(sum _(n) m_(n) dot(x)_(n) (partial dot(x)_(n) ) / (partial q_(k) ) - sum _(n) dot(x)_(n) (partial dot(x)_(n) ) / (partial q_(k) ), = 0) \
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= sum _(n) F_(n) (partial x_(n) ) / (partial q_(k) ) , space k = 1, ..., f.
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$
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#definition[
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Generalisierte Kraefte sind gegeben durch
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$
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Q_(k) = sum_(i=1)^(3 N) F_(n) (partial x_(n) ) / (partial q_(k) ) \
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=> dif / (dif t) (partial T) / (partial dot(q)_(k) ) - (partial T) / (partial q_(k) ) = Q_(k) , space k = 1,...,f.
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$
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]
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Es gilt fuer konservative Kraefte mit $L = L (q, dot(q), t)$
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$
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F_(n) = - (partial V) / (partial x_(n) ) => Q_(k) = sum _(n) F_(n) (partial x_(n) ) / (partial q_(k) ) = - sum _(n) (partial V) / (partial x_(n) ) (partial x_(n) ) / (partial q_(k) ) = - (partial V (q, t)) / (partial q_(k) ) \
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dif / (dif t) (partial (T-V)) / (partial dot(q)_(k) ) - (partial (T-V)) / (partial q_(k) ) = 0 \
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=> dif / (dif t) (partial L) / (partial dot(q)_(k) ) - (partial L) / (partial q_(k) ) = 0 , space 1, ..., f.
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$
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Die Grundaufgabe ist herrauszufinden welche Aussagen ueber Lagrangefunktionen gemacht werden koenne.
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#example[
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MP auf einer rotierenden Stange.
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Wir geben vor
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$
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arrow(omega) = omega arrow(e)_(z) \
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=> phi = omega t \
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f = 2 -1 = 1 \
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$
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Waehle generalisierte Koordinaten
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$
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r = r (t) \
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x = x (r, t) = r cos (omega t) \
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y = y (r, t) = r sin (omega t)
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$
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Fuer die Lagrangefunktion ergib sich
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$
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V &= 0 => L = T (r, dot(r), t) \
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T &= m/2 (dot(x)^2 + dot(y)^2 + dot(z)^2 ) = m/2 (dot(r)^2 + omega^2 r^2 ) \
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&= m/2 (dot(r)^2 + dot(phi)^2 r^2 ).
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$
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Dann bilde die 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) = omega ^2 m r \
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=> m dot.double(r) - omega^2 m r = 0.
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$
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Dadurch folgt fuer die Loesung
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$
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r (t) = a e ^(omega t) + b e ^(- omega t).
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$
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]
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200
S2/AnaMech/VL/AnMeVL12.typ
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200
S2/AnaMech/VL/AnMeVL12.typ
Normal file
@@ -0,0 +1,200 @@
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// Main VL template
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#import "../preamble.typ": *
|
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|
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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: 12,
|
||||
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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E: 2.6.25
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= Uebersicht
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+ Ein Beispiel fuer Lagrange II
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+ Erhaltungsgroessen in L I und L II
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+ Symetrien u. Erhaltungsgroessen
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= Lagrange Funktion
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Es gilt fuer die Lagrange Funktion
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$
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L (q_(k) dot(q)_(k), t) = T (q_(k), dot(q)_(k) , t) - V (q_(k) , t) , space k = 1, ..., f.
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$
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Lagrange II
|
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$
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dif / (dif t) (partial L) / (partial dot(q)_(k) ) - (partial L) / (partial q_(k) ) = 0 , space k = 1, ..., f.
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$
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#example[
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MP im Kegel.
|
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Es wirkt nur die Gravitationskraft.
|
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Hier gilt fuer die Koordinaten
|
||||
$
|
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R = 1 => f = 2 \
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rho = z tan alpha.
|
||||
$
|
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|
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Es gilt fuer eine Z.B.
|
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$
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g (x, y, z) = x ^2 + y ^2 - z ^2 tan ^2 alpha = 0 \
|
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x = r sin alpha cos phi , space theta = alpha \
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y = r sin alpha sin phi \
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z = r cos alpha.
|
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$
|
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Dann waehle fuer die generalisierten Koordinaten
|
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$
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q = {r, phi}.
|
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$
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Das ergibt fuer die Lagrange-Funktion
|
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$
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L (x,y,z,dot(x),dot(y), dot(z)) = m/2 (dot(x)^2 + dot(y)^2 + dot(z)^2 ) - m g z \
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=> L = m/2 (dot(r)^2 + r ^2 dot(phi)^2 sin ^2 alpha) - underbrace(m g cos alpha r, V (r)).
|
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$
|
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Hier gibt es eine zyklische Koordinate
|
||||
$
|
||||
(partial L) / (partial phi) = 0 => p_(phi) = (partial L) / (partial dot(phi)) = m r ^2 sin ^2 alpha dot(phi) prop L_(z)
|
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$
|
||||
#highlight[TODO: berechne Drehimpuls in Kugelkoordinaten]
|
||||
]
|
||||
|
||||
= Erhaltungsgroessen in L I und L II
|
||||
|
||||
Es gilt
|
||||
$
|
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m_(n) dot.double(x)_(n) = F_(n) sum_(alpha = 1)^(R) alpha_(alpha) (partial g_(alpha) (arrow(x), t)) / (partial x_(n) ) , space n = 1, ..., 3N \
|
||||
g_(alpha) (arrow(x), t) = 0 , space alpha = 1, ..., R \
|
||||
=> (x_(n) , lambda_(alpha) ).
|
||||
$
|
||||
|
||||
== Energieerhaltung
|
||||
|
||||
=== Lagrange I
|
||||
|
||||
Wir betrachten Zwangskraefte
|
||||
$
|
||||
g_(alpha) (arrow(x), t) = 0 \
|
||||
arrow(Z)_(alpha) * d arrow(r) = 0.
|
||||
$
|
||||
|
||||
Wir wollen Erhaltung von mit $V = V (x_(n) )$
|
||||
$
|
||||
E = T + V \
|
||||
(dif T) / (dif t) = dif / (dif t) sum_(n) 1/2 m_(n) dot(x)_(n) ^2 = sum _(n) m_(n) dot(x)_(n) dot.double(x)_(n) \
|
||||
(dif V) / (dif t) = sum _(n) (partial_(n) V)dot(x)_(n) = - sum _(n) F_(n) dot(x)_(n) , space "Konservativ" => partial_(n) V = - F_(n).
|
||||
$
|
||||
Nebenrechnung
|
||||
$
|
||||
(dif g_(alpha) ) / (dif t) = sum _(n) (partial g_(alpha) ) / (partial x_(n) ) dot(x)_(n) + (partial g_(alpha) ) / (partial t) = 0.
|
||||
$
|
||||
Das ergibt dann
|
||||
$
|
||||
dif / (dif t) (T + V) = sum _(n) sum _(alpha) lambda_(alpha) (partial g_(alpha) ) / (partial x_(n) ) dot(x)_(n) = sum _(alpha) lambda_(alpha) (- (partial g_(alpha) ) / (partial t) ) .
|
||||
$
|
||||
Wir erwarten die Energieerhaltung nur fuer abg. und kons. Systeme.
|
||||
|
||||
Falls die Zwangskraefte sich also nicht mit der Zeit aendern, dann gilt die Energieerhaltung
|
||||
$
|
||||
(partial g_(alpha) ) / (partial t) = 0 space forall alpha \
|
||||
=> E = T + V = "const".
|
||||
$
|
||||
|
||||
=== Lagrange II
|
||||
|
||||
Allgemeine Erhaltungsgroessen.
|
||||
|
||||
Erinnerung fuer mechanische Groessen
|
||||
$
|
||||
Q = Q (q, dot(q), t) \
|
||||
(dif Q) / (dif t) = 0 <=> Q "erhalten".
|
||||
$
|
||||
|
||||
Wie bekommt man im Lagrangeformalismus erhaltungsgroessen
|
||||
+ Zyklische Koordinaten
|
||||
$
|
||||
underbrace((partial L) / (partial q_(k) ) = 0, q_(k) "zyklisch") => dif / (dif t) (partial L) / (partial dot(q)_(k) ) = 0 => p_(k) "erhalten" , space p_(k) := "gen. Impuls".
|
||||
$
|
||||
Die Frage nach zyklischen Koordinaten darf nur gestellt werden, wenn man schon $f$ unabhaengige Koordinaten hat.
|
||||
|
||||
Nun
|
||||
$
|
||||
(partial g_(alpha) ) / (partial t) != 0 <=>^(!) (partial x_(n) ) / (partial t) != 0 and (partial L) / (partial t) = 0 \
|
||||
(dif L) / (dif t) = sum _(k) (partial L) / (partial q_(k) ) dot(q)_(k) + sum _(k) (partial L) / (partial dot(q)_(k) ) dot.double(q)_(k) + (partial L) / (partial t) \
|
||||
=> dif / (dif t) (sum _(k) (partial L) / (partial dot(q)_(k) ) - L) = - (partial L) / (partial t)
|
||||
$.
|
||||
|
||||
Betrachte als Nebenrechnung
|
||||
$
|
||||
dif / (dif t) sum_(i=1)^(t) (partial L) / (partial dot(q)_(i) ) dot(q)_(i) = sum _(k) dot(q)_(k) dif / (dif t) (partial L) / (partial dot(q)_(k) ) = sum _(k) dot.double(q)_(k) (partial L) / (partial dot(q)_(k) ) =^("L II") sum _(k) (dot(q)_(k) (partial L) / (partial q_(k) ) + dot.double(q)_(k) (partial L) / (partial dot(q)_(k) ) ).
|
||||
$
|
||||
|
||||
Es gilt also
|
||||
$
|
||||
(partial L) / (partial t) = 0 <=> sum _(k = 1) ^(f) p_(k) dot(q)_(k) - L "erhalten"
|
||||
$ <erh>
|
||||
|
||||
Jetzt der Fall, dass
|
||||
$
|
||||
(partial g_(alpha) ) / (partial t) &= 0 , space (partial L) / (partial t) = 0 \
|
||||
&=> x_(n) = x_(n) (q) \
|
||||
&=> T = sum _(i, i) m_(i k) (q) dot(q)_(i) dot(q)_(k) \
|
||||
&=> sum_(i=1)^(f) (partial L) / (partial dot(q)_(i) ) = sum _(k) (partial T) / (partial dot(q)_(k) ) dot(q)_(k) = 2 T (q, dot(q)) \
|
||||
(partial L) / (partial t) = 0 &=> V (q, t) = V (q) \
|
||||
sum_(k = 1)^(f) (partial L) / (partial dot(q)_(k) ) dot(q)_(k) - L &= 2 T - (T - V) = T + V = E.
|
||||
$
|
||||
|
||||
Nun der naechste Fall, dass
|
||||
$
|
||||
(partial L) / (partial t) != 0 => "keine Erhaltungsgroesse".
|
||||
$
|
||||
Zuletzt
|
||||
$
|
||||
(partial L) / (partial t) = 0 , space (partial g_(alpha) ) / (partial t) != 0 => (partial x_(n) ) / (partial t) != 0 \
|
||||
=> "Erhaltungsgroesse durch" #[@erh].
|
||||
$
|
||||
|
||||
= Beispiele
|
||||
|
||||
#example[
|
||||
MP im Kegel.
|
||||
|
||||
Hier gilt fuer die Zwangsbedingung
|
||||
$
|
||||
(partial g_(alpha) ) / (partial t) = 0 => (partial x_(n) ) / (partial t) = 0 \
|
||||
=> (partial L) / (partial t) = 0 => E = T + V.
|
||||
$
|
||||
]
|
||||
|
||||
#example[
|
||||
Perle auf rotierendem Draht.
|
||||
|
||||
Der Draht rotiert in der Ebene mit
|
||||
$
|
||||
phi = omega t , space omega = "const." \
|
||||
g (x, g, t) = tan ^(-1) (y/x) - omega t = phi - omega t = 0 \
|
||||
=> (partial g_(alpha) ) / (partial t) != 0 .. (alpha = 1).
|
||||
$
|
||||
Es gilt fuer die Trafo
|
||||
$
|
||||
x = r cos (omega t) \
|
||||
y = r sin (omega t) \
|
||||
=> x_(n) = x_(n) (q, t) \
|
||||
=> L = m/2 (dot(r)^2 + omega ^2 r ^2 ) , space V = 0 \
|
||||
(partial L) / (partial t) = 0 => underbrace(sum _(k) p _(k) dot(q)_(k) - L, = O) "erhalten" \
|
||||
O = m dot(r) dot(r) - m/2 dot(r)^2 - m/2 omega^2 r^2 = m/2 dot(r)^2 - m/2 omega^2 r^2
|
||||
$
|
||||
]
|
||||
|
||||
196
S2/AnaMech/VL/AnMeVL13.typ
Normal file
196
S2/AnaMech/VL/AnMeVL13.typ
Normal file
@@ -0,0 +1,196 @@
|
||||
// Main VL template
|
||||
#import "../preamble.typ": *
|
||||
|
||||
// Fix theorems to be shown the right way in this document
|
||||
#import "@preview/ctheorems:1.1.3": *
|
||||
#show: thmrules
|
||||
|
||||
// Main settings call
|
||||
#show: conf.with(
|
||||
// May add more flags here in the future
|
||||
num: 13,
|
||||
type: 0, // 0 normal, 1 exercise
|
||||
date: datetime.today().display(),
|
||||
//date: datetime(
|
||||
// year: 2025,
|
||||
// month: 5,
|
||||
// day: 1,
|
||||
//).display(),
|
||||
)
|
||||
|
||||
= Widerholung
|
||||
|
||||
== Erhaltungsgroessen in Lagrange II hier also Lagrangeformalismus
|
||||
|
||||
Was ist die Lagrangefunktion und wie leite ich aus dieser
|
||||
die Bewegungsgleichungen fuer die gen. Koordinaten ab?
|
||||
|
||||
Zyklische Koordinate $q_(k) :<=> $ zugehoeriger kanonischer Impuls ist erhalten.
|
||||
|
||||
Es gilt also
|
||||
$
|
||||
(partial L) / (partial q_(k) ) = 0 => p_(k) = (partial L) / (partial dot(q)_(k) ) = "const.".
|
||||
$
|
||||
|
||||
Eine weitere Erhaltungsgroesse ist
|
||||
$
|
||||
(partial L) / (partial t) =0 => sum _(k) dot(q)_(k) p_(k) - L .. "erhalten".
|
||||
$
|
||||
|
||||
Falls die Zwangsbedingungen zeitunabhaengig sind, dann ist diese Erhaltungsgroesse gleich der Gesamtenergie.
|
||||
Dies bekommt eine tiefere Bedeutung im Hamiltonformalismus.
|
||||
|
||||
= Symetrien und Erhaltungsgroessen
|
||||
|
||||
Ein Kreis hat eine $2 pi$ Rotationssymetrie. Ein Quadrad hat eine 4-fache Symetrie mit $phi_(n) = n pi/2$.
|
||||
Ein Dreieck hat eine 3-fache Symetrie.
|
||||
|
||||
== Symetrieoperationen
|
||||
- Rotation mit Winkel
|
||||
- Translation im Raum mit Vektor
|
||||
- Wahl des Inertialsystems mit Vektor
|
||||
- Zeittranslation mit der Referenz der Zeit
|
||||
- (Spiegelung)
|
||||
- (Skalierung)
|
||||
|
||||
Jedes mal wenn ich invarianz unter einer
|
||||
dieser Operation ist, dann ist dies eine Erhaltungsgroesse.
|
||||
Es lassen sich also so maximal 10 Erhaltungsgroessen in diesem System finden.
|
||||
Diese Klasse wird dann als allgemeine Gallileitrafo bezeichnet.
|
||||
|
||||
== Galileitrafo in Inertialsystemen
|
||||
|
||||
Wir betrachten N MP mit den Ortsvektoren mit $n = 1, ..., N$
|
||||
$
|
||||
arrow(r)_(n) = vec(r_(n 1), r_(n 2) , r_(n 3) )_(x y z).
|
||||
$
|
||||
Wir bleiben dabei voellig in kartesischen Koordinaten.
|
||||
Es gelten die Trafos
|
||||
$
|
||||
r'_(n, i) = sum_(j=1)^(3) D_(i j) r_(j) + v_("rel", i) t + a_(i) \
|
||||
t' = t + t_0.
|
||||
$
|
||||
Wobei fuer die Drehung gilt
|
||||
$
|
||||
arrow(r)' = D arrow(r) \
|
||||
det D = 1 , space D D^(T) = D^(T) D = E.
|
||||
$
|
||||
Skalarprodukte sind also invariant unter einer Drehung.
|
||||
Wir nehmen uns eine Drehachse
|
||||
$
|
||||
arrow(n) = arrow(e)_(z).
|
||||
$
|
||||
Dieser Vektor $arrow(n)$ legt dann die Drehachse fest.
|
||||
Es folgt fuer die Drehmatrix
|
||||
$
|
||||
D_(z) = mat(
|
||||
cos phi, -sin phi, 0;
|
||||
sin phi, cos phi, 0;
|
||||
0, 0, 1;
|
||||
).
|
||||
$
|
||||
Es gilt immer, dass die Rotationsmatrix zerlegt werden kann
|
||||
$
|
||||
D = D_(alpha arrow(n)_(alpha) ) D_(beta arrow(n)_(beta) ) D _(gamma arrow(n)_(gamma) ).
|
||||
$
|
||||
|
||||
= Symetrieinvarianz der Lagrangefkt. unter SYmetrietrafo
|
||||
|
||||
Ellipsengleichung
|
||||
$
|
||||
(x^2 ) / (a^2 ) + (y^2 ) / (b^2 ) = 1.
|
||||
$
|
||||
|
||||
#example[
|
||||
Allgemeine Galileitrafos
|
||||
- Inertialsystem
|
||||
- kart. Koordinaten $==>$ $f = 3 N$
|
||||
- Abg. Systeme $(arrow(L), arrow(p), E)$ sind 7 Erhaltungsgroessen
|
||||
]
|
||||
|
||||
Exkurs in die Festkoerperphysik unter dem Blickwinkel der Symetrien und der Betrachtung der Phasenuebergaenge von Materie.
|
||||
|
||||
Das Ziel ist hier die Invarianz unter kontinuierlichen Symetrietrafo $==>$ Erhaltungsgroesse.
|
||||
Die allgemeine Lagrange lautet
|
||||
$
|
||||
L = sum _(n) dot(arrow(r))_(n) ^2 m_(n) /2 - V (arrow(r)_(n), t ) , space m_(n = m).
|
||||
$
|
||||
Und in abgeschlossenen Systemen ist sie gegeben durch
|
||||
$
|
||||
L_(O) = sum _(n) dot(arrow(r))_(n) ^2 m_(n) /2 - 1/2 sum _(n, m) v_(n m ) (arrow(r)_(n) - arrow(r)_(m) ).
|
||||
$
|
||||
Betrachte die Homogenitaet der Zeit
|
||||
$
|
||||
arrow(r)'_(n) = arrow(r)_(n) => d arrow(r)' = d arrow(r) \
|
||||
t' = t + epsilon => d t' = d t \
|
||||
=> dot(arrow(r))'_(n) = arrow(r)_(n)
|
||||
$
|
||||
Es folgt
|
||||
$
|
||||
L' = L' (arrow(r)'_(n) , dot(arrow(r))'_(n), t' ) = L' (arrow(r)_(n), dot(arrow(r))_(n) , t + epsilon ).
|
||||
$
|
||||
Fuer eine infinitisimal kleine Symetrietrafo gilt, dass der Parameter $epsilon << 1$.
|
||||
Dadurch muss bei einer Entwicklung nur die fuehrende Ordnung betrachtet werden.
|
||||
Es gilt natuerlich auch, dass $L' |_(epsilon = 0) = L $.
|
||||
|
||||
Allgemeine Lagrangefunktion Invarianzbedingung
|
||||
$
|
||||
((dif L') / (dif epsilon) )(epsilon = 0) = 0 .. "fuer alle Faelle".
|
||||
$
|
||||
Zeittrafo der allgemeinen Lagrangefunktion
|
||||
$
|
||||
((dif L') / (dif epsilon) )(epsilon = 0) = ((partial L') / (partial t') (dif t') / (dif epsilon) )(epsilon = 0) = (partial L) / (partial t) * 1.
|
||||
$
|
||||
Fuer abgeschlossene Systeme gilt dann
|
||||
$
|
||||
((dif L'_(O) ) / (dif epsilon) )(epsilon = 0) = 0.
|
||||
$
|
||||
Allgemein gilt nur
|
||||
$
|
||||
((dif L') / (dif epsilon) )(epsilon = 0) = dif / (dif t) (sum _(n) dot(arrow(r))_(n) * arrow(p) - L).
|
||||
$
|
||||
Dann wieder fuer die abgeschlossenen
|
||||
$
|
||||
(partial L_(O) ) / (partial t) = 0 => sum _(n) dot(arrow(r))_(n) * arrow(p)_(n) - L = "const." = E = T + V.
|
||||
$
|
||||
Es gilt also: Homogenitaet der Zeit $<=> $ Gesamtenergie ist erhalten.
|
||||
Dies ergibt das Paar $E <--> t_0 $.
|
||||
|
||||
= Homogenitaet des Raumes
|
||||
Es gilt fuer die Transformation der Lagrangegleichung
|
||||
$
|
||||
arrow(r)'_(n) = arrow(r)_(n) + epsilon arrow(a) , space arrow(a): "fest, beliebig" \
|
||||
t' = t \
|
||||
dot(arrow(r))'_(n) = dot(arrow(r))_(n) \
|
||||
=> L' (arrow(r)'_(n) , dot(arrow(r))'_(n) , t') = L' (arrow(r)_(n) + epsilon arrow(a), dot(arrow(r))_(n) , t).
|
||||
$
|
||||
Wieder betrachte fuer die allgemeine Lagrangefunktion
|
||||
|
||||
|
||||
$
|
||||
((dif L') / (dif epsilon) )(epsilon = 0) = ((partial L') / (partial arrow(r)'_(n)) (dif arrow(r)'_(n) ) / (dif epsilon) )(epsilon = 0) = sum_n (partial L) / (partial arrow(r)_(n) ) *arrow(a) = sum _(n)( dif / (dif t) (partial L) / (partial dot(arrow(r))_(n) )) * arrow(a) \
|
||||
=> ((dif L') / (dif epsilon) )(epsilon = 0) = dif / (dif t) (sum _(n) p_(n) * arrow(a)) = dif / (dif t) (arrow(P) * arrow(a)) , space arrow(P): "Gesamtimpuls".
|
||||
$
|
||||
|
||||
Im abgeschlossenen System gilt
|
||||
$
|
||||
((dif L'_(O) ) / (dif epsilon) ) = 0 => dif / (dif t) (arrow(P) * arrow(a)) = 0 => dif / (dif t) arrow(P) = 0
|
||||
$
|
||||
wodurch im raeumlich invarianten System der Gesamtimpuls erhalten ist.
|
||||
Hier gilt
|
||||
$
|
||||
arrow(r)'_(n) - arrow(r)'_(m) = arrow(r)_(n) - arrow(r)_(m).
|
||||
$
|
||||
|
||||
=== Nebenrechnungen
|
||||
|
||||
Es gilt
|
||||
$
|
||||
dif / (dif t) (a b) = (dif a) / (dif t) b + a (dif b) / (dif t) \
|
||||
(partial f) / (partial arrow(r)) = arrow(nabla)_(x y z) f \
|
||||
arrow(r ) = vec(x, y, z).
|
||||
$
|
||||
Natuerlich gilt fuer die Lagrangefunktion
|
||||
$
|
||||
(partial L) / (partial t) = 0 => sum_(i=1)^(f) p_(i) dot(q)_(i) - L .. "erhalten".
|
||||
$
|
||||
Reference in New Issue
Block a user