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S2/AnaMech/VL/AnMeVL12.typ
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S2/AnaMech/VL/AnMeVL12.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: 12,
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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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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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$
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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
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$
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(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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$
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#highlight[TODO: berechne Drehimpuls in Kugelkoordinaten]
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]
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= Erhaltungsgroessen in L I und L II
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Es gilt
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$
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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 \
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g_(alpha) (arrow(x), t) = 0 , space alpha = 1, ..., R \
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=> (x_(n) , lambda_(alpha) ).
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$
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== Energieerhaltung
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=== Lagrange I
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Wir betrachten Zwangskraefte
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$
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g_(alpha) (arrow(x), t) = 0 \
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arrow(Z)_(alpha) * d arrow(r) = 0.
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$
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Wir wollen Erhaltung von mit $V = V (x_(n) )$
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$
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E = T + V \
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(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) \
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(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).
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$
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Nebenrechnung
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$
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(dif g_(alpha) ) / (dif t) = sum _(n) (partial g_(alpha) ) / (partial x_(n) ) dot(x)_(n) + (partial g_(alpha) ) / (partial t) = 0.
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$
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Das ergibt dann
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$
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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) ) .
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$
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Wir erwarten die Energieerhaltung nur fuer abg. und kons. Systeme.
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Falls die Zwangskraefte sich also nicht mit der Zeit aendern, dann gilt die Energieerhaltung
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$
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(partial g_(alpha) ) / (partial t) = 0 space forall alpha \
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=> E = T + V = "const".
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$
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=== Lagrange II
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Allgemeine Erhaltungsgroessen.
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Erinnerung fuer mechanische Groessen
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$
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Q = Q (q, dot(q), t) \
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(dif Q) / (dif t) = 0 <=> Q "erhalten".
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$
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Wie bekommt man im Lagrangeformalismus erhaltungsgroessen
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+ Zyklische Koordinaten
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$
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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".
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$
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Die Frage nach zyklischen Koordinaten darf nur gestellt werden, wenn man schon $f$ unabhaengige Koordinaten hat.
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Nun
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$
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(partial g_(alpha) ) / (partial t) != 0 <=>^(!) (partial x_(n) ) / (partial t) != 0 and (partial L) / (partial t) = 0 \
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(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) \
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=> dif / (dif t) (sum _(k) (partial L) / (partial dot(q)_(k) ) - L) = - (partial L) / (partial t)
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$.
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Betrachte als Nebenrechnung
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$
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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) ) ).
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$
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Es gilt also
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$
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(partial L) / (partial t) = 0 <=> sum _(k = 1) ^(f) p_(k) dot(q)_(k) - L "erhalten"
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$ <erh>
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Jetzt der Fall, dass
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$
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(partial g_(alpha) ) / (partial t) &= 0 , space (partial L) / (partial t) = 0 \
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&=> x_(n) = x_(n) (q) \
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&=> T = sum _(i, i) m_(i k) (q) dot(q)_(i) dot(q)_(k) \
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&=> 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)) \
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(partial L) / (partial t) = 0 &=> V (q, t) = V (q) \
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sum_(k = 1)^(f) (partial L) / (partial dot(q)_(k) ) dot(q)_(k) - L &= 2 T - (T - V) = T + V = E.
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$
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Nun der naechste Fall, dass
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$
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(partial L) / (partial t) != 0 => "keine Erhaltungsgroesse".
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$
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Zuletzt
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$
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(partial L) / (partial t) = 0 , space (partial g_(alpha) ) / (partial t) != 0 => (partial x_(n) ) / (partial t) != 0 \
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=> "Erhaltungsgroesse durch" #[@erh].
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$
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= Beispiele
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#example[
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MP im Kegel.
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Hier gilt fuer die Zwangsbedingung
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$
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(partial g_(alpha) ) / (partial t) = 0 => (partial x_(n) ) / (partial t) = 0 \
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=> (partial L) / (partial t) = 0 => E = T + V.
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$
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]
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#example[
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Perle auf rotierendem Draht.
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Der Draht rotiert in der Ebene mit
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$
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phi = omega t , space omega = "const." \
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g (x, g, t) = tan ^(-1) (y/x) - omega t = phi - omega t = 0 \
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=> (partial g_(alpha) ) / (partial t) != 0 .. (alpha = 1).
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$
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Es gilt fuer die Trafo
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$
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x = r cos (omega t) \
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y = r sin (omega t) \
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=> x_(n) = x_(n) (q, t) \
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=> L = m/2 (dot(r)^2 + omega ^2 r ^2 ) , space V = 0 \
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(partial L) / (partial t) = 0 => underbrace(sum _(k) p _(k) dot(q)_(k) - L, = O) "erhalten" \
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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
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$
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]
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