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S2/AnaMech/VL/AnMeVL9.typ
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169
S2/AnaMech/VL/AnMeVL9.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: 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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$
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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.
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$
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Jetzt werden beliebige Koordinaten gewaehlt
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$
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m arrow(g)_(i) * dot.double(arrow(r)) = arrow(g)_(i) * arrow(f) \
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<==> 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)) \
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<==> 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) ) \
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<==> 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) )
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$
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Nebenrechung
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$
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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) \
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= (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) )
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$
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Betrachtung der kinetischen Energie
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$
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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) ) \
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==> dif / (dif t) ((partial T) / (partial dot(q)_(i) ) ) = (partial T) / (partial q_(i) ) + arrow(g)_(i) * arrow(f).
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$
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Allgemein gilt fuer die Produktregel
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$
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(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) ) ).
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$
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Skalarprodukt ist eine Projektion.
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Verallgemeinere die kinetische Energie
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$
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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) \
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&= m/2 sum_(i,j) g_(i j) dot(q)_(i) dot(q)_(j), \
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&g_(i j) = g _(i j) (q_1, q_2, q_3 ).
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$
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= Konservative Kraftfelder
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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 $L (q_1, q_2, q_3, dot(q)_(1) , dot(q)_(2) , dot(q)_(3) ) := T (q_(i) , dot(q)_(i) ) - V (q_(i) )$
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$
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arrow(f) (arrow(r)) = - arrow(nabla) V (arrow(r)) = - (partial V) / (partial arrow(r)) \
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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) ) \
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==> dif / (dif t) ((partial T) / (partial dot(q)_(i) ) ) - (partial T) / (partial q_(i) ) + (partial V) / (partial dot(q)_(i) ) =^(!) 0 \
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V = V (q_1, q_2, q_3 ) ==> (partial V) / (partial dot(q)_(i) ) = 0 \
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==> dif / (dif t) (partial L) / (partial dot(q)_(i) ) - (partial L) / (partial q_(i) ) = 0
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$
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Diese Lagrangegleichung der II Art ist
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- Forminvariant
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- Nicht messbar
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- Nicht eindeutig
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Das meschanische System ist so definiert durch $q_(i) "und" L$.
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Der *verallgemeinerte Impuls* ist gegeben durch
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$
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p_(i) := (partial L) / (partial dot(q)_(i) ).
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$
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Eine *zyklische verallgemeinerte Koordinate* $q_(i) $ erfuellt
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$
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(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"
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$
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