// 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: 9,
	type: 0, // 0 normal, 1 exercise
	date: datetime.today().display(),
	//date: datetime(
	//	year: 2025,
	//	month: 5,
	//	day: 1,
	//).display(),
)

= Exkurs in die Geometrie

Zunaechst betrachten wir einen Massepunkt $m$  in $arrow(r) in RR^(3) $.

Wir kennen die kartesischen Raumkoordinaten.
Der Ursprung bleibt bei der Transformation gleich.

Es gilt fuer die Basisvektoren 
$
	arrow(e)_(i) * arrow(e)_(j) = delta_(i j).
$

Die Koordinatentransformation muss umkehrbar sein in fast jedem Punkt
$
	x_(i)  =  x_(i) (q_1, q_2, ..., q_n ).
$

Theoretische Physik geht los wenn alle griechischen Buchstabe fuer Indizes verbraucht sind.

$
	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".
$

Die $q_i $ Kurven koennen krummlinieg verlaufen. Die Basisvektoren im Punkt $P$ sind gegeben durch
$
	arrow(q)_(i)  =  (partial arrow(r) (p)) / (partial q_i) , arrow(r) =  x_(i) (q_1, q_2, q_3) arrow(e)_(i), \
	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).
$

In fast jedem Punkt sind diese linear unabhaengig. Dreibein ${arrow(q)_(1) , arrow(q)_(2) , arrow(q)_(3) }$.

#example[
	Kugelkoordinaten.

	$
		x_1 = r cos phi sin theta \
		x_2 =  r sin phi sin theta \
		x_3 = r cos theta \
		r >0 , space  theta in [0, pi] , space  phi in [0, 2 pi)
	$

	Durch Ableiten kann so das Dreibein gebildet werden. Dieses erfuellt die gefordeten Eigenschaften von linearer Unabhaengigkeit.
]

Es gilt
$
	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  \
	&= d x_1 arrow(e)_(2)  + d x_2 arrow(e)_(2)  + d x_3 arrow(e)_(3).
$

= Metrischer Tensor

Wird auch metrisches Dings genannt.

Es gilt
$
	g_(i j)  &=  arrow(g)_(i)  * arrow(g)_(j),  \
	g_(i j) &= mat(
		1, 0, 0;
		0, r^2 , 0;
		0, 0, r^2 sin^2 theta;
	), \
	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 ).
$

= Bewegungsgleichung fuer $q_i $

Hier sind $dot(q)_(j)$ die verallgemeinerten Geschwindigkeiten.

Berechne
$
	dif / (dif t) T =  m dot(x)_(i)  dot.double(x)_(i)  \
	T = m/2 dot(arrow(r)) * dot(arrow(r)) =  m/2 dot(x)_(i)  dot(x)_(i)  \
	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" \
	(partial dot(arrow(r))) / (partial dot(q)_(j) ) =  arrow(g)_(i); quad arrow(g)_(i)  =  (partial arrow(r)) / (partial q_(i) )  \
	arrow(r) = x_(i)  arrow(e)_(i) \
	(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).
$

Wir starten von Newton II
$
	m dot.double(arrow(r)) =  arrow(f) \
	<==> m arrow(e)_(i)  * dot.double(arrow(r)) =  arrow(e)_(i) * arrow(f) \
	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"
$