added anamech 3

S2/AnaMech/VL/AnMeVL3.typ

@@ -85,3 +85,58 @@ Fuer kleine Schwingungen entwickle $V (x)$ um $x_0 $ (Position des Minimums)
 $
 	v (x)= v (x_0 )+1/2 V'' (x_0 ) (x-x_0 )^2 +1/6 V''' (x_0 ) (x-x_0 )^3 +1/24 V'''' (x_0 ) (x-x_0 )^(4) + ...
 $
+
+Fuer kleine Amplituden sind die letzten Terme zu vernachlaessigen, da $O((x-x_0)^3 )$.
+
+$
+	V (x)=V (delta_(x)  )+1/2 V'' (x_0 )delta_(x) ^2 \
+	m dot.double(delta)_(n) =m dot.double(x)=-V' (x)=-V'' (x_0 )delta_(x) \
+	delta_(x) =x-x_0 ==> dot.double(delta)_(x) =dot.double(x)\
+	==> m delta_(x) +V'' (x_0 )delta_(x) = 0, omega^2 _(0) = (V'' (x_0 )) / (m) 
+$
+
+Periode allgemeine (beliebige Amplidtuden) geschlossene ahn, Umkehrpunkte $x_(3) ,x_(2) $
+
+$
+	t=T,x=x_2 ,x_0 = x_3 ,t_0 = 0, x_0 = x_3 
+$
+
+$
+	t-t_0 = +- integral_(x)^(x_0 ) d x' (1) / (sqrt(2/m (E-V (x)))) 	\
+	==> T=2 integral_(x_2 )^(x_3 ) d x (1) / (sqrt(2/m (E-V (x)))) 
+$
+
++ $
+		V (x)=1/2k x^2 
+		==> T=(2 pi) / (omega_0 ) , omega_(0) =k/m\
+		==> omega_0 != omega_0 (a)
+	$
+
++ Anharmonischer Fall
+	$
+		==> omega=omega (a)
+	$
+$
+	V (x)approx  k/2 x^2 +epsilon V_1 (x)+ "weitere Korrekturterme"
+$
+
+Dann wird dieser Ausdruck fuer das Potential in den fuer die Peiode eingesetzt
+
+$
+	==> T = T_0 epsilon^(0) +I_1 epsilon +I_2 epsilon^2 + O(epsilon^3).
+$
+
+Das Ziel ist dann $I_1 $ zu berechnen.
+
+$
+	[E-V (x)]^(-1/2) = [E-k/2x^2 -epsilon V_1 (x)]^(-1/2) \
+	E=V (a)= [k/2 (a^2 -x^2 )-epsilon (V_1 (x)-V_1 (a))]^(-1/2) \
+	= [k/2 (a^2 -x^2 )]^(-1/2) [1-epsilon A]^(-1/2) 
+$
+
+$
+	A=(V_1 (x)-V_1 (a)) / (k/2 (-x^2 +a^2 )) \
+	(1-u)^(-1/2) approx 1 +1/2 u, "fuer" u "klein"
+$
+
+#highlight[TODO: refactor the calculations]