From e37baa2b70b480d582ff46a514948d9ac3c64b92 Mon Sep 17 00:00:00 2001 From: Jonas Hahn Date: Thu, 24 Apr 2025 09:50:23 +0200 Subject: [PATCH] added anamech 3 --- S2/AnaMech/VL/AnMeVL3.typ | 55 ++++++++++++++++++++++++++++++++++++ S2/AnaMech/pdfs/AnMeVL3.pdf | Bin 41918 -> 55051 bytes book.typ | 28 +++++++++--------- 3 files changed, 69 insertions(+), 14 deletions(-) diff --git a/S2/AnaMech/VL/AnMeVL3.typ b/S2/AnaMech/VL/AnMeVL3.typ index 69cebb2..250d47b 100644 --- a/S2/AnaMech/VL/AnMeVL3.typ +++ b/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] diff --git a/S2/AnaMech/pdfs/AnMeVL3.pdf b/S2/AnaMech/pdfs/AnMeVL3.pdf index 8e33654e3a86496c8fe82a73fe21fb5080f0b938..83e1d67dce0d4ebd3d16d593eb3b49ef2810d0cd 100644 GIT binary patch delta 46975 zcmZs?1yEd1@GgpLaEIXT?(XgmL4&(HEbhLzy9Rf6cMI+sg1hs`?|<*R?^WHc@6^on z^z`)Bo;hcyr@!tIu&8Qq6mlgo33_G*PB?P^uMZp?X97wLB^MDBk%}H15fP)By}F&{ zPkR$nB4(z4LO1~dI8!?ldt)m&IIaW~GN2MKUV=0}kOeSVfKmCueGB?qWg2!O9NwrxE~qL%;zBq$UG~;s$XfNXcM` z9Mi%==Pq!J2P8-ol12i04Az%=t=JPh)ZEvX)ZJcoM@H=GneO`(78R!jY&DT=jVPc0 zcH{8#{W>)+`}KGG7yIuY4V$diYwagCGdJf}f5*2{?erW8qPu?kXx2BD&TqnlK~IEL z*U;v^(Y!Vw#iAGj>L{Dpb2`r$qXl0Q4Vur#*Z5g?`_|Gst3{kQEo8piyzEN5=iF)W zETE$CLTn|(PbP^gQu$#eYvN#DGU(Y?R`LFp%OC%Spw99sC-}ZyZO>@)9nafjm>|K^ zKE_to-FV-+;gif=bRo;uBJmo#X?k$m68W>ahAuO4M>k0gz~YGHtA{I#CkN>HeFO)g 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z_qS>FHrbTz?cWxPzgFo#MQJ-Td(uBa@Yfp7BrQ&=L-YTh{sIEDr2qe{zt%Bl7egl( V4<}PI1a@X_4rT;ON(lwY{|{6%2vPt5 diff --git a/book.typ b/book.typ index b0eef92..373eca1 100644 --- a/book.typ +++ b/book.typ @@ -4,33 +4,33 @@ #book-meta( // put metadata of your book like book.toml of mdbook - title: "JonasDocs", + title: "JonasDocs", - description: "University Typst Documents", + description: "University Typst Documents", - repository: "https://gitlab.gwdg.de/j.hahn02/university", + repository: "https://gitlab.gwdg.de/j.hahn02/university", - authors: ("JonasHahn", "Ascyii"), + authors: ("JonasHahn", "Ascyii"), - language: "de", + language: "de", - summary: [ // this field works like summary.md of mdbook + summary: [ // this field works like summary.md of mdbook - = ExPhy II - - #chapter("S2/ExPhyII/VL/ExIIVL1.typ")[VL1] + = ExPhy II + - #chapter("S2/ExPhyII/VL/ExIIVL1.typ")[VL1] - #chapter("S2/ExPhyII/VL/ExIIVL2.typ")[VL2] - = CWR - - #chapter("S2/CWR/VL/CwrVL1.typ")[VL1] + = CWR + - #chapter("S2/CWR/VL/CwrVL1.typ")[VL1] - #chapter("S2/CWR/VL/CwrVL2.typ")[VL2] - = AnaMech - - #chapter("S2/AnaMech/VL/AnMeVL1.typ")[VL1] + = AnaMech + - #chapter("S2/AnaMech/VL/AnMeVL1.typ")[VL1] - #chapter("S2/AnaMech/VL/AnMeVL3.typ")[VL3] - = Diff II + = Diff II - #chapter("S2/DiffII/VL/DiIIVL1.typ")[VL1] - #chapter("S2/DiffII/VL/DiIIVL2.typ")[VL2] - ] +] )