(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 157599, 2835] NotebookOptionsPosition[ 154689, 2786] NotebookOutlinePosition[ 155087, 2802] CellTagsIndexPosition[ 155044, 2799] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["\<\ Anzahl der Pendel in der Kette, alle haben der Einfachheit halber die gleiche \ L\[ADoubleDot]nge und an allen h\[ADoubleDot]ngt die gleiche Masse. Am besten \ mit kleinen M starten und dann sukzessiv vergr\[ODoubleDot]\[SZ]ern. Auf \ meinem Macbook funktioniert auch M=50 noch ganz gut, die numerische L\ \[ODoubleDot]sung dauert einige Minuten, bei M=70 10 Minuten. \ \>", "Text", CellChangeTimes->{{3.801203431446981*^9, 3.8012034645128508`*^9}, { 3.8012039771782007`*^9, 3.801204037223473*^9}, {3.801215641831389*^9, 3.8012156531461067`*^9}, {3.801216485195475*^9, 3.801216508279497*^9}},ExpressionUUID->"48241999-2ca0-4dea-a6a5-\ 50a6ebc926c2"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"M", " ", ":=", " ", "20"}], ";"}], " "}]], "Input", CellChangeTimes->{{3.801051633922699*^9, 3.801051649602048*^9}, 3.801051703458714*^9, {3.801052126551921*^9, 3.801052127510633*^9}, { 3.801052813971798*^9, 3.801052814373065*^9}, {3.8010529869154463`*^9, 3.80105298768517*^9}, {3.8010530262059393`*^9, 3.801053026326791*^9}, { 3.8010531743424873`*^9, 3.801053174482942*^9}, {3.801056587395136*^9, 3.801056587646593*^9}, {3.801058149843788*^9, 3.801058150290431*^9}, { 3.80105820466591*^9, 3.801058204988625*^9}, {3.801058384200509*^9, 3.801058384363002*^9}, {3.801058985100009*^9, 3.8010589851930933`*^9}, { 3.801061673339177*^9, 3.801061673481048*^9}, {3.8010727230656013`*^9, 3.8010727243367777`*^9}, {3.801200419200499*^9, 3.8012004204379263`*^9}, { 3.801202923291747*^9, 3.801202923822752*^9}, {3.8012030375813303`*^9, 3.801203037707313*^9}, {3.801203147662899*^9, 3.8012031488429337`*^9}, { 3.801203235145124*^9, 3.801203235326784*^9}, {3.801203968028109*^9, 3.801203973140295*^9}, {3.801204106178236*^9, 3.801204106678879*^9}, { 3.801215278426865*^9, 3.801215282657267*^9}, {3.801215389431986*^9, 3.801215389898101*^9}, {3.8012154310197973`*^9, 3.801215433166349*^9}, { 3.801215614726603*^9, 3.801215616084091*^9}, {3.801215705017695*^9, 3.801215711143207*^9}, {3.801216330556601*^9, 3.801216330700091*^9}, { 3.801216515551425*^9, 3.801216515631774*^9}, {3.8614397949434347`*^9, 3.86143979536292*^9}, {3.861672102108444*^9, 3.8616721029189997`*^9}, { 3.923582390723515*^9, 3.923582392881074*^9}, 3.923582530157583*^9, { 3.923582659289476*^9, 3.923582660812191*^9}, 3.92358309273853*^9, { 3.9235832523830748`*^9, 3.92358325577464*^9}, {3.923623807704669*^9, 3.9236238085887814`*^9}, {3.9236238915427227`*^9, 3.923623892805969*^9}, { 3.923624096988639*^9, 3.9236240987398367`*^9}, {3.92362813856192*^9, 3.923628138682946*^9}, 3.923628669290628*^9, 3.92362876486097*^9}, CellLabel-> "In[952]:=",ExpressionUUID->"f7b6c8da-be06-4aef-b8c2-7f6141f753a4"], Cell["\<\ Definition aller gew\[ODoubleDot]hnlichen, kartesischen Koordinaten durch die \ jeweiligen Winkel zur Vertikalen per Rekursion. Alle L\[ADoubleDot]ngen \ werden in der Einheitsl\[ADoubleDot]nge eines Pendels gemessen. \ \>", "Text", CellChangeTimes->{{3.8012034060051413`*^9, 3.8012034175707703`*^9}, { 3.8012034751046886`*^9, 3.8012035397003317`*^9}},ExpressionUUID->"9344e46c-7655-4cf6-8f2c-\ a4bdff202bcd"], Cell[BoxData[{ RowBox[{ RowBox[{"Array", "[", RowBox[{"x", ",", " ", "M"}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Array", "[", RowBox[{"y", ",", " ", "M"}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Array", "[", RowBox[{"phi", ",", " ", "M"}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"x", "[", "1", "]"}], " ", "=", " ", RowBox[{"Sin", "[", RowBox[{ RowBox[{"phi", "[", "1", "]"}], "[", "t", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"y", "[", "1", "]"}], " ", "=", " ", RowBox[{"-", " ", RowBox[{"Cos", "[", RowBox[{ RowBox[{"phi", "[", "1", "]"}], "[", "t", "]"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Do", "[", RowBox[{ RowBox[{ RowBox[{"x", "[", RowBox[{"i", "+", "1"}], "]"}], "=", RowBox[{ RowBox[{"x", "[", "i", "]"}], "+", RowBox[{"Sin", "[", RowBox[{ RowBox[{"phi", "[", RowBox[{"i", "+", "1"}], "]"}], "[", "t", "]"}], "]"}]}]}], ",", " ", RowBox[{"{", RowBox[{"i", ",", " ", "1", ",", " ", RowBox[{"M", "-", "1"}]}], "}"}]}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Do", "[", RowBox[{ RowBox[{ RowBox[{"y", "[", RowBox[{"i", "+", "1"}], "]"}], "=", RowBox[{ RowBox[{"y", "[", "i", "]"}], "-", RowBox[{"Cos", "[", RowBox[{ RowBox[{"phi", "[", RowBox[{"i", "+", "1"}], "]"}], "[", "t", "]"}], "]"}]}]}], ",", " ", RowBox[{"{", RowBox[{"i", ",", " ", "1", ",", " ", RowBox[{"M", "-", "1"}]}], "}"}]}], "]"}], ";"}]}], "Input", CellChangeTimes->{{3.800599527434259*^9, 3.8005996865672092`*^9}, { 3.8006096893586607`*^9, 3.800609718334867*^9}, {3.800610662400467*^9, 3.8006106682930326`*^9}, 3.800631781921908*^9, {3.80072170842453*^9, 3.800721802690838*^9}, {3.800721877778915*^9, 3.800721884399313*^9}, { 3.800721994879622*^9, 3.800722010504116*^9}, {3.800724136272974*^9, 3.800724151393523*^9}, {3.8010379502693443`*^9, 3.8010381037469378`*^9}, { 3.801038141801362*^9, 3.801038150572101*^9}, {3.801038368288249*^9, 3.80103838660013*^9}, {3.8010384209481277`*^9, 3.801038456169505*^9}, { 3.80103880204006*^9, 3.801038804938488*^9}, {3.801038882272339*^9, 3.8010390920959473`*^9}, {3.801039151857333*^9, 3.801039164405189*^9}, { 3.80103921268917*^9, 3.801039229792058*^9}, {3.801051654666874*^9, 3.801051671397532*^9}}, CellLabel-> "In[953]:=",ExpressionUUID->"5410642a-4017-415f-a43a-f2a771195534"], Cell["\<\ Definition der Lagrangefunktion. Auch hier werden Rationale Einheiten \ verwendet, Energien werden in Einheiten (mgl) gemessen. l ist die L\ \[ADoubleDot]nge und m die Masse eines Pendels. Die Erdbeschleunigung g kann \ durch Definition der Zeiteinheit 1/omega rationalisiert werden, die immer in \ den Zeitableitungen vorkommt. omega w\[ADoubleDot]re die Frequenz eines \ einzigen (\[OpenCurlyDoubleQuote]Faden-\[OpenCurlyDoubleQuote]) Pendels. Das \ ist f\[UDoubleDot]r numerische Rechnungen immer g\[UDoubleDot]nstig. \ Mechanische Probleme haben meist eine Skaleninvarianz. \ \>", "Text", CellChangeTimes->{{3.801203545894083*^9, 3.8012037433272457`*^9}, { 3.801203817039258*^9, 3.801203869154895*^9}, {3.801204113390791*^9, 3.8012041146503773`*^9}, 3.861672879011635*^9},ExpressionUUID->"0bcf7a7a-2e7d-412d-8bbe-\ 05ccbd98c164"], Cell[BoxData[{ RowBox[{ RowBox[{"T", "=", " ", RowBox[{ RowBox[{"1", "/", "2"}], " ", RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"x", "[", "i", "]"}], ",", " ", "t"}], "]"}], "^", "2"}], "+", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"y", "[", "i", "]"}], ",", " ", "t"}], "]"}], "^", "2"}]}], ",", " ", RowBox[{"{", RowBox[{"i", ",", " ", "1", ",", " ", "M"}], "}"}]}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"V", "=", " ", RowBox[{"Sum", "[", RowBox[{ RowBox[{"y", "[", "i", "]"}], ",", " ", RowBox[{"{", RowBox[{"i", ",", " ", "M"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"L", " ", "=", " ", RowBox[{"T", "-", "V"}]}], ";"}]}], "Input", CellChangeTimes->{{3.80103923445007*^9, 3.801039331879978*^9}, { 3.801051676969895*^9, 3.801051677772275*^9}, 3.801058201931137*^9, { 3.801203045198968*^9, 3.8012030627990637`*^9}}, CellLabel-> "In[960]:=",ExpressionUUID->"ec3d0053-4c2e-4b84-a849-325e1ad15d78"], Cell["\<\ Die Euler-Lagrange-Gleichungen und Definition der Anfangsbedingungen. Alle \ Pendel werden ruhend aus der Horizontalen gestartet. Hier ist \ nat\[UDoubleDot]rlich Raum zum spielen. Interessant sind auch kleine Winkel \ (eher intuitiv aus der Erfahrung bekannt) oder Start aus der Vertikalen. F\ \[UDoubleDot]r gro\[SZ]e M entspricht die Bewegung etwa der eines Seils oder \ eine langen Kette. Auch Situationen wie teils h\[ADoubleDot]ngend, teils \ vertikal stehend k\[ODoubleDot]nnen interessant sein. \ \>", "Text", CellChangeTimes->{{3.801203760307722*^9, 3.80120381313307*^9}, { 3.801203878782392*^9, 3.801203952456065*^9}, {3.801204063406011*^9, 3.8012040842112913`*^9}, 3.8616728876209393`*^9},ExpressionUUID->"1417a5d9-7ad1-45e7-8501-\ 0cdb47301bfd"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Do", "[", RowBox[{ RowBox[{ RowBox[{"eom", "[", "i", "]"}], " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"D", "[", RowBox[{"L", ",", " ", RowBox[{ RowBox[{ RowBox[{"phi", "[", "i", "]"}], "'"}], "[", "t", "]"}]}], "]"}], ",", " ", "t"}], "]"}], " ", "-", " ", RowBox[{"D", "[", RowBox[{"L", ",", " ", RowBox[{ RowBox[{"phi", "[", "i", "]"}], "[", "t", "]"}]}], "]"}]}], "\[Equal]", "0"}]}], ",", " ", RowBox[{"{", RowBox[{"i", ",", " ", "1", ",", " ", "M"}], "}"}]}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Array", "[", RowBox[{ RowBox[{"fi", "[", "i", 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Ruhig mal die \ Dokumentation auf der Mathematica Seite nachschauen, insbesondere f\ \[UDoubleDot]r Method etc. Da hilft manchmal schon etwas ausprobieren, wenn \ man nicht zu einer L\[ODoubleDot]sung kommt. NDSolve ist der Flaschenhals in \ dieser Rechnung, daher ist auch die Rechenzeit interessant, auch um M \ hochzudrehen. 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