(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 28162, 816] NotebookOptionsPosition[ 25790, 746] NotebookOutlinePosition[ 26192, 763] CellTagsIndexPosition[ 26149, 760] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Lecture 20 Appendix A (coupled oscillators)", "Section", CellChangeTimes->{{3.4327524790146*^9, 3.4327524881636*^9}, { 3.43275419150321*^9, 3.43275420312871*^9}, 3.43275810901661*^9, { 3.433346355209504*^9, 3.4333463629875045`*^9}, {3.4333463976275043`*^9, 3.4333464029315042`*^9}, {3.4340462570965147`*^9, 3.4340462628165145`*^9}, {3.4345644078982973`*^9, 3.4345644249172974`*^9}, {3.4437276280645456`*^9, 3.443727632864546*^9}}], Cell["\<\ Here we want to analyze a system of 3 springs and 2 masses from the Appendix \ which is described by the matrix (the kinetic energy is proportional to the \ unit matrix so the structure comes from the potential - \ \>", "Text", CellChangeTimes->{{3.4327542331307096`*^9, 3.43275424572021*^9}, { 3.4333464201875043`*^9, 3.4333464314035044`*^9}, 3.4333502801535044`*^9, { 3.4340463292185144`*^9, 3.4340463706805143`*^9}, {3.4340464262875147`*^9, 3.4340464335365143`*^9}, {3.4345644585407977`*^9, 3.4345644992917976`*^9}, {3.434564529907798*^9, 3.4345645798282976`*^9}, { 3.443727654560546*^9, 3.4437276591685457`*^9}}], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"V", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"3", ",", RowBox[{"-", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "6"}], "}"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.43275425515021*^9, 3.43275430266521*^9}, 3.43275446416171*^9, {3.4333464388455043`*^9, 3.433346539933504*^9}, { 3.4340464420275145`*^9, 3.4340464819455147`*^9}, {3.4345645927227974`*^9, 3.434564625845298*^9}, {3.4437276852035456`*^9, 3.443727697817546*^9}}], Cell["So the eigenvalues are", "Text", CellChangeTimes->{{3.4345646610252976`*^9, 3.4345646693072977`*^9}}], Cell[BoxData[ RowBox[{"Eigenvalues", "[", "V", "]"}]], "Input", CellChangeTimes->{{3.4345646712412977`*^9, 3.4345646786292973`*^9}}], Cell[BoxData[ RowBox[{"Eigenvectors", "[", "V", "]"}]], "Input", CellChangeTimes->{{3.4345646912127976`*^9, 3.434564695510298*^9}}], Cell["\<\ Of course, we are normalized to unit vectors or with the same phases. 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