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Notebook[{
Cell["Start with two modes:", "Text"],

Cell[BoxData[{
    \(eq1\  = \ \(x1''\)[t]\ \  \[Equal] \ \(-2\)\ x1[t]\  + \ 
          x2[t]\), "\[IndentingNewLine]", 
    \(eq2\  = \ \(x2''\)[t]\ \  \[Equal] \ x1[t]\  - \ 2  x2[t]\)}], "Input"],

Cell["\<\
Let's solve with certain boundary conditions and solve for the motion:\
\>", "Text"],

Cell[BoxData[
    \(s = 
      DSolve[{eq1, eq2, x1[0] \[Equal] 0, 
          x2[0] \[Equal] 0, \(x1'\)[0] \[Equal] 1, \(x2'\)[0] \[Equal] 
            0}, {x1[t], x2[t]}, t]\)], "Input"],

Cell["Here is a function to draw the state:", "Text"],

Cell[BoxData[
    \(L = 12\)], "Input"],

Cell[BoxData[
    \(state[x_, y_] := \ 
      Show[{ParametricPlot[{0 + \((t*x)\), Sin[2\ Pi\ 6\ t]}, {t, 0, 1}, 
            Axes \[Rule] True, PlotStyle \[Rule] RGBColor[1, 0, 0], 
            DisplayFunction \[Rule] Identity, 
            PlotRange \[Rule] {{0, L}, {\(-1\), 1}}], 
          Graphics[
            Rectangle[{x - 1/2, \(-1\)/4}, {x + 1/2, 
                1/4}]], \[IndentingNewLine]ParametricPlot[{x + t*\((y - x)\), 
              Sin[2\ Pi\ 6\ t]}, {t, 0, 1}, Axes \[Rule] True, 
            PlotStyle \[Rule] RGBColor[1, 0, 0], 
            DisplayFunction \[Rule] Identity, 
            PlotRange \[Rule] {{0, L}, {\(-1\), 
                  1}}], \[IndentingNewLine]Graphics[
            Rectangle[{y - 1/2, \(-1\)/4}, {y + 1/2, 
                1/4}]], \[IndentingNewLine]ParametricPlot[{y + t*\((L - y)\), 
              Sin[2\ Pi\ 6\ t]}, {t, 0, 1}, Axes \[Rule] True, 
            PlotStyle \[Rule] RGBColor[1, 0, 0], 
            DisplayFunction \[Rule] Identity, 
            PlotRange \[Rule] {{0, L}, {\(-1\), 1}}]}, 
        DisplayFunction \[Rule] $DisplayFunction]\)], "Input"],

Cell["And here is how we use it to animate the motion.", "Text"],

Cell[BoxData[
    \(Table[
      state[4 + x1[t] /. s[\([1]\)], 8 + x2[t] /. s[\([1]\)]], {t, 0, 
        10,  .1}]\)], "Input"],

Cell["\<\
Your first task: play with the boundary conditions until you get the \"nice\" \
modes with only one frequency.\
\>", "Text"],

Cell["\<\
Your next task: add a third mass.  That is, write the equations for three \
coordinates, DSolve, and animate with:\
\>", "Text"],

Cell[BoxData[
    \(state[x_, y_, z_] := \ 
      Show[{\[IndentingNewLine]ParametricPlot[{0 + \((t*x)\), 
              Sin[2\ Pi\ 6\ t]}, {t, 0, 1}, Axes \[Rule] True, 
            PlotStyle \[Rule] RGBColor[1, 0, 0], 
            DisplayFunction \[Rule] Identity, 
            PlotRange \[Rule] {{0, L}, {\(-1\), 1}}], 
          Graphics[
            Rectangle[{x - 1/2, \(-1\)/4}, {x + 1/2, 
                1/4}]], \[IndentingNewLine]Graphics[
            Rectangle[{x - 1/2, \(-1\)/4}, {x + 1/2, 
                1/4}]], \[IndentingNewLine]ParametricPlot[{x + t*\((y - x)\), 
              Sin[2\ Pi\ 6\ t]}, {t, 0, 1}, Axes \[Rule] True, 
            PlotStyle \[Rule] RGBColor[1, 0, 0], 
            DisplayFunction \[Rule] Identity, 
            PlotRange \[Rule] {{0, L}, {\(-1\), 
                  1}}], \[IndentingNewLine]Graphics[
            Rectangle[{y - 1/2, \(-1\)/4}, {y + 1/2, 
                1/4}]], \[IndentingNewLine]ParametricPlot[{y + t*\((z - y)\), 
              Sin[2\ Pi\ 6\ t]}, {t, 0, 1}, Axes \[Rule] True, 
            PlotStyle \[Rule] RGBColor[1, 0, 0], 
            DisplayFunction \[Rule] Identity, 
            PlotRange \[Rule] {{0, L}, {\(-1\), 
                  1}}], \[IndentingNewLine]Graphics[
            Rectangle[{z - 1/2, \(-1\)/4}, {z + 1/2, 
                1/4}]], \[IndentingNewLine]ParametricPlot[{z + t*\((L - z)\), 
              Sin[2\ Pi\ 6\ t]}, {t, 0, 1}, Axes \[Rule] True, 
            PlotStyle \[Rule] RGBColor[1, 0, 0], 
            DisplayFunction \[Rule] Identity, 
            PlotRange \[Rule] {{0, L}, {\(-1\), 1}}]}, 
        DisplayFunction \[Rule] $DisplayFunction]\)], "Input"],

Cell["\<\
Next task: write the matrix which describes this problem.  For the 2x2 case \
it would be:\
\>", "Text"],

Cell[BoxData[
    \(\((m\  = \ {{2, \(-1\)}, {\(-1\), 2}})\)\  // \ MatrixForm\)], "Input"],

Cell["\<\
and we could ask for the Eigenvectors.  Your job: extend to the 3x3 problem \
at hand, find the eigenvectors and animate.  As you do so,
form the charateristic polynomial by taking\
\>", "Text"],

Cell[BoxData[
    \(\((id\  = \ {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}})\) // 
      MatrixForm\)], "Input"],

Cell[BoxData[
    \(Det[m\  - \ \[Lambda]\ id]\)], "Input"],

Cell["\<\
and Solving for the roots.  For good measure, pick one of the eigenvalues, \
and then Solve appropriate equations to find a corresponding eigenvector.\
\>", "Text"],

Cell["\<\
Further extensions:  (a) add damping ( a term -\[Gamma] x'[t]),  (b) add yet \
more masses (best to do this for general n at some point), (c) add forcing.
\
\>", "Text"]
},
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