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Cell[CellGroupData[{
Cell["Reproducing BTM figure 3.1", "Subsubsection",
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Cell["Define a function", "Text",
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Cell["Make a ContourPlot like Shankar's:", "Text",
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}, Open  ]],

Cell["Next draw the parametric plot for a circle of radius 2:", "Text",
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To do Shankar one better, find the extrema by the Lagrange multiplier method. \
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points where g(x,y)==0, then construct F(x,y,\[Lambda]):\
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Then form the equations which say that F is extremized, and solve them:\
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Cell["\<\
Form a list of points from these solutions and then ListPlot those points:\
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And the acceleration---let's display this result in column vector format:\
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Specifically at the top (t=0) the velocity and acceleration are:\
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Cell["\<\
For extra fun, let's animate the motion.
First define our field of view, then define a function to draw our point:\
\>", "Text",
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Cell[BoxData[
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     RowBox[{"pt", "[", "t", "]"}], "}"}], ",", 
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    RowBox[{"PlotStyle", "\[Rule]", 
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Cell[CellGroupData[{

Cell[BoxData[
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   RowBox[{"drawPt", "[", "t", "]"}], ",", 
   RowBox[{"{", 
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     RowBox[{"2", "\[Pi]"}]}], "}"}]}], "]"}]], "Input",
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    Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = 
    "\"untitled\"", Typeset`specs$$ = {{
      Hold[$CellContext`t$$], 0, 2 Pi}}, Typeset`size$$ = {
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      "OtherVariables" :> {
       Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 
        Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 
        Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$,
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      "Body" :> $CellContext`drawPt[$CellContext`t$$], 
      "Specifications" :> {{$CellContext`t$$, 0, 2 Pi, 
         AppearanceElements -> {
          "ProgressSlider", "PlayPauseButton", "FasterSlowerButtons", 
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      "Options" :> {
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        SynchronousUpdating -> True, ShrinkingDelay -> 10.}, 
      "DefaultOptions" :> {}],
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    DynamicModuleValues:>{},
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}, Open  ]],

Cell["If we want to add the wheel and the cycloid:", "Text",
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And finally let's add in the instantaneous velocity and acceleration vectors.
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which defines the function ListVectorFieldPlot among others.
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We drop a ball from height h, and it hits a redirector at height y0.  From \
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Let's take the positive one and ask for the motion from there:\
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Tell yourself, \"oh yeah, I know how long it takes to fall a height y0 \
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And now some fun, make a parametric plot of the full motion.  First figure \
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Then form piecewise continuous expressions for x[t] and y[t] by gluing \
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To use this expression we of course need to definre the speed at the \
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And here is a widget for finding the optimimum diverter height \"by hand\".\
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A spring of length L is compressed to 0 length, so we have energy (1/2) k L^2 \
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A mass m is attached and launched around a circle of radius R.  What is the \
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Energy conservation:\
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