THE MATHEMATICAL PENDULUM 2
Elliptical Integrals and Functions
K.6.3.3: Figs. 6.8 and 6.9

Franz Michel, Bernhard Schnizer

Institut fuer Theoretische Physik
Technische Universitaet Graz
A-8010  Graz, Austria
Projekt Multimedial Lehre des
Zentralen Informatikdienstes der TU Graz

2002-01-31

Needs["Graphics`Graphics`"]
Needs["Graphics`Colors`"]

Introduction

Complete Integral of the First Kind. 
Period of the Pendulum.  Fig.6.8.

Clear["@"]

\(\(pkt\  = \ 
        Plot[Evaluate[
            EllipticK[Sin[π/180  α]^2] {1, 2/π}], {α, 
            0, 90}, 
		PlotRange\  -> {0, 
              8}, \ 
		AxesLabel\  -> \ {SequenceForm["α =
", Subscript[δ, 0], "/2"], 
              None}, 
		Ticks\  -> \ {10 
                 Range[0, 9], 
              Join[{π/2}, 
                Range[1, 
                  8]]}, \ 
		Epilog\  -> \ {Line[{{90, 0}, {90, 8}}], 
			Text[
                SequenceForm["T/", Subscript[T, 0]], {82, 
                  1.5}], \ 
			Text["K(k=sinα)", {45, 2.5}]}];\)\)

Jacobi's Amplitude Function and Sine Amplitude.  
Fig.6.9.

Jacobi's Amplitude Function

The corresponding values of the paramter  m  are given in list lm: those of the complete 
elliptic integral of the first kind in list lgk.

lm   =  Sin[π la/360. ]^2  ;   
lgk  =  EllipticK[lm];
th  =  Thickness[0.005];
lc  =  {{th, Hue[0]}, {th, GrayLevel[0]}, {th, 
            Hue[ .65]}};

pam  =  
        Table[Plot[
            Evaluate[JacobiAmplitude[4 u lgk[[i]], lm[[i]]]], {u, 
              0, 1},  PlotStyle  ->  lc[[i]]], {i, 
            Length[la]}];

\(\(pa\  = \ 
        Show[pam, \ Ticks\  -> \ {Range[0, 4]/4. , PiScale}, 
          ImageSize\  -> \ 
            600, 
	\ \ \ \ \ \ \ AxesLabel\  -> \ {"t/T", SequenceForm["am(t/T,k=sin(", 
                Subscript[δ, 
                  0], "/2))
"]}, \ \ 
	\ \ \ \ \ \ \ Epilog\  -> \ {Hue[0], 
              Text[SequenceForm[
                  Subscript[δ, 0], "=5°"], {0.65, 
                  3}], \ 
			\ \ \ \ \ \ GrayLevel[0], 
              Text["90°", { .75, 
                  4}], 
			\ \ \ \ \ \ Hue[0.65], 
              Text[SequenceForm[
                  Subscript[δ, 0], "=175°"], {0.6, 
                  5.2}]}, 
	\ \ \ \ \ \ \ TextStyle\ \  -> \ {FontSize\ \  -> \ \ 14}];\)\)

Jacobi's Sine Amplitude.  Fig.6.9.

psn  =  
        Table[Plot[
            Evaluate[JacobiSN[4 u lgk[[i]], lm[[i]]]], {u, 0, 1},  
            PlotStyle  ->  lc[[i]]], {i, 
            Length[la]}];

\(\(ps\  = \ 
        Show[psn, 
          Ticks\  -> \ {Range[0, 4]/4, 
              Range[\(-1\), 
                1]}, \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ AxesLabel\  -> \ {"t/T", 
              SequenceForm["     sn(t/T,k=sin(", 
                Subscript[δ, 
                  0], "/2))

"]}, 
	\ \ \ \ \ \ PlotRange\  -> \ All, \ 
          ImageSize\  -> \ 
            600, 
	\ \ \ \ \ \ Epilog\  -> \ {Hue[0], 
              Text[SequenceForm[
                  Subscript[δ, 
                    0], "=5°"], {0.2,  .5}], \ 
	\ \ \ \ \ \ \ GrayLevel[0], 
              Text[SequenceForm[
                  Subscript[δ, 
                    0], "=90°"], { .70,  .95}], 
	\ \ \ \ \ \ \ Line[{{ .44, 
                    JacobiSN[4\ 0.43\ lgk[\([2]\)], 
                      lm[\([2]\)]]}, { .54,  .9}}], 
	\ \ \ \ \ \ \ Hue[0.65], 
              Text[SequenceForm[
                  Subscript[δ, 0], "=175°"], {0.25, 
                  1.2}]}, 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ TextStyle\ \  -> \ {FontSize\ \  -> \ \ 14}];\)\)

Final Form of  Fig.6.9.

Show[GraphicsArray[{pa, ps}],  
        ImageSize  ->  700];

Motions of a Mathematical Pendulum for Various Values of Total Energy.
Fig.6.12.

Oscillation

amplitude 90°

kk  =  Sin[α π/180] // N;
t0    =  EllipticK[kk^2];
δ  =  
        2 180/π  ArcSin[
            kk  JacobiSN[2  π tt,  kk^2]];

tl  = 
        Table[{i, ""}, {i, 0, 5, 1/4}];
tn  =  Range[0, 5];
ta  =  Join[tl, tn];

\(\(po1\  = \ 
        Plot[δ, \ {tt, 0, 5}, \ 
          PlotRange\  -> 
            90\ {\(-1\), 
                1}, \ 
AxesLabel\  -> \ {SequenceForm["t/", 
                Subscript[T, 
                  0]], "δ°"}, 
Ticks\  -> \ {ta, 
              Range[\(-90\), 90, 
                90]}, \ 
Ticks\  -> \ {ta, 
              Range[\(-90\), 90, 
                90]}, \ 
PlotStyle\  -> {{th, 
                Red}}, 
AspectRatio\  -> \ \  .3, \ 
          ImageSize\  -> \ 600];\)\)

amplitude 175°

kk  =  Sin[α π/180] // N;
t0    =  EllipticK[kk^2];
δ  =  
        2 180/π  ArcSin[
            kk  JacobiSN[2  π tt,  kk^2]];

tl  = 
        Table[{i, ""}, {i, 0, 5, 1/4}];
tn  =  Range[0, 5];
ta  =  Join[tl, tn];

\(\(po2\  = \ 
        Plot[δ, \ {tt, 0, 5}, \ 
          PlotRange\  -> 
            180\ {\(-1\), 
                1}, \ 
AxesLabel\  -> \ {SequenceForm["t/", 
                Subscript[T, 
                  0]], "δ°"}, 
Ticks\  -> \ {ta, 
              Range[\(-180\), 180, 
                90]}, \ 
PlotStyle\  -> \ 
            Thickness[
              0.005], 
AspectRatio\  -> \ \  .3, \ 
          ImageSize\  -> \ 600];\)\)

Show[po2, po1];

Limit Case δ = 180°

δ  =  
        4 180/π  ArcTan[  Tanh[2  π tt] ];

tl  = 
        Table[{i, ""}, {i, 0, 5, 1/4}];
tn  =  Range[0, 5];
ta  =  Join[tl, tn];

\(\(pl\  = \ 
        Plot[δ, \ {tt, 0, 5}, \ 
          PlotRange\  -> 
            180\ {\(-1\), 
                1}, \ 
AxesLabel\  -> \ {SequenceForm["t/", 
                Subscript[T, 
                  0]], "δ°"}, 
Ticks\  -> \ {ta, 
              Range[\(-180\), 180, 
                90]}, \ 
AspectRatio\  -> \ \  .3, \ 
          ImageSize\  -> \ 600];\)\)

Rotation

Slow Rotation

kk  =  
        Sqrt[2/(1  +  ε)];
δ  =  
        2 180/π  JacobiAmplitude[2  π tt/kk,  
            kk^2];

tl  = 
        Table[{i, ""}, {i, 0, 5, 1/4}];
tn  =  Range[0, 5];
ta  =  Join[tl, tn];

\(\(pr1\  = \ 
        Plot[δ, \ {tt, 0, 5}, \ 
          PlotRange\  -> {0, 
              1170}, \ 
AxesLabel\  -> \ {SequenceForm["t/", 
                Subscript[T, 
                  0]], "δ°"}, 
Ticks\  -> \ {ta, 
              Range[\(-180\), 1170, 
                90]}, \ 
Ticks\  -> \ {ta, 
              Range[\(-180\), 1170, 
                90]}, \ 
PlotStyle\  -> \ {{th, 
                Blue}}, 
AspectRatio\  -> \ \  .6, \ 
          ImageSize\  -> \ 600];\)\)

Fast Rotation

kk  =  
        Sqrt[2/(1  +  ε)];
δ  =  
        2 180/π  JacobiAmplitude[2  π tt/kk,  
            kk^2];

tl  = 
        Table[{i, ""}, {i, 0, 5, 1/4}];
tn  =  Range[0, 5];
ta  =  Join[tl, tn];

\(\(pr2\  = \ 
        Plot[δ, \ {tt, 0, 5}, \ 
          PlotRange\  -> {0, 
              1170}, \ 
AxesLabel\  -> \ {SequenceForm["t/", 
                Subscript[T, 
                  0]], "δ°"}, 
Ticks\  -> \ {ta, 
              Range[\(-180\), 1170, 
                90]}, \ 
Ticks\  -> \ {ta, 
              Range[\(-180\), 1170, 
                90]}, \ 
AspectRatio\  -> \ \  .6, \ 
          ImageSize\  -> \ 600];\)\)

Show[pr1, pr2];

All Motions Together, Fig.6.12

\(\(ptx\  = \ {Text["ε=20", { .57, 
              700}], 
Text[
            SequenceForm[
              Subscript["δ", 0], "=180°"], {2.15, 
              215}], 
Text[
            SequenceForm[
              Subscript["δ", 
                0], "=175°"], {2.15, \(-135\)}], 
Red, 
          Text[SequenceForm[
              Subscript["δ", 
                0], "=90°"], { .9, \(-135\)}], 
Text[
            SequenceForm[
              Subscript["δ", 
                0], "=90°"], {3.25, \(-135\)}], 
Blue, \ 
          Text["ε=1.01", {2.15, 
              450}]};\)\)
Show[pr1, pr2, pl, po1, po2,  
        PlotRange  ->  {-180, 
            1170},  
Epilog  ->  ptx,  
        AspectRatio  ->    .75];