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Brad Klee

Isoperiodic time helices and 3D models

Brad Klee

Posted 1 year ago

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Introduction In general the period of an oscillation can depend on a variable parameter such as energy, here $\alpha$. Then two distinct oscillators of variable energy are called isoperiodic whenever periods equal for all values of $\alpha$ : $T_{1}(\alpha)=T_{2}(\alpha)$. Even in the simple setting of Hamiltonian oscillation, there are a wide range of possibilities as explored in the following Demonstration: http://demonstrations.wolfram.com/IsoperiodicPotentialsViaSeriesExpansion This demonstration focuses solely on construction of isoperiodic potentials, which are curves in two dimensions. Exact solutions of the time-dependent oscillation make for beautiful geometry in a three-dimensional space where time goes along the vertical, above the horizontal plane, phase space. Using the Morse and Pöschl–Teller example, as in the demonstration above, we construct the following models and export to Shapeways: https://www.shapeways.com/product/GXBPX8F24/morse-oscillator-time-helices?optionId=62983286 https://www.shapeways.com/product/BB74864YX/poschl-teller-oscillator-time-helices?optionId=62983299 When the printed models stand side-by-side on a flat surface, the upper-most level sets are at equal heights, a stunning and tangible demonstration of isoperiodicity. Contrasting demonstration with models, it's rewarding to imagine that the continuous deformation of potentials implies a continuous deformation between the two models. Expect to hear more about symmetry breaking in the near future, but for now we will discuss an example where symmetry is added rather than broken ! Simple Pendulum ( Again ) First we construct the phase space trajectory by applying series inversion as in numerous previous references ( Cf. W.C. 1 , W.C. 2 ). We then run a couple of validation test to make sure the general primitives look roughly okay: vSet[n_, m_] := Map[Range[n] /. Append[Rule @@ # & /@ Tally[#], x_Integer :> 0] &, DeleteCases[ Select[IntegerPartitions[n + m], Length[#] > m - 1 &], {n + 1, 1 ...}]] GExp[n_] := yTotal[y^# g[#] & /@ Range[0, n - 1]] gCalc[0, _] = 1; gCalc[n_, m_] := With[{vs = vSet[n, m]}, Total@ReplaceAll[Times[-1/m, Multinomial @@ #, c[Total[#] - m], Times @@ Power[gSet[#] & /@ Range[0, n - 1], #]] & /@ vs, {c[0] -> 1}]] MultinomialExpand[n_, m_] := Module[{}, Clear@gSet; Set[gSet[#], Expand@gCalc[#, m]] & /@ Range[0, n - 1]; Expand[GExp[n + 1] /. g[n2_] :> gCalc[n2, m]]] \[Psi]Test = MultinomialExpand[10, 2] /. c[x_] :> c[x] Q^(x + 2); TrigReduce[Normal@Series[ (p^2 + q^2 + Total[c[#] q^(# + 2) & /@ Range[5] ] ) /. {p -> \[Psi]Test Sin[\[Phi]], q -> \[Psi]Test Cos[\[Phi]] } /. Q -> Cos[\[Phi]], {y, 0, 5}]] SameQ[ With[{exp = TrigReduce[Normal@Series[ -Q D[\[Psi]Test, Q]/\[Psi]Test - 1, {y, 0, 2}] /. Q -> Cos[\[Phi]]]}, Expand[Coefficient[exp, y, #] & /@ Range[0, 2]]], With[{exp = TrigReduce[Normal@Series[ D[ Expand[-(1/2)\[Psi]Test^2 /. y -> (2 \[Alpha])^(1/2)], \[Alpha]] /. \[Alpha] -> (1/ 2) y^2, {y, 0, 2}] /. Q -> Cos[\[Phi]]]}, Expand[Coefficient[exp, y, #] & /@ Range[0, 2]]]] The first prints $y^2$, suggesting the correct series inversion. The second prints "True" suggesting that the time-dependence is calculated correctly. All that remains is to fill in values for the expansion coefficients, and then to plot. Here we limit ourselves to half the total energy range: \[Psi]Pendulum = Sqrt[2 \[Alpha]] Expand[ (MultinomialExpand[20, 2] /. c[x_] :> c[x] Q^(x + 2) /. c[x_ /; OddQ[x]] :> 0 /. {c[2] -> (-1/12), c[4] -> 1/360, c[6] -> (-1/20160)} /. y -> Sqrt[4 \[Alpha]])/ Sqrt[4 \[Alpha]]] /. \[Alpha]^n_ /; n > 3 :> 0 dt = TrigReduce[ D[Expand[(1/2)*\[Psi]Pendulum^2], \[Alpha]] /. Q -> Cos[\[Phi]]] /. \[Alpha]^n_ /; n > 5 :> 0; t[\[Phi]1_, \[Phi]2_] := Expand[(1/2/Pi) Integrate[dt, {\[Phi], \[Phi]1, \[Phi]2}]] tA\[Phi] = t[Pi/2, Pi/2 + \[Phi]]; tB\[Phi] = t[3 Pi/2, 3 Pi/2 + \[Phi]]; tC\[Phi] = t[Pi/2 + \[Phi], \[Phi] + Pi + Pi/2]; tB\[Phi] + tA\[Phi] /. {\[Phi] -> Pi} PST = (\[Psi]Pendulum /. Q -> Cos[\[Phi] + Pi/2]) {Cos[\[Phi] + Pi/2], Sin[\[Phi] + Pi/2], 0}; PST2 = (\[Psi]Pendulum /. Q -> Cos[\[Phi] + Pi/2 + Pi]) {Cos[\[Phi] + Pi/2 + Pi], Sin[\[Phi] + Pi/2 + Pi], 0}; DoubleHelixPendulum = Show[ ParametricPlot3D[Evaluate[ Plus[PST, {0, 0, -2 tA\[Phi]}] /. \[Alpha] -> .5 #/5 & /@ Range[5]], {\[Phi], 0, -2 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100], ParametricPlot3D[Evaluate[ Plus[PST2, {0, 0, -2 tB\[Phi]}] /. \[Alpha] -> .5 #/5 & /@ Range[5]], {\[Phi], 0, -2 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100], ParametricPlot3D[Evaluate[ Plus[PST, {0, 0, 2 tC\[Phi]}] /. \[Alpha] -> .5 #/5 & /@ Range[5]], {\[Phi], 0, 1.1 2 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100], ParametricPlot3D[Evaluate[ Plus[PST2, {0, 0, 2 tB\[Phi] /. \[Phi] -> 2 Pi}] /. \[Alpha] -> .5 #/5 & /@ Range[5]], {\[Phi], 0, -1.1 2 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100], ParametricPlot3D[Evaluate[ Plus[PST2, {0, 0, 2 tB\[Phi] /. \[Phi] -> 0}] /. \[Alpha] -> .5 #/5 & /@ Range[5]], {\[Phi], 0, -1.1 2 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100], ParametricPlot3D[Evaluate[ Plus[PST2, {0, 0, 2 tB\[Phi] /. \[Phi] -> 0}] /. \[Phi] -> 2 Pi #/6 & /@ Range[6]], {\[Alpha], .1, 0.5}, PlotStyle -> Tube[1/32], PlotPoints -> 100], ParametricPlot3D[Evaluate[ Plus[PST, {0, 0, 2 tC\[Phi]}] /. \[Phi] -> 2 Pi #/6 & /@ Range[6]], {\[Alpha], .1, 0.5}, PlotStyle -> Tube[1/32], PlotPoints -> 100], ParametricPlot3D[Evaluate[ Plus[PST2, {0, 0, 2 tB\[Phi] /. \[Phi] -> 2 Pi}] /. \[Phi] -> 2 Pi #/6 & /@ Range[6]], {\[Alpha], .1, 0.5}, PlotStyle -> Tube[1/32], PlotPoints -> 100], Boxed -> False, Axes -> False, PlotRange -> All, ImageSize -> 300 ] This model can be saved as an ".stl" file and exported directly to shapeways. Shapeways: Simple Pendulum Time Helices Edward's Curve As recently announced on seqfans, it's relatively easy to apply polar coordinates in an analysis of the Edward's Curve ( Cf. Edwards & Bernstein & Lange ), and this approach readily yields a simple exact form for the "time dependence" of the addition rules. First we calculate a radius function for the genus-one solution: Edwards = x^2 + y^2 - (1 + d x^2 y^2); r[\[Phi]_] := Sqrt[(1 - Sqrt[1 - d Sin[2 \[Phi]]^2]) 2 Csc[2 \[Phi]]^2/d] r[\[Phi]] And check this: TrigReduce[ Edwards /. {x -> r[\[Phi]] Cos[\[Phi]], y -> r[\[Phi]] Sin[\[Phi]]}] Yields Zero, as necessary. TrigReduce can be replaced by a set of replacement rules. Next we write the addition rule in the form of a tangent function, Tan\[Phi]3[\[Phi]1_, 0] := Tan[\[Phi]1] Tan\[Phi]3[\[Phi]1_, \[Phi]2_] := Tan[\[Phi]1 + \[Phi]2] (1 - d z)/(1 + d z) /. z -> r[\[Phi]1]^2 r[\[Phi]2]^2 Cos[\[Phi]1] Sin[\[Phi]1] Cos[\[Phi]2] Sin[\[Phi]2] And calculate the derivative \[Phi]dot = Times[ Normal@Series[ Tan\[Phi]3[\[Phi], \[Omega]dt], {\[Omega]dt, 0, 1}] - Tan\[Phi]3[\[Phi], 0], Cos[\[Phi]]^2/\[Omega]dt ] d\[Phi]/TrigReduce[Expand[\[Phi]dot]] The closed form result is , $$\omega \; dt = \frac{d\phi}{\sqrt{1-d \sin^2(2\phi)}} = \frac{d\phi}{\sqrt{1-4 \;d \big( \sin(\phi)\cos(\phi)\big)^2}} .$$ Clearly the complete elliptic integral of the first kind is given by any integral of the form $$K(d) \propto \frac{1}{2\pi} \int_{0}^{2\pi} \frac{d\phi}{\sqrt{1-d \sin^2(n\;\phi)}}, $$ with integer $n$. As is well known the pendulum has $n=1$, and we see here that Edward's curve has $n=2$, proving the two systems are isoperiodic. Let's now exploit the square symmetry of Edward's curve by making a quadruple-helix, 3D-printable Model. First we expand the radius to avoid singular points, and integrate the time dependence ( this is done naively, and could be optimized with a little more effort ), rEdwards = Normal[Series[Sqrt[2 d] r[\[Phi]0], {d, 0, 20}]]; dt = 1/Sqrt[Expand[1 - 4 d x^2]] d\[Phi] t = 1/2/Pi Integrate[ Evaluate[ Expand[TrigReduce[ Normal@Series[dt, {d, 0, 20}] /. x -> Cos[\[Phi]] Sin[\[Phi]]]/d\[Phi]]], {\[Phi], Pi, \[Phi]0}]; The extra factor of $\sqrt{d}$ corresponds to a coordinate change where Edward's equation takes a form $d = x^2+y^2-x^2y^2$, which allows us to plot the time spirals, EdwardsQuadrupleHelix = Show[ Function[{a}, ParametricPlot3D[ Evaluate[{rEdwards Cos[\[Phi]0 + Pi/2 a], rEdwards Sin[\[Phi]0 + Pi/2 a], 2 t} /. d -> #/10 & /@ Range[5]], {\[Phi]0, Pi, 3 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100] ] /@ Range[0, 3], Function[{a}, ParametricPlot3D[ Evaluate[{rEdwards Cos[\[Phi]0 + Pi/2 a], rEdwards Sin[\[Phi]0 + Pi/2 a], 2 t} /. \[Phi]0 -> Pi], {d, 0.1, .5}, PlotStyle -> Tube[1/32], PlotPoints -> 100] ] /@ Range[0, 3], Function[{a}, ParametricPlot3D[ Evaluate[{rEdwards Cos[\[Phi]0 + Pi/2 a], rEdwards Sin[\[Phi]0 + Pi/2 a], 2 t} /. \[Phi]0 -> 2 Pi], {d, 0.1, .5}, PlotStyle -> Tube[1/32], PlotPoints -> 100] ] /@ Range[0, 3], Function[{a}, ParametricPlot3D[ Evaluate[{rEdwards Cos[\[Phi]0 + Pi/2 a], rEdwards Sin[\[Phi]0 + Pi/2 a], 2 t} /. \[Phi]0 -> 3 Pi], {d, 0.1, .5}, PlotStyle -> Tube[1/32], PlotPoints -> 100] ] /@ Range[0, 3], ParametricPlot3D[ Evaluate[{rEdwards Cos[\[Phi]0 + Pi], rEdwards Sin[\[Phi]0 + Pi], 2 t /. \[Phi]0 -> Pi} /. d -> #/10 & /@ Range[5]], {\[Phi]0, Pi, 3 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100], ParametricPlot3D[ Evaluate[{rEdwards Cos[\[Phi]0 + Pi], rEdwards Sin[\[Phi]0 + Pi], 2 t /. \[Phi]0 -> 2 Pi} /. d -> #/10 & /@ Range[5]], {\[Phi]0, Pi, 3 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100], ParametricPlot3D[ Evaluate[{rEdwards Cos[\[Phi]0 + Pi], rEdwards Sin[\[Phi]0 + Pi], 2 t /. \[Phi]0 -> 3 Pi} /. d -> #/10 & /@ Range[5]], {\[Phi]0, Pi, 3 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100] , PlotRange -> All, Boxed -> False, Axes -> False, ImageSize -> 800 ] Again, can be exported directly to shapeways. Finally let's take a closer look at isoperiodicity, by comparing the upper-most level sets, Show[Function[{a}, ParametricPlot3D[ Evaluate[{rEdwards Cos[\[Phi]0 + Pi/2 a], rEdwards Sin[\[Phi]0 + Pi/2 a], 2 t} /. \[Phi]0 -> 3 Pi], {d, 0.1, .5}, PlotStyle -> Tube[1/32], PlotPoints -> 100] ] /@ Range[0, 3], ParametricPlot3D[ Evaluate[{rEdwards Cos[\[Phi]0 + Pi], rEdwards Sin[\[Phi]0 + Pi], 2 t /. \[Phi]0 -> 3 Pi} /. d -> #/10 & /@ Range[5]], {\[Phi]0, Pi, 3 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100], Show[ParametricPlot3D[Evaluate[ Plus[PST2, {0, 0, 2 tB\[Phi] /. \[Phi] -> 2 Pi}] /. \[Alpha] -> .5 #/5 & /@ Range[5]], {\[Phi], 0, -1.1 2 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100], ParametricPlot3D[Evaluate[ Plus[PST2, {0, 0, 2 tB\[Phi] /. \[Phi] -> 2 Pi}] /. \[Phi] -> 2 Pi #/6 & /@ Range[6]], {\[Alpha], .1, 0.5}, PlotStyle -> Tube[1/32], PlotPoints -> 100]], PlotRange -> All, Boxed -> False, Axes -> False, ImageSize -> 800 ] If you look closely, on the outermost trajectory, the effects of overly-liberal series truncation are narrowly observable. However, the physical scale for the error is about 1/100 of an inch, around the precision limit of the printer, so why worry? Conclusion We have shown that, with a little more work, computer-based calculus could be exported to a formal pen-and-paper proof of isoperiodicity between the simple pendulum and the Edward's Curve with addition rules. This is well demonstrated by the level sets of 3D printed models. Work is on-going regardless of anything else...

Introduction

In general the period of an oscillation can depend on a variable parameter such as energy, here $\alpha$. Then two distinct oscillators of variable energy are called isoperiodic whenever periods equal for all values of $\alpha$ : $T_{1}(\alpha)=T_{2}(\alpha)$. Even in the simple setting of Hamiltonian oscillation, there are a wide range of possibilities as explored in the following Demonstration:

http://demonstrations.wolfram.com/IsoperiodicPotentialsViaSeriesExpansion

This demonstration focuses solely on construction of isoperiodic potentials, which are curves in two dimensions. Exact solutions of the time-dependent oscillation make for beautiful geometry in a three-dimensional space where time goes along the vertical, above the horizontal plane, phase space. Using the Morse and Pöschl–Teller example, as in the demonstration above, we construct the following models and export to Shapeways:

https://www.shapeways.com/product/GXBPX8F24/morse-oscillator-time-helices?optionId=62983286

https://www.shapeways.com/product/BB74864YX/poschl-teller-oscillator-time-helices?optionId=62983299

When the printed models stand side-by-side on a flat surface, the upper-most level sets are at equal heights, a stunning and tangible demonstration of isoperiodicity. Contrasting demonstration with models, it's rewarding to imagine that the continuous deformation of potentials implies a continuous deformation between the two models. Expect to hear more about symmetry breaking in the near future, but for now we will discuss an example where symmetry is added rather than broken !

Simple Pendulum ( Again )

First we construct the phase space trajectory by applying series inversion as in numerous previous references ( Cf. W.C. 1 , W.C. 2 ). We then run a couple of validation test to make sure the general primitives look roughly okay:

vSet[n_, m_] := 
 Map[Range[n] /. Append[Rule @@ # & /@ Tally[#], x_Integer :> 0] &,
  DeleteCases[
   Select[IntegerPartitions[n + m], Length[#] > m - 1 &], {n + 1, 
    1 ...}]]

GExp[n_] := y*Total[y^# g[#] & /@ Range[0, n - 1]]

gCalc[0, _] = 1;
gCalc[n_, m_] := With[{vs = vSet[n, m]},
  Total@ReplaceAll[Times[-1/m, Multinomial @@ #, c[Total[#] - m],
       Times @@ Power[gSet[#] & /@ Range[0, n - 1], #]] & /@ vs,
    {c[0] -> 1}]]

MultinomialExpand[n_, m_] := Module[{},
  Clear@gSet; Set[gSet[#], Expand@gCalc[#, m]] & /@ Range[0, n - 1];
  Expand[GExp[n + 1] /. g[n2_] :> gCalc[n2, m]]]

\[Psi]Test = MultinomialExpand[10, 2] /. c[x_] :> c[x] Q^(x + 2);

TrigReduce[Normal@Series[
   (p^2 + q^2 + Total[c[#] q^(# + 2) & /@ Range[5] ]
      ) /. {p -> \[Psi]Test Sin[\[Phi]], q -> \[Psi]Test Cos[\[Phi]]
      } /. Q -> Cos[\[Phi]], {y, 0, 5}]]

SameQ[ 
 With[{exp = TrigReduce[Normal@Series[
        -Q D[\[Psi]Test, Q]/\[Psi]Test - 1,
        {y, 0, 2}] /. Q -> Cos[\[Phi]]]},
  Expand[Coefficient[exp, y, #] & /@ Range[0, 2]]],
 With[{exp = TrigReduce[Normal@Series[
        D[
          Expand[-(1/2)*\[Psi]Test^2 /. 
            y -> (2 \[Alpha])^(1/2)], \[Alpha]] /. \[Alpha] -> (1/
             2) y^2,
        {y, 0, 2}] /. Q -> Cos[\[Phi]]]},
  Expand[Coefficient[exp, y, #] & /@ Range[0, 2]]]]

The first prints $y^2$, suggesting the correct series inversion. The second prints "True" suggesting that the time-dependence is calculated correctly.

All that remains is to fill in values for the expansion coefficients, and then to plot. Here we limit ourselves to half the total energy range:

\[Psi]Pendulum = Sqrt[2 \[Alpha]] Expand[
    (MultinomialExpand[20, 2] /. c[x_] :> c[x] Q^(x + 2) /. 
         c[x_ /; OddQ[x]] :> 0 /. {c[2] -> (-1/12), c[4] -> 1/360, 
         c[6] -> (-1/20160)} /. y -> Sqrt[4 \[Alpha]])/
     Sqrt[4 \[Alpha]]] /. \[Alpha]^n_ /; n > 3 :> 0

dt = TrigReduce[
    D[Expand[(1/2)*\[Psi]Pendulum^2], \[Alpha]] /. 
     Q -> Cos[\[Phi]]] /. \[Alpha]^n_ /; n > 5 :> 0;

t[\[Phi]1_, \[Phi]2_] := 
 Expand[(1/2/Pi) Integrate[dt, {\[Phi], \[Phi]1, \[Phi]2}]]

tA\[Phi] = t[Pi/2, Pi/2 + \[Phi]];

tB\[Phi] = t[3 Pi/2, 3 Pi/2 + \[Phi]];

tC\[Phi] = t[Pi/2 + \[Phi], \[Phi] + Pi + Pi/2];

tB\[Phi] + tA\[Phi] /. {\[Phi] -> Pi}


PST = (\[Psi]Pendulum /. Q -> Cos[\[Phi] + Pi/2]) {Cos[\[Phi] + Pi/2],
     Sin[\[Phi] + Pi/2], 0};
PST2 = (\[Psi]Pendulum /. 
     Q -> Cos[\[Phi] + Pi/2 + Pi]) {Cos[\[Phi] + Pi/2 + Pi], 
    Sin[\[Phi] + Pi/2 + Pi], 0};

DoubleHelixPendulum = Show[
  ParametricPlot3D[Evaluate[
    Plus[PST, {0, 0, -2 tA\[Phi]}] /. \[Alpha] -> .5 #/5 & /@ 
     Range[5]],
   {\[Phi], 0, -2 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100],
  ParametricPlot3D[Evaluate[
    Plus[PST2, {0, 0, -2 tB\[Phi]}] /. \[Alpha] -> .5 #/5 & /@ 
     Range[5]],
   {\[Phi], 0, -2 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100],
  ParametricPlot3D[Evaluate[
    Plus[PST, {0, 0, 2 tC\[Phi]}] /. \[Alpha] -> .5 #/5 & /@ Range[5]],
   {\[Phi], 0, 1.1 2 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100],

  ParametricPlot3D[Evaluate[
    Plus[PST2, {0, 0, 
         2 tB\[Phi] /. \[Phi] -> 2 Pi}] /. \[Alpha] -> .5 #/5 & /@ 
     Range[5]],
   {\[Phi], 0, -1.1 2 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100],
  ParametricPlot3D[Evaluate[
    Plus[PST2, {0, 0, 
         2 tB\[Phi] /. \[Phi] -> 0}] /. \[Alpha] -> .5 #/5 & /@ 
     Range[5]],
   {\[Phi], 0, -1.1 2 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100],
  ParametricPlot3D[Evaluate[
    Plus[PST2, {0, 0, 2 tB\[Phi] /. \[Phi] -> 0}] /. \[Phi] -> 
        2 Pi #/6 & /@ Range[6]],
   {\[Alpha], .1, 0.5}, PlotStyle -> Tube[1/32], PlotPoints -> 100],

  ParametricPlot3D[Evaluate[
    Plus[PST, {0, 0, 2 tC\[Phi]}] /. \[Phi] -> 2 Pi #/6 & /@ Range[6]],
   {\[Alpha], .1, 0.5}, PlotStyle -> Tube[1/32], PlotPoints -> 100],

  ParametricPlot3D[Evaluate[
    Plus[PST2, {0, 0, 2 tB\[Phi] /. \[Phi] -> 2 Pi}] /. \[Phi] -> 
        2 Pi #/6 & /@ Range[6]],
   {\[Alpha], .1, 0.5}, PlotStyle -> Tube[1/32], PlotPoints -> 100],

  Boxed -> False, Axes -> False, PlotRange -> All, ImageSize -> 300
  ]

time helices

This model can be saved as an ".stl" file and exported directly to shapeways.

Shapeways: Simple Pendulum Time Helices

Edward's Curve

As recently announced on seqfans, it's relatively easy to apply polar coordinates in an analysis of the Edward's Curve ( Cf. Edwards & Bernstein & Lange ), and this approach readily yields a simple exact form for the "time dependence" of the addition rules. First we calculate a radius function for the genus-one solution:

Edwards = x^2 + y^2 - (1 + d x^2 y^2);
r[\[Phi]_] :=  Sqrt[(1 - Sqrt[1 - d Sin[2 \[Phi]]^2]) 2  Csc[2 \[Phi]]^2/d]
r[\[Phi]]

r expression

And check this:

TrigReduce[ Edwards /. {x -> r[\[Phi]] Cos[\[Phi]], y -> r[\[Phi]] Sin[\[Phi]]}]

Yields Zero, as necessary. TrigReduce can be replaced by a set of replacement rules. Next we write the addition rule in the form of a tangent function,

Tan\[Phi]3[\[Phi]1_, 0] := Tan[\[Phi]1]
Tan\[Phi]3[\[Phi]1_, \[Phi]2_] :=  Tan[\[Phi]1 + \[Phi]2] (1 - d z)/(1 + d z) /. 
  z -> r[\[Phi]1]^2 r[\[Phi]2]^2 Cos[\[Phi]1] Sin[\[Phi]1] Cos[\[Phi]2] Sin[\[Phi]2]

And calculate the derivative

\[Phi]dot = Times[
  Normal@Series[ Tan\[Phi]3[\[Phi],   \[Omega]dt], {\[Omega]dt, 0, 1}] -  Tan\[Phi]3[\[Phi], 0],
  Cos[\[Phi]]^2/\[Omega]dt  ]
d\[Phi]/TrigReduce[Expand[\[Phi]dot]]

The closed form result is ,

$$\omega \; dt = \frac{d\phi}{\sqrt{1-d \sin^2(2\phi)}} = \frac{d\phi}{\sqrt{1-4 \;d \big( \sin(\phi)\cos(\phi)\big)^2}} .$$

Clearly the complete elliptic integral of the first kind is given by any integral of the form

$$K(d) \propto \frac{1}{2\pi} \int_{0}^{2\pi} \frac{d\phi}{\sqrt{1-d \sin^2(n\;\phi)}}, $$

with integer $n$. As is well known the pendulum has $n=1$, and we see here that Edward's curve has $n=2$, proving the two systems are isoperiodic. Let's now exploit the square symmetry of Edward's curve by making a quadruple-helix, 3D-printable Model.

First we expand the radius to avoid singular points, and integrate the time dependence ( this is done naively, and could be optimized with a little more effort ),

rEdwards = Normal[Series[Sqrt[2 d] r[\[Phi]0], {d, 0, 20}]];
dt = 1/Sqrt[Expand[1 - 4 d x^2]] d\[Phi] 
t = 1/2/Pi Integrate[ Evaluate[  Expand[TrigReduce[   Normal@Series[dt, {d, 0, 20}] /. 
         x -> Cos[\[Phi]] Sin[\[Phi]]]/d\[Phi]]], {\[Phi],   Pi, \[Phi]0}];

The extra factor of $\sqrt{d}$ corresponds to a coordinate change where Edward's equation takes a form $d = x^2+y^2-x^2y^2$, which allows us to plot the time spirals,

EdwardsQuadrupleHelix = Show[
  Function[{a},
    ParametricPlot3D[
     Evaluate[{rEdwards Cos[\[Phi]0 + Pi/2 a], 
          rEdwards Sin[\[Phi]0 + Pi/2 a], 2 t} /. d -> #/10 & /@ 
       Range[5]], {\[Phi]0, Pi, 3 Pi}, PlotStyle -> Tube[1/32], 
     PlotPoints -> 100]
    ] /@ Range[0, 3],

  Function[{a},
    ParametricPlot3D[
     Evaluate[{rEdwards Cos[\[Phi]0 + Pi/2 a], 
        rEdwards Sin[\[Phi]0 + Pi/2 a], 2 t} /. \[Phi]0 -> Pi], {d, 
      0.1, .5}, PlotStyle -> Tube[1/32], PlotPoints -> 100]
    ] /@ Range[0, 3],

  Function[{a},
    ParametricPlot3D[
     Evaluate[{rEdwards Cos[\[Phi]0 + Pi/2 a], 
        rEdwards Sin[\[Phi]0 + Pi/2 a], 2 t} /. \[Phi]0 -> 2 Pi], {d, 
      0.1, .5}, PlotStyle -> Tube[1/32], PlotPoints -> 100]
    ] /@ Range[0, 3],

  Function[{a},
    ParametricPlot3D[
     Evaluate[{rEdwards Cos[\[Phi]0 + Pi/2 a], 
        rEdwards  Sin[\[Phi]0 + Pi/2 a], 2 t} /. \[Phi]0 -> 3 Pi], {d,
       0.1, .5}, PlotStyle -> Tube[1/32], PlotPoints -> 100]
    ] /@ Range[0, 3],


  ParametricPlot3D[
   Evaluate[{rEdwards Cos[\[Phi]0 + Pi], rEdwards Sin[\[Phi]0 + Pi], 
        2 t /. \[Phi]0 -> Pi} /. d -> #/10 & /@ Range[5]], {\[Phi]0, 
    Pi, 3 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100],
  ParametricPlot3D[
   Evaluate[{rEdwards Cos[\[Phi]0 + Pi], rEdwards Sin[\[Phi]0 + Pi], 
        2 t /. \[Phi]0 -> 2 Pi} /. d -> #/10 & /@ 
     Range[5]], {\[Phi]0, Pi, 3 Pi}, PlotStyle -> Tube[1/32], 
   PlotPoints -> 100],
  ParametricPlot3D[
   Evaluate[{rEdwards Cos[\[Phi]0 + Pi], rEdwards Sin[\[Phi]0 + Pi], 
        2 t /. \[Phi]0 -> 3 Pi} /. d -> #/10 & /@ 
     Range[5]], {\[Phi]0, Pi, 3 Pi}, PlotStyle -> Tube[1/32], 
   PlotPoints -> 100]

  , PlotRange -> All, Boxed -> False, Axes -> False, ImageSize -> 800
  ]

Edward's isoperiodicity

Again, can be exported directly to shapeways. Finally let's take a closer look at isoperiodicity, by comparing the upper-most level sets,

Show[Function[{a},
   ParametricPlot3D[
    Evaluate[{rEdwards Cos[\[Phi]0 + Pi/2 a], 
       rEdwards  Sin[\[Phi]0 + Pi/2 a], 2 t} /. \[Phi]0 -> 3 Pi], {d, 
     0.1, .5}, PlotStyle -> Tube[1/32], PlotPoints -> 100]
   ] /@ Range[0, 3],
 ParametricPlot3D[
  Evaluate[{rEdwards Cos[\[Phi]0 + Pi], rEdwards Sin[\[Phi]0 + Pi], 
       2 t /. \[Phi]0 -> 3 Pi} /. d -> #/10 & /@ Range[5]], {\[Phi]0, 
   Pi, 3 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100],
 Show[ParametricPlot3D[Evaluate[
    Plus[PST2, {0, 0, 
         2 tB\[Phi] /. \[Phi] -> 2 Pi}] /. \[Alpha] -> .5 #/5 & /@ 
     Range[5]],
   {\[Phi], 0, -1.1 2 Pi}, PlotStyle -> Tube[1/32], PlotPoints -> 100],
  ParametricPlot3D[Evaluate[
    Plus[PST2, {0, 0, 2 tB\[Phi] /. \[Phi] -> 2 Pi}] /. \[Phi] -> 
        2 Pi #/6 & /@ Range[6]],
   {\[Alpha], .1, 0.5}, PlotStyle -> Tube[1/32], PlotPoints -> 100]],
 PlotRange -> All, Boxed -> False, Axes -> False, ImageSize -> 800
 ]

comparison

If you look closely, on the outermost trajectory, the effects of overly-liberal series truncation are narrowly observable. However, the physical scale for the error is about 1/100 of an inch, around the precision limit of the printer, so why worry?

Conclusion

We have shown that, with a little more work, computer-based calculus could be exported to a formal pen-and-paper proof of isoperiodicity between the simple pendulum and the Edward's Curve with addition rules. This is well demonstrated by the level sets of 3D printed models. Work is on-going regardless of anything else...

POSTED BY: Brad Klee

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Brad Klee

Posted 1 year ago

Addition from notes on July 9, 2017. Define radius function: $$r(\phi) =\sqrt{ 2 \csc(2\phi)^2\bigg(1-\sqrt{1-\alpha \sin(2\phi)^2}\bigg) },$$ The solution of $$ \alpha = x^2 + y^2 - x^2 y^2 , $$ Then compute, as above, $$\frac{dt}{d\phi} = \frac{d}{d\alpha} r(\phi,\alpha)^2= \frac{1}{\sqrt{1-\alpha \sin^2(2\phi)}},$$ thus completing the Hamiltonian Analogy. This provides an alternative, geometric answer to the question: how should the addition law be analyzed? ( cf. HALES ). More on this later... As for now, how about another picture? Generated from a slight modification of the algorithm above, negative $\alpha$ solutions along the Edward's curve: Available on Shapeways soon ! !

POSTED BY: Brad Klee

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Moderation Team

Moderation Team, WOLFRAM

Posted 1 year ago

- Congratulations! This post is now a Staff Pick as distinguished by a badge on your profile! Thank you, keep it coming!

POSTED BY: Moderation Team

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Brad Klee

Posted 1 year ago

Hey thanks for the plaudit. I realize that these elliptic functions involve some complicated calculus, but think that the effort is worthwhile. Though elliptic curves are easy-enough to understand, especially in Edward's normal form, some of the younger curve-calculators may want an easier example to start out with. I hate to sound trite, but maybe it's time to look at the harmonic oscillator again? Fibration1 = Show[ ParametricPlot3D[ Evaluate[{Cos[t], -Sin[t], (1/2) # Pi/6} & /@ Range[0, 12]], {t, 0, 2 Pi}, Boxed -> False, Axes -> False, PlotStyle -> Directive[Black, Thickness[.005]]], ParametricPlot3D[ Evaluate[{Cos[# Pi/6], -Sin[# Pi/6], #/12 (1/2) t} & /@ Range[0, 12]], {t, 0, 2 Pi}, Boxed -> False, Axes -> False, PlotStyle -> Directive[Black, Thickness[0.005]]], ParametricPlot3D[ Evaluate[{1.01 Cos[t], -1.01 Sin[t], (1/2) t}], {t, 0, 2 Pi}, Boxed -> False, Axes -> False, PlotStyle -> Directive[Thickness[.015], Black]], Graphics3D[{ PointSize[Large], Point[{1.01 Cos[# Pi/6], -1.01 Sin[# Pi/6], 0}] & /@ Range[12] }], PlotRange -> All, ImageSize -> {500, 500}] The time helix of the harmonic oscillator is a proper helix in the sense that intersections along the vertical, at equal time intervals, project into the phase plane, as points spaced by equal-angle intervals. The connection between continuous and discrete time evolution can be expressed in terms of the well known trigonometric addition rules: $$ (X_3,Y_3) \leftarrow (X_1 X_2-Y_1Y_2, X_1 Y_2 + Y_1X_2) . $$ With a little bit of differential geometry, it's relatively easy to motivate and derive the invariant differentials of the harmonic oscillator: $$ dt= \frac{dX}{Y} = -\frac{dY}{X} $$ But are they truly invariant? What will happen if we transform these differentials by the trigonometric addition rules? rep = { X3[t] -> X1[t] X2[t] - Y1[t] Y2[t], Y3[t] -> X1[t] Y2[t] + Y1[t] X2[t] }; dt == D[X3[t], t]/(Y3[t]) == Expand@Factor[D[X3[t] /. rep, t]/(Y3[t] /. rep) /. { D[Y1[t], t] -> -(X1[t]/Y1[t]) D[X1[t], t], D[Y2[t], t] -> -(X2[t]/Y2[t]) D[X2[t], t] }] dt == D[-Y3[t], t]/(X3[t]) == Expand@Factor[D[-Y3[t] /. rep, t]/(X3[t] /. rep) /. { D[X1[t], t] -> -(Y1[t]/X1[t]) D[Y1[t], t], D[X2[t], t] -> -(Y2[t]/X2[t]) D[Y2[t], t] }] In terms of group theory--I will explain more later--this validation code proves associative and closure property of the trigonometric addition rules at once. Generalizing to elliptic curve addition laws, this approach provides a viable alternative to the zero-calculus method advocated by HALES. However, his notes can be useful because more complicated calculations do require use of a function like PolynomialReduce.