How to plot on a curved plane?












4












$begingroup$


I'm ploting the phase space of a pendulum problem using a symplectic Euler scheme.



$qquad H = frac{1}{2}p^2 - cos q$, where $dot{p}=-sin q$ and $dot{q}=p$



h=0.2; (*time step*)

p[0]=0.0; (*initial conditions*)

q[0]=0.5;



p[i_] := p[i] = p[i - 1] - h*Sin[q[i - 1]];

q[i_] := q[i] = q[i - 1] + h*p[i - 1] - h^2*Sin[q[i - 1]];



ListPlot[Table[{p[i], q[i]}, {i, 0, 100}], Frame -> True]


gives



phase space plot



Since the vector field is $2π$-periodic in q, it is natural to consider q as a variable on the circle $S^1$, I'd expect it to look something like



enter image description here



Any suggest how to do it?






plotting







share|improve this question











$endgroup$












  • $begingroup$
    There is no such thing as a "curved plane". If want to plot on a 2-manifold, please give a description of the manifold in Wolfram Language code.
    $endgroup$
    – m_goldberg
    15 hours ago
















4












$begingroup$


I'm ploting the phase space of a pendulum problem using a symplectic Euler scheme.



$qquad H = frac{1}{2}p^2 - cos q$, where $dot{p}=-sin q$ and $dot{q}=p$



h=0.2; (*time step*)

p[0]=0.0; (*initial conditions*)

q[0]=0.5;



p[i_] := p[i] = p[i - 1] - h*Sin[q[i - 1]];

q[i_] := q[i] = q[i - 1] + h*p[i - 1] - h^2*Sin[q[i - 1]];



ListPlot[Table[{p[i], q[i]}, {i, 0, 100}], Frame -> True]


gives



phase space plot



Since the vector field is $2π$-periodic in q, it is natural to consider q as a variable on the circle $S^1$, I'd expect it to look something like



enter image description here



Any suggest how to do it?






plotting







share|improve this question











$endgroup$












  • $begingroup$
    There is no such thing as a "curved plane". If want to plot on a 2-manifold, please give a description of the manifold in Wolfram Language code.
    $endgroup$
    – m_goldberg
    15 hours ago














4












4








4





$begingroup$


I'm ploting the phase space of a pendulum problem using a symplectic Euler scheme.



$qquad H = frac{1}{2}p^2 - cos q$, where $dot{p}=-sin q$ and $dot{q}=p$



h=0.2; (*time step*)

p[0]=0.0; (*initial conditions*)

q[0]=0.5;



p[i_] := p[i] = p[i - 1] - h*Sin[q[i - 1]];

q[i_] := q[i] = q[i - 1] + h*p[i - 1] - h^2*Sin[q[i - 1]];



ListPlot[Table[{p[i], q[i]}, {i, 0, 100}], Frame -> True]


gives



phase space plot



Since the vector field is $2π$-periodic in q, it is natural to consider q as a variable on the circle $S^1$, I'd expect it to look something like



enter image description here



Any suggest how to do it?






plotting







share|improve this question











$endgroup$




I'm ploting the phase space of a pendulum problem using a symplectic Euler scheme.



$qquad H = frac{1}{2}p^2 - cos q$, where $dot{p}=-sin q$ and $dot{q}=p$



h=0.2; (*time step*)

p[0]=0.0; (*initial conditions*)

q[0]=0.5;



p[i_] := p[i] = p[i - 1] - h*Sin[q[i - 1]];

q[i_] := q[i] = q[i - 1] + h*p[i - 1] - h^2*Sin[q[i - 1]];



ListPlot[Table[{p[i], q[i]}, {i, 0, 100}], Frame -> True]


gives



phase space plot



Since the vector field is $2π$-periodic in q, it is natural to consider q as a variable on the circle $S^1$, I'd expect it to look something like



enter image description here



Any suggest how to do it?






plotting




plotting






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 15 hours ago









m_goldberg

88.2k872199




88.2k872199










asked 19 hours ago









GvxfjørtGvxfjørt

986




986












  • $begingroup$
    There is no such thing as a "curved plane". If want to plot on a 2-manifold, please give a description of the manifold in Wolfram Language code.
    $endgroup$
    – m_goldberg
    15 hours ago


















  • $begingroup$
    There is no such thing as a "curved plane". If want to plot on a 2-manifold, please give a description of the manifold in Wolfram Language code.
    $endgroup$
    – m_goldberg
    15 hours ago
















$begingroup$
There is no such thing as a "curved plane". If want to plot on a 2-manifold, please give a description of the manifold in Wolfram Language code.
$endgroup$
– m_goldberg
15 hours ago




$begingroup$
There is no such thing as a "curved plane". If want to plot on a 2-manifold, please give a description of the manifold in Wolfram Language code.
$endgroup$
– m_goldberg
15 hours ago










1 Answer
1






active

oldest

votes


















6












$begingroup$


h = 0.2;
p[0, a_] := 0;
q[0, a_] := a
p[i_, a_] := p[i, a] = p[i - 1, a] - h*Sin[q[i - 1, a]];
q[i_, a_] :=
q[i, a] = q[i - 1, a] + h*p[i - 1, a] - h^2*Sin[q[i - 1, a]];




plots = Table[
ListPointPlot3D[
Table[{Sin[q[i, a]], Cos[q[i, a]], p[i, a]}, {i, 0, 100}],
PlotStyle -> PointSize[0.008],
PlotRange -> {{-1, 1}, {-1, 1}, {-3, 3}}], {a, 0.5, 3, 0.5}];



Show[plots,
Graphics3D[{Opacity[0.1], Cylinder[{{0, 0, -3}, {0, 0, 3}}]}]]



enter image description here






share|improve this answer









$endgroup$














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    1 Answer
    1






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    active

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    active

    oldest

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    6












    $begingroup$


    h = 0.2;
    p[0, a_] := 0;
    q[0, a_] := a
    p[i_, a_] := p[i, a] = p[i - 1, a] - h*Sin[q[i - 1, a]];
    q[i_, a_] :=
    q[i, a] = q[i - 1, a] + h*p[i - 1, a] - h^2*Sin[q[i - 1, a]];




    plots = Table[
    ListPointPlot3D[
    Table[{Sin[q[i, a]], Cos[q[i, a]], p[i, a]}, {i, 0, 100}],
    PlotStyle -> PointSize[0.008],
    PlotRange -> {{-1, 1}, {-1, 1}, {-3, 3}}], {a, 0.5, 3, 0.5}];



    Show[plots,
    Graphics3D[{Opacity[0.1], Cylinder[{{0, 0, -3}, {0, 0, 3}}]}]]



    enter image description here






    share|improve this answer









    $endgroup$


















      6












      $begingroup$


      h = 0.2;
      p[0, a_] := 0;
      q[0, a_] := a
      p[i_, a_] := p[i, a] = p[i - 1, a] - h*Sin[q[i - 1, a]];
      q[i_, a_] :=
      q[i, a] = q[i - 1, a] + h*p[i - 1, a] - h^2*Sin[q[i - 1, a]];




      plots = Table[
      ListPointPlot3D[
      Table[{Sin[q[i, a]], Cos[q[i, a]], p[i, a]}, {i, 0, 100}],
      PlotStyle -> PointSize[0.008],
      PlotRange -> {{-1, 1}, {-1, 1}, {-3, 3}}], {a, 0.5, 3, 0.5}];



      Show[plots,
      Graphics3D[{Opacity[0.1], Cylinder[{{0, 0, -3}, {0, 0, 3}}]}]]



      enter image description here






      share|improve this answer









      $endgroup$
















        6












        6








        6





        $begingroup$


        h = 0.2;
        p[0, a_] := 0;
        q[0, a_] := a
        p[i_, a_] := p[i, a] = p[i - 1, a] - h*Sin[q[i - 1, a]];
        q[i_, a_] :=
        q[i, a] = q[i - 1, a] + h*p[i - 1, a] - h^2*Sin[q[i - 1, a]];




        plots = Table[
        ListPointPlot3D[
        Table[{Sin[q[i, a]], Cos[q[i, a]], p[i, a]}, {i, 0, 100}],
        PlotStyle -> PointSize[0.008],
        PlotRange -> {{-1, 1}, {-1, 1}, {-3, 3}}], {a, 0.5, 3, 0.5}];



        Show[plots,
        Graphics3D[{Opacity[0.1], Cylinder[{{0, 0, -3}, {0, 0, 3}}]}]]



        enter image description here






        share|improve this answer









        $endgroup$




        h = 0.2;
        p[0, a_] := 0;
        q[0, a_] := a
        p[i_, a_] := p[i, a] = p[i - 1, a] - h*Sin[q[i - 1, a]];
        q[i_, a_] :=
        q[i, a] = q[i - 1, a] + h*p[i - 1, a] - h^2*Sin[q[i - 1, a]];




        plots = Table[
        ListPointPlot3D[
        Table[{Sin[q[i, a]], Cos[q[i, a]], p[i, a]}, {i, 0, 100}],
        PlotStyle -> PointSize[0.008],
        PlotRange -> {{-1, 1}, {-1, 1}, {-3, 3}}], {a, 0.5, 3, 0.5}];



        Show[plots,
        Graphics3D[{Opacity[0.1], Cylinder[{{0, 0, -3}, {0, 0, 3}}]}]]



        enter image description here







        share|improve this answer












        share|improve this answer



        share|improve this answer










        answered 15 hours ago









        ulviulvi

        1,166612




        1,166612






























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