Initial values of position (x) and speed (v) of a particle visualizing using Mathematica












2














$$vec{F}(vec{r})=-momega^2begin{pmatrix}x\4yend{pmatrix}$$



I have the force $F$ shown above. How could I specify the initial values of position ($x$) and speed ($v$) in Mathematica using the Manipulate command to try finding the initial values so the particle could move on a parabolic trajectory ($alpha$) and a eight-shaped trajectory ($beta$) ?



The initial values are not exact, just one solution each is enough.



Unfortunately I have no idea how to realize this problem, would be thankful for help!






differential-equations manipulate physics simulation







share|improve this question





























    2














    $$vec{F}(vec{r})=-momega^2begin{pmatrix}x\4yend{pmatrix}$$



    I have the force $F$ shown above. How could I specify the initial values of position ($x$) and speed ($v$) in Mathematica using the Manipulate command to try finding the initial values so the particle could move on a parabolic trajectory ($alpha$) and a eight-shaped trajectory ($beta$) ?



    The initial values are not exact, just one solution each is enough.



    Unfortunately I have no idea how to realize this problem, would be thankful for help!






    differential-equations manipulate physics simulation







    share|improve this question



























      2












      2








      2


      1





      $$vec{F}(vec{r})=-momega^2begin{pmatrix}x\4yend{pmatrix}$$



      I have the force $F$ shown above. How could I specify the initial values of position ($x$) and speed ($v$) in Mathematica using the Manipulate command to try finding the initial values so the particle could move on a parabolic trajectory ($alpha$) and a eight-shaped trajectory ($beta$) ?



      The initial values are not exact, just one solution each is enough.



      Unfortunately I have no idea how to realize this problem, would be thankful for help!






      differential-equations manipulate physics simulation







      share|improve this question















      $$vec{F}(vec{r})=-momega^2begin{pmatrix}x\4yend{pmatrix}$$



      I have the force $F$ shown above. How could I specify the initial values of position ($x$) and speed ($v$) in Mathematica using the Manipulate command to try finding the initial values so the particle could move on a parabolic trajectory ($alpha$) and a eight-shaped trajectory ($beta$) ?



      The initial values are not exact, just one solution each is enough.



      Unfortunately I have no idea how to realize this problem, would be thankful for help!






      differential-equations manipulate physics simulation




      differential-equations manipulate physics simulation






      share|improve this question















      share|improve this question













      share|improve this question




      share|improve this question








      edited yesterday









      J. M. is computer-less♦

      96.2k10300460




      96.2k10300460










      asked yesterday









      TomTom

      534




      534






















          2 Answers
          2






          active

          oldest

          votes


















          3














          Here is an interactive Manipulate using ParametricNDSolveValue to solve the differential equation; you can interact with it by dragging the locators to the desired sites and by adjusting the time horizon T by dragg the control bar at the top:



          F[{x_, y_}] := {x, 4 y};
          
traj = ParametricNDSolveValue[
          
   {
          
    Y''[t] == -F[Y[t]],
          
    Y[0] == {x0, y0},
          
    Y'[0] == {v0, w0}
          
    },
          
   Y,
          
   {t, 0, T},
          
   {x0, y0, v0, w0, T}
          
   ];
          

          
Manipulate[
          
 Show[
          
  Graphics[Arrow[{X[[1]], X[[2]]}]],
          
  ParametricPlot[
          
   traj[X[[1, 1]], X[[1, 2]], X[[2, 1]] - X[[1, 1]], X[[2, 2]] - X[[1, 2]], T][t], 
          
   {t, 0, T}
          
   ],
          
  PlotRange -> {{-1, 1}, {-1, 1}} 2
          
  ],
          
 {{X, {{1, 0}, {1, 1}}}, Locator},
          
 {{T, 5}, 0, 10}
          
 ]





          share|improve this answer





















          • looks amazing! thank you very much!
            – Tom
            yesterday










          • You're welcome. Have fun!
            – Henrik Schumacher
            yesterday










          • @Tom Btw.: Don't forget to upvote answers that you found helpful... That's what drives the community.
            – Henrik Schumacher
            yesterday






          • 1




            @Tom Just in case that you wonder: You can accept only one answer per question. ;) And I can live with it if you choose David's one...
            – Henrik Schumacher
            yesterday






          • 1




            @Tom: you can upvote both answers, but you can only accept one.
            – J. M. is computer-less♦
            yesterday



















          5














          Solve



          x[t] /. DSolve[ x''[t] == - w^2 x[t], x[t], t]
          

          
y[t] /. DSolve[ y''[t] == - w^2 4 y[t], y[t], t]


          to find $x(t) = cos (omega t) + sin (omega t)$ and $y(t) = cos (2 omega t) + sin (2 omega t)$, with arbitrary constants that depend upon the initial conditions. Then plot:



          w = 1;
          
ParametricPlot[{Cos[w t] + Sin[w t], Cos[ 2 w t] + Sin[2 w t]}, {t, 0,
          
   5}]


          enter image description here






          share|improve this answer





















          • thank you very much David!
            – Tom
            yesterday










          • Note that you can also include x[0] == x0, y[0] == y0 etc. in the list of equations sent to DSolve. This will yield a functional form for the solution that explicitly contains the initial conditions.
            – Michael Seifert
            yesterday











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          2 Answers
          2






          active

          oldest

          votes








          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          3














          Here is an interactive Manipulate using ParametricNDSolveValue to solve the differential equation; you can interact with it by dragging the locators to the desired sites and by adjusting the time horizon T by dragg the control bar at the top:



          F[{x_, y_}] := {x, 4 y};
          
traj = ParametricNDSolveValue[
          
   {
          
    Y''[t] == -F[Y[t]],
          
    Y[0] == {x0, y0},
          
    Y'[0] == {v0, w0}
          
    },
          
   Y,
          
   {t, 0, T},
          
   {x0, y0, v0, w0, T}
          
   ];
          

          
Manipulate[
          
 Show[
          
  Graphics[Arrow[{X[[1]], X[[2]]}]],
          
  ParametricPlot[
          
   traj[X[[1, 1]], X[[1, 2]], X[[2, 1]] - X[[1, 1]], X[[2, 2]] - X[[1, 2]], T][t], 
          
   {t, 0, T}
          
   ],
          
  PlotRange -> {{-1, 1}, {-1, 1}} 2
          
  ],
          
 {{X, {{1, 0}, {1, 1}}}, Locator},
          
 {{T, 5}, 0, 10}
          
 ]





          share|improve this answer





















          • looks amazing! thank you very much!
            – Tom
            yesterday










          • You're welcome. Have fun!
            – Henrik Schumacher
            yesterday










          • @Tom Btw.: Don't forget to upvote answers that you found helpful... That's what drives the community.
            – Henrik Schumacher
            yesterday






          • 1




            @Tom Just in case that you wonder: You can accept only one answer per question. ;) And I can live with it if you choose David's one...
            – Henrik Schumacher
            yesterday






          • 1




            @Tom: you can upvote both answers, but you can only accept one.
            – J. M. is computer-less♦
            yesterday
















          3














          Here is an interactive Manipulate using ParametricNDSolveValue to solve the differential equation; you can interact with it by dragging the locators to the desired sites and by adjusting the time horizon T by dragg the control bar at the top:



          F[{x_, y_}] := {x, 4 y};
          
traj = ParametricNDSolveValue[
          
   {
          
    Y''[t] == -F[Y[t]],
          
    Y[0] == {x0, y0},
          
    Y'[0] == {v0, w0}
          
    },
          
   Y,
          
   {t, 0, T},
          
   {x0, y0, v0, w0, T}
          
   ];
          

          
Manipulate[
          
 Show[
          
  Graphics[Arrow[{X[[1]], X[[2]]}]],
          
  ParametricPlot[
          
   traj[X[[1, 1]], X[[1, 2]], X[[2, 1]] - X[[1, 1]], X[[2, 2]] - X[[1, 2]], T][t], 
          
   {t, 0, T}
          
   ],
          
  PlotRange -> {{-1, 1}, {-1, 1}} 2
          
  ],
          
 {{X, {{1, 0}, {1, 1}}}, Locator},
          
 {{T, 5}, 0, 10}
          
 ]





          share|improve this answer





















          • looks amazing! thank you very much!
            – Tom
            yesterday










          • You're welcome. Have fun!
            – Henrik Schumacher
            yesterday










          • @Tom Btw.: Don't forget to upvote answers that you found helpful... That's what drives the community.
            – Henrik Schumacher
            yesterday






          • 1




            @Tom Just in case that you wonder: You can accept only one answer per question. ;) And I can live with it if you choose David's one...
            – Henrik Schumacher
            yesterday






          • 1




            @Tom: you can upvote both answers, but you can only accept one.
            – J. M. is computer-less♦
            yesterday














          3












          3








          3






          Here is an interactive Manipulate using ParametricNDSolveValue to solve the differential equation; you can interact with it by dragging the locators to the desired sites and by adjusting the time horizon T by dragg the control bar at the top:



          F[{x_, y_}] := {x, 4 y};
          
traj = ParametricNDSolveValue[
          
   {
          
    Y''[t] == -F[Y[t]],
          
    Y[0] == {x0, y0},
          
    Y'[0] == {v0, w0}
          
    },
          
   Y,
          
   {t, 0, T},
          
   {x0, y0, v0, w0, T}
          
   ];
          

          
Manipulate[
          
 Show[
          
  Graphics[Arrow[{X[[1]], X[[2]]}]],
          
  ParametricPlot[
          
   traj[X[[1, 1]], X[[1, 2]], X[[2, 1]] - X[[1, 1]], X[[2, 2]] - X[[1, 2]], T][t], 
          
   {t, 0, T}
          
   ],
          
  PlotRange -> {{-1, 1}, {-1, 1}} 2
          
  ],
          
 {{X, {{1, 0}, {1, 1}}}, Locator},
          
 {{T, 5}, 0, 10}
          
 ]





          share|improve this answer












          Here is an interactive Manipulate using ParametricNDSolveValue to solve the differential equation; you can interact with it by dragging the locators to the desired sites and by adjusting the time horizon T by dragg the control bar at the top:



          F[{x_, y_}] := {x, 4 y};
          
traj = ParametricNDSolveValue[
          
   {
          
    Y''[t] == -F[Y[t]],
          
    Y[0] == {x0, y0},
          
    Y'[0] == {v0, w0}
          
    },
          
   Y,
          
   {t, 0, T},
          
   {x0, y0, v0, w0, T}
          
   ];
          

          
Manipulate[
          
 Show[
          
  Graphics[Arrow[{X[[1]], X[[2]]}]],
          
  ParametricPlot[
          
   traj[X[[1, 1]], X[[1, 2]], X[[2, 1]] - X[[1, 1]], X[[2, 2]] - X[[1, 2]], T][t], 
          
   {t, 0, T}
          
   ],
          
  PlotRange -> {{-1, 1}, {-1, 1}} 2
          
  ],
          
 {{X, {{1, 0}, {1, 1}}}, Locator},
          
 {{T, 5}, 0, 10}
          
 ]






          share|improve this answer












          share|improve this answer



          share|improve this answer










          answered yesterday









          Henrik SchumacherHenrik Schumacher

          49.8k469142




          49.8k469142












          • looks amazing! thank you very much!
            – Tom
            yesterday










          • You're welcome. Have fun!
            – Henrik Schumacher
            yesterday










          • @Tom Btw.: Don't forget to upvote answers that you found helpful... That's what drives the community.
            – Henrik Schumacher
            yesterday






          • 1




            @Tom Just in case that you wonder: You can accept only one answer per question. ;) And I can live with it if you choose David's one...
            – Henrik Schumacher
            yesterday






          • 1




            @Tom: you can upvote both answers, but you can only accept one.
            – J. M. is computer-less♦
            yesterday


















          • looks amazing! thank you very much!
            – Tom
            yesterday










          • You're welcome. Have fun!
            – Henrik Schumacher
            yesterday










          • @Tom Btw.: Don't forget to upvote answers that you found helpful... That's what drives the community.
            – Henrik Schumacher
            yesterday






          • 1




            @Tom Just in case that you wonder: You can accept only one answer per question. ;) And I can live with it if you choose David's one...
            – Henrik Schumacher
            yesterday






          • 1




            @Tom: you can upvote both answers, but you can only accept one.
            – J. M. is computer-less♦
            yesterday
















          looks amazing! thank you very much!
          – Tom
          yesterday




          looks amazing! thank you very much!
          – Tom
          yesterday












          You're welcome. Have fun!
          – Henrik Schumacher
          yesterday




          You're welcome. Have fun!
          – Henrik Schumacher
          yesterday












          @Tom Btw.: Don't forget to upvote answers that you found helpful... That's what drives the community.
          – Henrik Schumacher
          yesterday




          @Tom Btw.: Don't forget to upvote answers that you found helpful... That's what drives the community.
          – Henrik Schumacher
          yesterday




          1




          1




          @Tom Just in case that you wonder: You can accept only one answer per question. ;) And I can live with it if you choose David's one...
          – Henrik Schumacher
          yesterday




          @Tom Just in case that you wonder: You can accept only one answer per question. ;) And I can live with it if you choose David's one...
          – Henrik Schumacher
          yesterday




          1




          1




          @Tom: you can upvote both answers, but you can only accept one.
          – J. M. is computer-less♦
          yesterday




          @Tom: you can upvote both answers, but you can only accept one.
          – J. M. is computer-less♦
          yesterday











          5














          Solve



          x[t] /. DSolve[ x''[t] == - w^2 x[t], x[t], t]
          

          
y[t] /. DSolve[ y''[t] == - w^2 4 y[t], y[t], t]


          to find $x(t) = cos (omega t) + sin (omega t)$ and $y(t) = cos (2 omega t) + sin (2 omega t)$, with arbitrary constants that depend upon the initial conditions. Then plot:



          w = 1;
          
ParametricPlot[{Cos[w t] + Sin[w t], Cos[ 2 w t] + Sin[2 w t]}, {t, 0,
          
   5}]


          enter image description here






          share|improve this answer





















          • thank you very much David!
            – Tom
            yesterday










          • Note that you can also include x[0] == x0, y[0] == y0 etc. in the list of equations sent to DSolve. This will yield a functional form for the solution that explicitly contains the initial conditions.
            – Michael Seifert
            yesterday
















          5














          Solve



          x[t] /. DSolve[ x''[t] == - w^2 x[t], x[t], t]
          

          
y[t] /. DSolve[ y''[t] == - w^2 4 y[t], y[t], t]


          to find $x(t) = cos (omega t) + sin (omega t)$ and $y(t) = cos (2 omega t) + sin (2 omega t)$, with arbitrary constants that depend upon the initial conditions. Then plot:



          w = 1;
          
ParametricPlot[{Cos[w t] + Sin[w t], Cos[ 2 w t] + Sin[2 w t]}, {t, 0,
          
   5}]


          enter image description here






          share|improve this answer





















          • thank you very much David!
            – Tom
            yesterday










          • Note that you can also include x[0] == x0, y[0] == y0 etc. in the list of equations sent to DSolve. This will yield a functional form for the solution that explicitly contains the initial conditions.
            – Michael Seifert
            yesterday














          5












          5








          5






          Solve



          x[t] /. DSolve[ x''[t] == - w^2 x[t], x[t], t]
          

          
y[t] /. DSolve[ y''[t] == - w^2 4 y[t], y[t], t]


          to find $x(t) = cos (omega t) + sin (omega t)$ and $y(t) = cos (2 omega t) + sin (2 omega t)$, with arbitrary constants that depend upon the initial conditions. Then plot:



          w = 1;
          
ParametricPlot[{Cos[w t] + Sin[w t], Cos[ 2 w t] + Sin[2 w t]}, {t, 0,
          
   5}]


          enter image description here






          share|improve this answer












          Solve



          x[t] /. DSolve[ x''[t] == - w^2 x[t], x[t], t]
          

          
y[t] /. DSolve[ y''[t] == - w^2 4 y[t], y[t], t]


          to find $x(t) = cos (omega t) + sin (omega t)$ and $y(t) = cos (2 omega t) + sin (2 omega t)$, with arbitrary constants that depend upon the initial conditions. Then plot:



          w = 1;
          
ParametricPlot[{Cos[w t] + Sin[w t], Cos[ 2 w t] + Sin[2 w t]}, {t, 0,
          
   5}]


          enter image description here







          share|improve this answer












          share|improve this answer



          share|improve this answer










          answered yesterday









          David G. StorkDavid G. Stork

          23.4k22051




          23.4k22051












          • thank you very much David!
            – Tom
            yesterday










          • Note that you can also include x[0] == x0, y[0] == y0 etc. in the list of equations sent to DSolve. This will yield a functional form for the solution that explicitly contains the initial conditions.
            – Michael Seifert
            yesterday


















          • thank you very much David!
            – Tom
            yesterday










          • Note that you can also include x[0] == x0, y[0] == y0 etc. in the list of equations sent to DSolve. This will yield a functional form for the solution that explicitly contains the initial conditions.
            – Michael Seifert
            yesterday
















          thank you very much David!
          – Tom
          yesterday




          thank you very much David!
          – Tom
          yesterday












          Note that you can also include x[0] == x0, y[0] == y0 etc. in the list of equations sent to DSolve. This will yield a functional form for the solution that explicitly contains the initial conditions.
          – Michael Seifert
          yesterday




          Note that you can also include x[0] == x0, y[0] == y0 etc. in the list of equations sent to DSolve. This will yield a functional form for the solution that explicitly contains the initial conditions.
          – Michael Seifert
          yesterday


















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