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

Question

$$\vec{F}(\vec{r})=-m\omega^2\begin{pmatrix}x\\4y\end{pmatrix}$$

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!

Answer 1

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}
 ]

Answer 2

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}]