Plot the elliptical orbit from a Lagrangian?

Questions
, , , , , ,
1

Can anyone help me?
I want to make the orbit ellipse.
Please help me to fix my code.

lagrangian = 1/2 (x'[t]^2 + y'[t]^2) + 1/Sqrt[x[t]^2 + y[t]^2];
eq = Table[
  D[lagrangian, x1] - D[D[lagrangian, D[x1, t]], t] == 
   0, {x1, {x[t], y[t]}}];
sol = NDSolve[Join[eq, {x[0] == 1, x'[0] == 0, y[0] == 0, y'[0] == 1}], {x[t], 
   y[t]}, {t, 0, 5}];
ParametricPlot[{x[t], y[t]} /. sol[[1]], {t, 0, 5}]

enter image description here

2

it’s easier to trouble shoot code if you look at it one line at a time instead of all at once. also, it’s easier for other people to read the code if you use the first icon to paste in your code.

3

your orbit is circular because your initial velocity is perpendicular to the line from the origin to the starting position.