As a starter
In[14]:= rorb[X, 0]
Out[14]= X^2/(4 \[Pi]^2 (1 - Sqrt[1 + (X^2 (-4 \[Pi]^2 + X^2/2))/(8 \[Pi]^4)]))
from which one sees, that rorb[0,0] seems to be indeterminate (0/0) but the numerator vanishs with second oder, the denominator vanishes with first order, so L’Hospital would mean rorb[0,0]=1
In[15]:= Limit[rorb[X, 0], X -> 0]
Out[15]= 1
but Mma says
In[16]:= rorb[0, 0]
During evaluation of In[16]:= Power::infy: Infinite expression 1/0 encountered. >>
During evaluation of In[16]:= Infinity::indet: Indeterminate expression (0 ComplexInfinity)/(4 \[Pi]^2) encountered. >>
Out[16]= Indeterminate
one could easily fix that by defining rorb[0,0] to the correct limit.In either case this
Manipulate[
ParametricPlot[{rorb[vy0, \[CurlyPhi]] Cos[\[CurlyPhi]],
rorb[vy0, \[CurlyPhi]] Sin[\[CurlyPhi]]}, {\[CurlyPhi], 0,
2 \[Pi]}, PlotStyle -> Black, AspectRatio -> Automatic,
FrameLabel -> {"\!\(\*
StyleBox[\"x\",\nFontFamily->\"cmmi12\",\nFontSize->14,\nFontWeight\
->\"Plain\",\nFontSlant->\"Italic\",\n\
FontVariations->{\"CompatibilityType\"->0}]\)\!\(\*
StyleBox[
StyleBox[
RowBox[{
StyleBox[\" \",\nFontFamily->\"cmmi12\"],
StyleBox[\" \",\nFontFamily->\"Arial Narrow\"]}]],\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontVariations->{\"CompatibilityType\"->0}]\)\
\!\(\*
StyleBox[\"[\",\nFontFamily->\"Arial Narrow\",\nFontSize->12,\n\
FontWeight->\"Plain\",\nFontVariations->{\"CompatibilityType\"->0}]\)\
\!\(\*
StyleBox[\"AU\",\nFontFamily->\"Arial Narrow\",\nFontSize->12,\n\
FontWeight->\"Plain\",\nFontVariations->{\"CompatibilityType\"->0}]\)\
\!\(\*
StyleBox[\"]\",\nFontFamily->\"Arial Narrow\",\nFontSize->12,\n\
FontWeight->\"Plain\",\n\
FontVariations->{\"CompatibilityType\"->0}]\)", "\!\(\*
StyleBox[\"y\",\nFontFamily->\"cmmi12\",\nFontSize->14,\nFontWeight\
->\"Plain\",\nFontSlant->\"Italic\",\n\
FontVariations->{\"CompatibilityType\"->0}]\)\!\(\*
StyleBox[\" \",\nFontFamily->\"cmmi12\",\nFontSize->14,\nFontWeight\
->\"Plain\",\nFontVariations->{\"CompatibilityType\"->0}]\)\!\(\*
StyleBox[\"[\",\nFontFamily->\"Arial Narrow\",\nFontSize->12,\n\
FontWeight->\"Plain\",\nFontVariations->{\"CompatibilityType\"->0}]\)\
\!\(\*
StyleBox[\"AU\",\nFontFamily->\"Arial Narrow\",\nFontSize->12,\n\
FontWeight->\"Plain\",\nFontVariations->{\"CompatibilityType\"->0}]\)\
\!\(\*
StyleBox[\"]\",\nFontFamily->\"Arial Narrow\",\nFontSize->12,\n\
FontWeight->\"Plain\",\n\
FontVariations->{\"CompatibilityType\"->0}]\)"}, PlotRange -> All,
PlotLabel -> "Elipti?na orbita, egzaktno"], {vy0, 0, 20,
Appearance -> "Labeled"}]
starts with an error without taking care about the limit.
Now about the EC thing: that will not work because - as the error message states - the iterator for the Do[…, {i,nt}] has no upper limit nt, nt is a module local variable and is not set inside the Module[]. You have set it before the module, but the Module EC
Euler - Cromerov metod
x0 = 1.; y0 = 0.; vx0 = 0.; vy0 = 2 \[Pi]; t0 = 0.; tf = 10.;
r0 = Sqrt[x0^2 + y0^2]; v02 = vx0^2 + vy0^2; e0 = v02/2 - (4 \[Pi]^2)/r0;
l0 = vy0 x0 - vx0 y0;
nt = 100000; dt = (tf - t0)/nt;
tt = {t0}; xx = {x0}; yy = {y0}; ee = {e0}; ll = {l0};
t = t0; x = x0; y = y0; vx = vx0; vy = vy0;
EC[vy_] := Module[{vx, x, dt, r, y, t, xx, yy, tt, v2, e, ee, ll, l, i, nt},
Do[(<snip> ....];
has all this variables as local entities, see Module, the fix is to initialize that local variables to the values you want.
Then there is another error, defining EC you cannot assign another value to vy! Starting EC[2 Pi] you say vy = 2 Pi and then vy = vy - (4 \^2*y*dt)/r^3 means to assign 2 Pi to become something different from 2 Pi - forbidden with very sound reason. Use another name for the input, e.g. EC then it works and gives this picture
last but not least find please my modifications - which should be checked against further errors - in the appendix
