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I am plotting geodesics on an ellipsoid. My problem is that I cannot do the manipulation, from a, b, c. From the starting point and the initial vector, the manipulation works well, but from the parameters of the ellipsoid (a, b, c) does not work.
My code (it's working, i manipulate l1, l2, v1, v2):

Clear[a, b, c];
    r[u_, v_] := {a*Cos[u]*Cos[v], b*Cos[u]*Sin[v], c*Sin[u]};
    Subscript[r, 1][u_, v_] := \!\(
    \*SubscriptBox[\(\[PartialD]\), \(u\)]\(r[u, v]\)\);
    Subscript[r, 2][u_, v_] := \!\(
    \*SubscriptBox[\(\[PartialD]\), \(v\)]\(r[u, v]\)\);
    g = Table[
       Subscript[r, i][Subscript[u, 1], Subscript[u, 2]].Subscript[r, j][
         Subscript[u, 1], Subscript[u, 2]], {i, 2}, {j, 2}];
    invg = Inverse[g] // Simplify;
    G[i_, j_, k_] := 1/2 \!\(
    \*UnderoverscriptBox[\(\[Sum]\), \(l = 1\), \(2\)]\(
    \*SubscriptBox[\(invg\), \(\([[\)\(l, k\)\(]]\)\)] \((
    \*SubscriptBox[\(\[PartialD]\), 
    SubscriptBox[\(u\), \(j\)]]
    \*SubscriptBox[\(g\), \(\([[\)\(i, l\)\(]]\)\)]\  + 
    \*SubscriptBox[\(\[PartialD]\), 
    SubscriptBox[\(u\), \(i\)]]
    \*SubscriptBox[\(g\), \(\([[\)\(j, l\)\(]]\)\)]\  - 
    \*SubscriptBox[\(\[PartialD]\), 
    SubscriptBox[\(u\), \(l\)]]
    \*SubscriptBox[\(g\), \(\([[\)\(i, j\)\(]]\)\)])\)\)\) /. 
        Subscript[u, m_] :> Subscript[u, m][t] // Simplify;

    eqns = Table[ Subscript[u, k]'' [t] + \!\(
    \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(2\)]\(
    \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(2\)]G[i, j, k]*\(
    \*SubscriptBox[\(u\), \(j\)]'\)[t]*\(
    \*SubscriptBox[\(u\), \(i\)]'\)[t]\)\) == 0, {k, 2}];
    Manipulate[
     geod = NDSolve[
       Join[eqns, {Subscript[u, 1][0] == Subscript[l, 1], 
         Subscript[u, 2][0] == Subscript[l, 2], 
         Subscript[u, 1]'[0] == Subscript[v, 1], 
         Subscript[u, 2]'[0] == Subscript[v, 2]}], 
       {Subscript[u, 1][t], Subscript[u, 2][t]}, {t, 0, T}];
     a = 1;
     b = 2;
     c = 1;
     Show[Graphics3D[{Purple, Ellipsoid[{0, 0, 0}, {a, b, c}], Green, 
        PointSize[Large], Point[r[Subscript[l, 1], Subscript[l, 2]]]}],
       ParametricPlot3D[
       r[Evaluate[Subscript[u, 1][t] /. geod], 
        Evaluate[Subscript[u, 2][t] /. geod]], {t, 0, T},
        PlotStyle -> Pink], 
      ImageSize -> 500, 
      Boxed -> False], {{Subscript[l, 1], 1}, -(\[Pi]/2), \[Pi]/
      2},    {{Subscript[l, 2], 1}, -\[Pi], \[Pi]}, {{T, 1}, 3, 
      100}, {{Subscript[v, 1], 1}, -3, 3}, {{Subscript[v, 2], 1}, -3, 3}]

I need to manipulate l1, l2, v1, v2, a, b, c. With l1, l2, v1, v2 it's ok. But some problems if i manipulate a, b, c:

Clear[a, b, c];
        r[u_, v_] := {a*Cos[u]*Cos[v], b*Cos[u]*Sin[v], c*Sin[u]};
        Subscript[r, 1][u_, v_] := \!\(
        \*SubscriptBox[\(\[PartialD]\), \(u\)]\(r[u, v]\)\);
        Subscript[r, 2][u_, v_] := \!\(
        \*SubscriptBox[\(\[PartialD]\), \(v\)]\(r[u, v]\)\);
        g = Table[
           Subscript[r, i][Subscript[u, 1], Subscript[u, 2]].Subscript[r, j][
             Subscript[u, 1], Subscript[u, 2]], {i, 2}, {j, 2}];
        invg = Inverse[g] // Simplify;
        G[i_, j_, k_] := 1/2 \!\(
        \*UnderoverscriptBox[\(\[Sum]\), \(l = 1\), \(2\)]\(
        \*SubscriptBox[\(invg\), \(\([[\)\(l, k\)\(]]\)\)] \((
        \*SubscriptBox[\(\[PartialD]\), 
        SubscriptBox[\(u\), \(j\)]]
        \*SubscriptBox[\(g\), \(\([[\)\(i, l\)\(]]\)\)]\  + 
        \*SubscriptBox[\(\[PartialD]\), 
        SubscriptBox[\(u\), \(i\)]]
        \*SubscriptBox[\(g\), \(\([[\)\(j, l\)\(]]\)\)]\  - 
        \*SubscriptBox[\(\[PartialD]\), 
        SubscriptBox[\(u\), \(l\)]]
        \*SubscriptBox[\(g\), \(\([[\)\(i, j\)\(]]\)\)])\)\)\) /. 
            Subscript[u, m_] :> Subscript[u, m][t] // Simplify;

        eqns = Table[ Subscript[u, k]'' [t] + \!\(
        \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(2\)]\(
        \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(2\)]G[i, j, k]*\(
        \*SubscriptBox[\(u\), \(j\)]'\)[t]*\(
        \*SubscriptBox[\(u\), \(i\)]'\)[t]\)\) == 0, {k, 2}];
        Manipulate[
         geod = NDSolve[
           Join[eqns, {Subscript[u, 1][0] == Subscript[l, 1], 
             Subscript[u, 2][0] == Subscript[l, 2], 
             Subscript[u, 1]'[0] == Subscript[v, 1], 
             Subscript[u, 2]'[0] == Subscript[v, 2]}], 
           {Subscript[u, 1][t], Subscript[u, 2][t]}, {t, 0, T}];

         Show[Graphics3D[{Purple, Ellipsoid[{0, 0, 0}, {a, b, c}], Green, 
            PointSize[Large], Point[r[Subscript[l, 1], Subscript[l, 2]]]}],
           ParametricPlot3D[
           r[Evaluate[Subscript[u, 1][t] /. geod], 
            Evaluate[Subscript[u, 2][t] /. geod]], {t, 0, T},
            PlotStyle -> Pink], 
          ImageSize -> 500, 
          Boxed -> False], {{Subscript[l, 1], 1}, -(\[Pi]/2), \[Pi]/
          2},    {{Subscript[l, 2], 1}, -\[Pi], \[Pi]}, {{T, 1}, 3, 
          100}, {{Subscript[v, 1], 1}, -3, 3}, {{Subscript[v, 2], 1}, -3, 3}, [a, 1, 5], [b, 1, 5], [c, 1, 5]]

Now it isn't working now. Can somebody help me?

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  • $\begingroup$ It seems to me you have a problem before you get to the 'Manipulate' code. I can't get your function 'G[...]' to evaluate. I have not had time to investigate why. $\endgroup$ – Jagra Jun 13 at 1:05
  • $\begingroup$ I put it wrong, I need to manipulate l1, l2, v1, v2, a, b, c. With l1, l2, v1, v2 it's ok. $\endgroup$ – Сalendula Jun 13 at 9:58
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Just an extended comment and question...w/additional analysis.

Do you want to adjust the size and shape of the presented ellipsoid by varying the values for a, b, & c? (You seem to have confirmed this).

Your code sets those values as constants at: 1, 2, & 1 respectively.

...
 a = 1;
 b = 2;
 c = 1;
...

If you do want to manipulate the size and shape of the ellipsoid, you can do so by adding three controls to the Manipulate.

A toy version follows:

Manipulate[    
 Graphics3D[
  {Purple, Ellipsoid[{0, 0, 0}, {a, b, c}],
   Green, PointSize[Large]}],

 {{a, 1}, 1, 2},
 {{b, 2}, 1, 4},
 {{c, 1}, 1, 2}  
 ]

Addition to the original response follows:

To better identify problems in your code, I suggest that you modify it by placing all the working code inside of a Module, itself inside the Manipulate.

This does a couple of things, which can help you debug the code.

You have a number of additional problems. Let's look at some simple ones first:

In the update to your question, you've inserted code to add controls for a, b, & c at the end of the Manipulate:

[a, 1, 5], [b, 1, 5], [c, 1, 5]]

You'll need to put this in the proper control format:

{{a, 1}, 1, 5, 0.1},
{{b, 2}, 1, 5, 0.1},
{{c, 1}, 1, 5, 0.1}  

Note, I've added a step, as Mathematica didn't seem to like running a simpler version of these controls. Maybe just me.

Next, placing all of your code inside the suggested Module, shows that the code has a number of undefined symbols. If you follow this suggestion, you will readily see the following undefined symbols showing in blue:

  • u in your function g[...];
  • u & t in your function G[...];
  • u & t in the assignment to your variable eqns (inside Table);
  • u, l, & v in your assignment to geod;
  • l in Graphics3D; and
  • u in ParametricPlot3d.

Note, in my Notebooks your symbol l looks very very close to 1, but if your run:

Head[l] it returns... Symbol


As you can likely gather from above, as a personal practice, I generally prefer to localize everything I can within a Module in a Manipulate. Doing so can help simplify debugging the code.

Doing so enables you to declare your symbols (variables & constants) within the list following the first bracket of the Manipulate.

Gathering them all thus:

Module[{r, g, invg, G, eqns, geod, u, t, l, v }

reveals more problems,

l & v still show up in blue in the code for the "controls" of the Manipulate showing again they have no values assigned to them.

This happens because in this part of the code, l & v, fall outside of the Module.

That's on me, I introduced Module, but it does reveal important information about the structure of the code.

I suggest that you try assigning values to l & v outside of the Manipulate.


In your definition of the function, G[i_, j_, k_], your code uses the shorthand for Part in four places:

[[l, k]]
[[i, l]]
[[j, l]]
[[i, j]]

AND does so as Subscripts.

I really don't understand the usage, perhaps someone else can offer an explanation.

In my Notebook the doubled [[ & ]] Part brackets show up in pink or red. In my experience this typically indicates that something has gone awry in the code.

Your function, G[i_, j_, k_]:= ..., looks to me like the center of your issue.

If you know a specific value(s) that you want G[i_, j_, k_] to return, I suggest you hard code it in a version of the function. This will enable you to run a test of the rest of your code.

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  • $\begingroup$ I edited my question. Your answer isn't exactly what i meant. $\endgroup$ – Сalendula Jun 13 at 10:03

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