3
$\begingroup$
Clear["`."]
rm = 1; al = Pi/6; csal = Cos[al];
DSolve[{rm ph'[s] == Sin[si[s]] Tan[si[s]]^2 csal, 
  rm si'[s] == -Sin[ph[s]] Sin[si[s]]^2}, {ph, si}, s]

How to Reduce in order to obtain the solution in a reasonable time?

EDIT1:

These are differential equations of straight (geodesic) and asymptotic lines on a one sheeted hyperboloid of revolution. Angles (si,ph) are angles made by meridians to the geodesic and symmetry axis respectively. A numerical solution looks like:

1 sht Hyperbld

I expected that the closed form solution to be extremely simple just as the shell shape is. But it turns out so hard to Reduce it !

EDIT2:

Obtained the following relations analytically:

$$ ph[s]=\phi,si[s]=\psi\,;$$

$$ \frac{1}{\sin^2\psi}=\frac{1}{\sin^2\alpha}+\frac{s^2}{rm^2} \,;$$

$$\tan \phi= \tan \alpha \sin \psi\cdot \frac{s}{rm}\,;$$

$\endgroup$
8
  • 1
    $\begingroup$ what is reasonable? I could calculate within 15s $\endgroup$ Mar 27, 2020 at 13:41
  • $\begingroup$ @morbo Could you demonstrate your solution? In version 11.2 the system returned the input unevaluated. $\endgroup$
    – Artes
    Mar 27, 2020 at 14:44
  • $\begingroup$ Mathematica 12.1 gives a solution within 20s. $\endgroup$ Mar 27, 2020 at 16:07
  • $\begingroup$ Yes mma 12 and above gives a huge result in short order, $\endgroup$ Mar 27, 2020 at 16:50
  • $\begingroup$ @morbo: Can you please check my EDIT2 analytical results? I wished to avoid these laborious calculations by employing Mathematica for the same purpose. $\endgroup$
    – Narasimham
    Apr 6, 2020 at 22:49

1 Answer 1

1
$\begingroup$

With "Weierstrass" -substitution si[s]=2 ArcTan[us[s],ph[s]=2 ArcTan[up[s],

you might find simple equation and a fast solution:

ode = {rm ph'[s] == Sin[si[s]] Tan[si[s]]^2 csal,rm si'[s] == -Sin[ph[s]] Sin[
     si[s]]^2} /. {ph -> (2 ArcTan[up[#]] &),si -> (2 ArcTan[us[#]] &)} // TrigExpand // FullSimplify
(*{(2 Derivative[1][up][s])/(1 + up[s]^2) == (4 Sqrt[3] us[s]^3)/((-1 + us[s]^2)^2 (1 + us[s]^2)),
(4 up[s] us[s]^2)/((1 + up[s]^2) (1 + us[s]^2)) +Derivative[1][us][s] == 0}*)

These equations are solved in 1.2s:

DSolve[ode, {up, us}, s] // FullSimplify
(* {us->…, up->... *)  


{{us -> Function[{s}, -((\[Sqrt](-1 + csal - 2 csal C[1] + 
       csal InverseFunction[-(((2 (1 + 2 csal C[1]) #1 (1 + 
                2 csal C[1] (1 + #1^2)))/(1 + 
                csal (-1 + 2 C[1])) + (I (1 + 
                csal (-1 + 4 C[1])) (EllipticE[
                I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(2 C[1] (1 + csal (-1 + 2 C[1])))] -
                 EllipticF[
                I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(
                2 C[1] (1 + csal (-1 + 2 C[1])))]) Sqrt[
                1 + (csal (-1 + 2 C[1]) #1^2)/(
                1 + csal (-1 + 2 C[1]))] Sqrt[(
                2 + 4 csal C[1] (1 + #1^2))/(
                1 + 2 csal C[1])])/((-1 + 2 C[1]) Sqrt[(
                csal C[1])/(1 + 2 csal C[1])]) - 
                2 #1 (1 + 
                csal (-1 + 2 C[1]) (1 + #1^2)))/(2 csal (1 + 
                2 csal C[1]) Sqrt[-1 - 2 csal C[1] (1 + #1^2)]
                 Sqrt[-1 - csal (-1 + 2 C[1]) (1 + #1^2)])) &][-((
           4 s)/(csal rm)) + C[2]]^2 - 
       2 csal C[
        1] InverseFunction[-(((2 (1 + 2 csal C[1]) #1 (1 + 
                2 csal C[1] (1 + #1^2)))/(1 + 
                csal (-1 + 2 C[1])) + (I (1 +         csal (-1 + 4 C[1])) (EllipticE[
                I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(2 C[1] (1 + csal (-1 + 2 C[1])))] -
                 EllipticF[
                I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(
                2 C[1] (1 + csal (-1 + 2 C[1])))]) Sqrt[
                1 + (csal (-1 + 2 C[1]) #1^2)/(
                1 + csal (-1 + 2 C[1]))] Sqrt[(
                2 + 4 csal C[1] (1 + #1^2))/(
                1 + 2 csal C[1])])/((-1 + 2 C[1]) Sqrt[(
                csal C[1])/(1 + 2 csal C[1])]) - 
                2 #1 (1 + 
                csal (-1 + 2 C[1]) (1 + #1^2)))/(2 csal (1 + 
                2 csal C[1]) Sqrt[-1 - 2 csal C[1] (1 + #1^2)]
                 Sqrt[-1 - csal (-1 + 2 C[1]) (1 + #1^2)])) &][-((
           4 s)/(csal rm)) + C[2]]^2))/(\[Sqrt](-1 - 
       2 csal C[1] - 
       2 csal C[
        1] InverseFunction[-(((2 (1 + 2 csal C[1]) #1 (1 + 
                2 csal C[1] (1 + #1^2)))/(1 + 
                csal (-1 + 2 C[1])) + (I (1 + 
                csal (-1 + 4 C[1])) (EllipticE[
                I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(2 C[1] (1 + csal (-1 + 2 C[1])))] -
                 EllipticF[
                I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(
                2 C[1] (1 + csal (-1 + 2 C[1])))]) Sqrt[
                1 + (csal (-1 + 2 C[1]) #1^2)/(
                1 + csal (-1 + 2 C[1]))] Sqrt[(
                2 + 4 csal C[1] (1 + #1^2))/(
                1 + 2 csal C[1])])/((-1 + 2 C[1]) Sqrt[(
                csal C[1])/(1 + 2 csal C[1])]) - 
                2 #1 (1 + 
                csal (-1 + 2 C[1]) (1 + #1^2)))/(2 csal (1 + 
                2 csal C[1]) Sqrt[-1 - 2 csal C[1] (1 + #1^2)]
                 Sqrt[-1 - csal (-1 + 2 C[1]) (1 + #1^2)])) &][-((
           4 s)/(csal rm)) + C[2]]^2)))], 
up -> Function[{s}, 
InverseFunction[-(((
        2 (1 + 2 csal C[1]) #1 (1 + 2 csal C[1] (1 + #1^2)))/(
        1 + csal (-1 + 
            2 C[1])) + (I (1 + 
             csal (-1 + 4 C[1])) (EllipticE[
              I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(2 C[1] (1 + csal (-1 + 2 C[1])))] -
              EllipticF[
              I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(
              2 C[1] (1 + csal (-1 + 2 C[1])))]) Sqrt[
           1 + (csal (-1 + 2 C[1]) #1^2)/(1 + csal (-1 + 2 C[1]))]
            Sqrt[(2 + 4 csal C[1] (1 + #1^2))/(
           1 + 2 csal C[1])])/((-1 + 2 C[1]) Sqrt[(csal C[1])/(
           1 + 2 csal C[1])]) - 
        2 #1 (1 + csal (-1 + 2 C[1]) (1 + #1^2)))/(2 csal (1 + 
          2 csal C[1]) Sqrt[-1 - 2 csal C[1] (1 + #1^2)]
         Sqrt[-1 - csal (-1 + 2 C[1]) (1 + #1^2)])) &][-((4 s)/(
   csal rm)) + C[2]]]}, {us -> 
Function[{s}, (\[Sqrt](-1 + csal - 2 csal C[1] + 
     csal InverseFunction[-(((2 (1 + 2 csal C[1]) #1 (1 + 
                2 csal C[1] (1 + #1^2)))/(1 + 
                csal (-1 + 2 C[1])) + (I (1 + 
                csal (-1 + 4 C[1])) (EllipticE[
                I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(2 C[1] (1 + csal (-1 + 2 C[1])))] -
                 EllipticF[
                I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(
                2 C[1] (1 + csal (-1 + 2 C[1])))]) Sqrt[
                1 + (csal (-1 + 2 C[1]) #1^2)/(
                1 + csal (-1 + 2 C[1]))] Sqrt[(
                2 + 4 csal C[1] (1 + #1^2))/(
                1 + 2 csal C[1])])/((-1 + 2 C[1]) Sqrt[(
                csal C[1])/(1 + 2 csal C[1])]) - 
              2 #1 (1 + 
                csal (-1 + 2 C[1]) (1 + #1^2)))/(2 csal (1 + 
                2 csal C[1]) Sqrt[-1 - 2 csal C[1] (1 + #1^2)]
               Sqrt[-1 - csal (-1 + 2 C[1]) (1 + #1^2)])) &][(
        4 s)/(csal rm) + C[2]]^2 - 
     2 csal C[
      1] InverseFunction[-(((2 (1 + 2 csal C[1]) #1 (1 + 
                2 csal C[1] (1 + #1^2)))/(1 + 
                csal (-1 + 2 C[1])) + (I (1 + 
                csal (-1 + 4 C[1])) (EllipticE[
                I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(2 C[1] (1 + csal (-1 + 2 C[1])))] -
                 EllipticF[
                I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(
                2 C[1] (1 + csal (-1 + 2 C[1])))]) Sqrt[
                1 + (csal (-1 + 2 C[1]) #1^2)/(
                1 + csal (-1 + 2 C[1]))] Sqrt[(
                2 + 4 csal C[1] (1 + #1^2))/(
                1 + 2 csal C[1])])/((-1 + 2 C[1]) Sqrt[(
                csal C[1])/(1 + 2 csal C[1])]) - 

              2 #1 (1 + 
                csal (-1 + 2 C[1]) (1 + #1^2)))/(2 csal (1 + 
                2 csal C[1]) Sqrt[-1 - 2 csal C[1] (1 + #1^2)]
               Sqrt[-1 - csal (-1 + 2 C[1]) (1 + #1^2)])) &][(
        4 s)/(csal rm) + C[2]]^2))/(\[Sqrt](-1 - 2 csal C[1] - 
     2 csal C[
      1] InverseFunction[-(((2 (1 + 2 csal C[1]) #1 (1 + 
                2 csal C[1] (1 + #1^2)))/(1 + 
                csal (-1 + 2 C[1])) + (I (1 + 
                csal (-1 + 4 C[1])) (EllipticE[
                I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(2 C[1] (1 + csal (-1 + 2 C[1])))] -
                 EllipticF[
                I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(
                2 C[1] (1 + csal (-1 + 2 C[1])))]) Sqrt[
                1 + (csal (-1 + 2 C[1]) #1^2)/(
                1 + csal (-1 + 2 C[1]))] Sqrt[(
                2 + 4 csal C[1] (1 + #1^2))/(
                1 + 2 csal C[1])])/((-1 + 2 C[1]) Sqrt[(
                csal C[1])/(1 + 2 csal C[1])]) - 
              2 #1 (1 + 
                csal (-1 + 2 C[1]) (1 + #1^2)))/(2 csal (1 + 
                2 csal C[1]) Sqrt[-1 - 2 csal C[1] (1 + #1^2)]
               Sqrt[-1 - csal (-1 + 2 C[1]) (1 + #1^2)])) &][(
        4 s)/(csal rm) + C[2]]^2))], 
up -> Function[{s}, 
InverseFunction[-(((
        2 (1 + 2 csal C[1]) #1 (1 + 2 csal C[1] (1 + #1^2)))/(
        1 + csal (-1 + 
            2 C[1])) + (I (1 + 
             csal (-1 + 4 C[1])) (EllipticE[
              I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(2 C[1] (1 + csal (-1 + 2 C[1])))] -
              EllipticF[
              I ArcSinh[
                Sqrt[2] Sqrt[(csal C[1])/(
                1 + 2 csal C[1])] #1], ((-1 + 2 C[1]) (1 + 
                2 csal C[1]))/(
              2 C[1] (1 + csal (-1 + 2 C[1])))]) Sqrt[
           1 + (csal (-1 + 2 C[1]) #1^2)/(1 + csal (-1 + 2 C[1]))]
            Sqrt[(2 + 4 csal C[1] (1 + #1^2))/(
           1 + 2 csal C[1])])/((-1 + 2 C[1]) Sqrt[(csal C[1])/(
           1 + 2 csal C[1])]) - 
        2 #1 (1 + csal (-1 + 2 C[1]) (1 + #1^2)))/(2 csal (1 + 
          2 csal C[1]) Sqrt[-1 - 2 csal C[1] (1 + #1^2)]
         Sqrt[-1 - csal (-1 + 2 C[1]) (1 + #1^2)])) &][(4 s)/(
  csal rm) + C[2]]]}}
$\endgroup$
4
  • $\begingroup$ Why don't you provide these solutions? In version 11.2 no solutions were obtained with your substiution. $\endgroup$
    – Artes
    Mar 27, 2020 at 17:42
  • $\begingroup$ @Artes The result is a huge expression. What's the problem with 11.2? $\endgroup$ Mar 27, 2020 at 18:49
  • $\begingroup$ @Ulrich Neumann: How does the result simplify with initial condition $ ph=0, si= \alpha $? $\endgroup$
    – Narasimham
    Apr 5, 2020 at 18:17
  • $\begingroup$ @Narasimham Look at the transformation rules: For example ph]0]==0 implies up[0]=Tan[0/2]=0 $\endgroup$ Apr 5, 2020 at 20:36

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