*** nonlinear system of ODE + Runge Kutta ?!!!! ***

• To: mathgroup at smc.vnet.net
• Subject: [mg26658] *** nonlinear system of ODE + Runge Kutta ?!!!! ***
• Date: Wed, 17 Jan 2001 00:47:14 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,
I have some problems with my nonlinear system.
It was everything OK til Runge-Kutta, I don´t know how to solve it?
With NDSolve it wasn't problems, but how to solve it with Runge
Kutta??
Thanks a lot.

That´s the NB.

\!\(y\_tt = \ \[PartialD]\_\(t, t\)\((A[t] x[t])\)\)
\!\(\*
RowBox[{
RowBox[{"2", " ",
RowBox[{
SuperscriptBox["A", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}], " ",
RowBox[{
SuperscriptBox["x", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}], "+",
RowBox[{\(x[t]\), " ",
RowBox[{
SuperscriptBox["A", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}]}], "+",
RowBox[{\(A[t]\), " ",
RowBox[{
SuperscriptBox["x", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}]}]}]\)
\!\(y\_tt\  = %\  /. \ \ {\(\(A'\)'\)[t]\  \[Rule] \ \(-A[t]\)}\)
\!\(\*
RowBox[{\(\(-A[t]\)\ x[t]\), "+",
RowBox[{"2", " ",
RowBox[{
SuperscriptBox["A", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}], " ",
RowBox[{
SuperscriptBox["x", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}], "+",
RowBox[{\(A[t]\), " ",
RowBox[{
SuperscriptBox["x", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}]}]}]\)
\!\(\(gl1\  = \n\
\(\(y'\)'\)[t]\  == \
\(-\(\(\[Mu]\ \((y[t] - \ b\_1)\)\)\/\((D\_2)\)\^3\)\) -
\(\[Mu]'\ \((y[t] - \ b\_2)\)\)\/\((D\_1)\)\^3; \)\)
\!\(gl2\ \  =
\ %\  /. \ {y[t] \[Rule] A[t]\ x[t], \ \(\(y'\)'\)[t] \[Rule]
y\_tt, \n
\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \
b\_1 \[Rule] A[t]\ a\_1, \ \ b\_2 \[Rule] A[t] a\_2\ }\)
\!\(\*
RowBox[{
RowBox[{\(\(-A[t]\)\ x[t]\), "+",
RowBox[{"2", " ",
RowBox[{
SuperscriptBox["A", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}], " ",
RowBox[{
SuperscriptBox["x", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}], "+",
RowBox[{\(A[t]\), " ",
RowBox[{
SuperscriptBox["x", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}]}]}], "==",
RowBox[{
\(-\(\(\[Mu]\ \((\(-A[t]\)\ a\_1 + A[t]\ x[t])\)\)\/D\_2\%3\)\),
"-",
FractionBox[
RowBox[{\((\(-A[t]\)\ a\_2 + A[t]\ x[t])\), " ",
SuperscriptBox["\[Mu]", "\[Prime]",
MultilineFunction->None]}], \(D\_1\%3\)]}]}]\)
Map[Simplify,Map[ ((1/A[t])#)&, gl2]] /.
{A[t] \[Rule] 1/Inverse[A[t]]}
\!\(\*
RowBox[{
RowBox[{\(-x[t]\), "+",
RowBox[{"2", " ", \(Inverse[A[t]]\), " ",
RowBox[{
SuperscriptBox["A", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}], " ",
RowBox[{
SuperscriptBox["x", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}], "+",
RowBox[{
SuperscriptBox["x", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}]}], "==",
RowBox[{
\(\(\[Mu]\ a\_1\)\/D\_2\%3\), "-", \(\(\[Mu]\ x[t]\)\/D\_2\%3\),
"+",
FractionBox[
RowBox[{\((a\_2 - x[t])\), " ",
SuperscriptBox["\[Mu]", "\[Prime]",
MultilineFunction->None]}], \(D\_1\%3\)]}]}]\)
\!\(\(\ Map[\((\ Collect[#, {D\_2, D\_1}])\)&, %]\)\)
\!\(\*
RowBox[{
RowBox[{\(-x[t]\), "+",
RowBox[{"2", " ", \(Inverse[A[t]]\), " ",
RowBox[{
SuperscriptBox["A", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}], " ",
RowBox[{
SuperscriptBox["x", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}], "+",
RowBox[{
SuperscriptBox["x", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}]}], "==",
RowBox[{\(\(\[Mu]\ a\_1 - \[Mu]\ x[t]\)\/D\_2\%3\), "+",
FractionBox[
RowBox[{\((a\_2 - x[t])\), " ",
SuperscriptBox["\[Mu]", "\[Prime]",
MultilineFunction->None]}], \(D\_1\%3\)]}]}]\)
gl3 = Map[( Collect[#,{\[Mu],\[Mu]'}])&,%,{3}]
\!\(\*
RowBox[{
RowBox[{\(-x[t]\), "+",
RowBox[{"2", " ", \(Inverse[A[t]]\), " ",
RowBox[{
SuperscriptBox["A", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}], " ",
RowBox[{
SuperscriptBox["x", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}], "+",
RowBox[{
SuperscriptBox["x", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}]}], "==",
RowBox[{\(\(\[Mu]\ \((a\_1 - x[t])\)\)\/D\_2\%3\), "+",
FractionBox[
RowBox[{\((a\_2 - x[t])\), " ",
SuperscriptBox["\[Mu]", "\[Prime]",
MultilineFunction->None]}], \(D\_1\%3\)]}]}]\)
\!\(\*
RowBox[{
RowBox[{\(A[t_]\), " ", "=", "  ",
RowBox[{"(", GridBox[{
{\(Cos[t]\), \(-Sin[t]\)},
{\(Sin[t]\), \(Cos[t]\)}
}], ")"}]}], ";"}]\)
\!\(Simplify[Inverse[A[t]].\[PartialD]\_t\ A[t]]\)
{{0,-1},{1,0}}
\!\(\*
RowBox[{
RowBox[{\(x[t_]\), "=", " ",
RowBox[{"(", "\[NegativeThinSpace]", GridBox[{
{\(x\_1[t]\)},
{\(x\_2[t]\)}
}], "\[NegativeThinSpace]", ")"}]}], ";", "   ",
RowBox[{\(a\_1\), " ", "=", " ",
RowBox[{"(", "\[NegativeThinSpace]", GridBox[{
{\(\[Mu]'\)},
{"0"}
}], "\[NegativeThinSpace]", ")"}]}], ";", "  ",
RowBox[{\(a\_2\), " ", "=", " ",
RowBox[{"(", "\[NegativeThinSpace]", GridBox[{
{\(-\[Mu]\)},
{"0"}
}], "\[NegativeThinSpace]", ")"}]}], ";"}]\)
\!\(\*
RowBox[{"gl4", " ", "=", " ", "\n", "\t",
RowBox[{
RowBox[{\(-x[t]\), " ", "+",
RowBox[{"2", " ",
RowBox[{
RowBox[{"(", GridBox[{
{"0", \(-1\)},
{"1", "0"}
}], ")"}], ".", \(\(x'\)[t]\)}]}], " ", "+", " ",
\(\(\(x'\)'\)[t]\)}], " ", "==", " ", \(Last[gl3]\)}]}]\)
\!\(\*
RowBox[{
RowBox[{"{",
RowBox[{
RowBox[{"{",
RowBox[{\(-x\_1[t]\), "-",
RowBox[{"2", " ",
RowBox[{
SubsuperscriptBox["x", "2", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}], "+",
RowBox[{
SubsuperscriptBox["x", "1", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}]}], "}"}],
",",
RowBox[{"{",
RowBox[{\(-x\_2[t]\), "+",
RowBox[{"2", " ",
RowBox[{
SubsuperscriptBox["x", "1", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}], "+",
RowBox[{
SubsuperscriptBox["x", "2", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}]}], "}"}]}],
"}"}],
"==",
RowBox[{"{",
RowBox[{
RowBox[{"{",
RowBox[{
FractionBox[
RowBox[{
SuperscriptBox["\[Mu]", "\[Prime]",
MultilineFunction->None], " ", \((\(-\[Mu]\) -
x\_1[t])\)}],
\(D\_1\%3\)], "+",
FractionBox[
RowBox[{"\[Mu]", " ",
RowBox[{"(",
RowBox[{
SuperscriptBox["\[Mu]", "\[Prime]",
MultilineFunction->None], "-", \(x\_1[t]\)}],
")"}]}],
\(D\_2\%3\)]}], "}"}], ",",
RowBox[{"{",
RowBox[{\(-\(\(\[Mu]\ x\_2[t]\)\/D\_2\%3\)\), "-",
FractionBox[
RowBox[{
SuperscriptBox["\[Mu]", "\[Prime]",
MultilineFunction->None], " ", \(x\_2[t]\)}],
\(D\_1\%3\)]}], "}"}]}], "}"}]}]\)
\!\(\(\ ode\  = \n\t\
Table[\(\(x\_i'\)'\)[t]\  == \ \
First[\((
\(\(x\_i'\)'\)[t]\  /.
Solve[\ gl4[\([1, i, 1]\)] == \ gl4[\([2, i, 1]\)], \n
\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\(\(x\_i'\)'\)[t]\ ])\)], \ {i, 1, 2}]\)\)
\!\(\*
RowBox[{"{",
RowBox[{
RowBox[{
RowBox[{
SubsuperscriptBox["x", "1", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}], "==",
RowBox[{
FractionBox[
RowBox[{
SuperscriptBox["\[Mu]", "\[Prime]",
MultilineFunction->None], " ", \((\(-\[Mu]\) -
x\_1[t])\)}],
\(D\_1\%3\)], "+",
FractionBox[
RowBox[{"\[Mu]", " ",
RowBox[{"(",
RowBox[{
SuperscriptBox["\[Mu]", "\[Prime]",
MultilineFunction->None], "-", \(x\_1[t]\)}],
")"}]}],
\(D\_2\%3\)], "+", \(x\_1[t]\), "+",
RowBox[{"2", " ",
RowBox[{
SubsuperscriptBox["x", "2", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}]}]}], ",",
RowBox[{
RowBox[{
SubsuperscriptBox["x", "2", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}], "==",
RowBox[{\(x\_2[t]\), "-", \(\(\[Mu]\ x\_2[t]\)\/D\_2\%3\),
"-",
FractionBox[
RowBox[{
SuperscriptBox["\[Mu]", "\[Prime]",
MultilineFunction->None], " ", \(x\_2[t]\)}],
\(D\_1\%3\)],
"-",
RowBox[{"2", " ",
RowBox[{
SubsuperscriptBox["x", "1", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}]}]}]}],
"}"}]\)
\!\(Clear[x, y]\n
x\_1[t_]\  = x[t]; \ \nx\_2[t_]\  = y[t]; \)
ode
\!\(\*
RowBox[{"{",
RowBox[{
RowBox[{
RowBox[{
SuperscriptBox["x", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}], "==",
RowBox[{\(x[t]\), "+",
FractionBox[
RowBox[{\((\(-\[Mu]\) - x[t])\), " ",
SuperscriptBox["\[Mu]", "\[Prime]",
MultilineFunction->None]}], \(D\_1\%3\)], "+",
FractionBox[
RowBox[{"\[Mu]", " ",
RowBox[{"(",
RowBox[{\(-x[t]\), "+",
SuperscriptBox["\[Mu]", "\[Prime]",
MultilineFunction->None]}], ")"}]}], \(D\_2\%3\)],
"+",
RowBox[{"2", " ",
RowBox[{
SuperscriptBox["y", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}]}]}], ",",
RowBox[{
RowBox[{
SuperscriptBox["y", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}], "==",
RowBox[{\(y[t]\), "-", \(\(\[Mu]\ y[t]\)\/D\_2\%3\), "-",
FractionBox[
RowBox[{\(y[t]\), " ",
SuperscriptBox["\[Mu]", "\[Prime]",
MultilineFunction->None]}], \(D\_1\%3\)], "-",
RowBox[{"2", " ",
RowBox[{
SuperscriptBox["x", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}]}]}]}],
"}"}]\)
\!\(\(D\_1 = \@\(\((x[t] + \[Mu])\)\^2\  + \ y[t]\^2\)\ ; \ \
(*\ Entfernung\ zur\ Erde\ *) \n
D\_2 = \@\(\((x[t] - \[Mu]')\)\^2\  + \ y[t]\^2\); \ \
(*\ Entfernung\ zum\ Mond\ *) \)\)
\[Mu]=0.012277471;  (* Masse vom Mond *)
\[Mu]'=1-\[Mu];  (* Masse der Erde *)
ode
\!\(\*
RowBox[{"{",
RowBox[{
RowBox[{
RowBox[{
SuperscriptBox["x", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}], "==",
RowBox[{\(x[t]\), "+",
FractionBox[
RowBox[{
StyleBox["0.0122774709999999998`",
StyleBoxAutoDelete->True,
PrintPrecision->8], " ",
\((\(0.987722529000000015`\[InvisibleSpace]\) -
x[t])\)}],
\(\((\((\(-0.987722529000000015`\) + x[t])\)\^2 +
y[t]\^2)\)\^\(3/
2\)\)], "+",

FractionBox[
\(0.987722529000000015`\
\((\(-0.0122774709999999998`\) - x[t])\)\),
SuperscriptBox[
RowBox[{"(",
RowBox[{
SuperscriptBox[
RowBox[{"(",
RowBox[{
RowBox[{
StyleBox["0.0122774709999999998`",
StyleBoxAutoDelete->True,
PrintPrecision->8], "\[InvisibleSpace]"}],
"+",
\(x[t]\)}], ")"}], "2"], "+", \(y[t]\^2\)}],
")"}],
\(3/2\)]], "+",
RowBox[{"2", " ",
RowBox[{
SuperscriptBox["y", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}]}]}], ",",
RowBox[{
RowBox[{
SuperscriptBox["y", "\[DoublePrime]",
MultilineFunction->None], "[", "t", "]"}], "==",
RowBox[{
\(y[t]\), "-",
\(\(0.0122774709999999998`\
y[t]\)\/\((
\((\(-0.987722529000000015`\) + x[t])\)\^2 +
y[t]\^2)
\)\^\(3/2\)\), "-",
FractionBox[\(0.987722529000000015`\ y[t]\),
SuperscriptBox[
RowBox[{"(",
RowBox[{
SuperscriptBox[
RowBox[{"(",
RowBox[{
RowBox[{
StyleBox["0.0122774709999999998`",
StyleBoxAutoDelete->True,
PrintPrecision->8], "\[InvisibleSpace]"}],
"+",
\(x[t]\)}], ")"}], "2"], "+", \(y[t]\^2\)}],
")"}],
\(3/2\)]], "-",
RowBox[{"2", " ",
RowBox[{
SuperscriptBox["x", "\[Prime]",
MultilineFunction->None], "[", "t", "]"}]}]}]}]}],
"}"}]\)
init2 = {x[0] ==0.994, y[0] == 0,
x'[0] == 0,
y'[0]==-2.00158510637908252240537862224};
t2 =17.0652165601579625588917206249;
ode2 = ode\[Union]init2;
soln2=NDSolve[ ode2, {x,y}, {t,0,t2}]
{{x\[Rule]InterpolatingFunction[{{0.,17.0652}},"<>"],
y\[Rule]InterpolatingFunction[{{0.,17.0652}},"<>"]}}

```

• Prev by Date: Re: Partial Differential Equation
• Next by Date: Shortcut for magnitude of a Vector?
• Previous by thread: Re: Infinite loop while generating gcc.1
• Next by thread: Shortcut for magnitude of a Vector?