Hi guys, Aaron the short here and I have a long problem with ODE’s,
I formed 3 equations dealing with 3 generalized coordinates for a lagrangian and the code is as below.
{A, L, p, mu, MM, MP, rm, rp, Eo, n} = {62.83*10^-6, 10000, 1140,
3.9877848*10^14, 5000, 1000, 0.5, 0.5, 5*10^9, 5};
rt = Sqrt[(A/\[Pi])];
T1 = 0;
R[t] = 6728000;
R'[t] = 0;
R''[t] = 0;
\[Theta]'[t] = Sqrt[(mu/R[t]^3)];
\[Theta][t] = \[Theta]'[t]*t;
\[Theta]''[t] = 0;
eqnq2 = (\[CurlyPhi]^\[Prime]\[Prime])[
t] ((2 A L^2 p)/\[Pi] + A L p q1[t]) + (3 A Eo \[Pi]^4 q2[t])/(
16 L^3) - (3 A Eo \[Pi]^4 q1[t]^2 q2[t])/(4 L^3) +
A L p (q2^\[Prime]\[Prime])[t] + (\[Pi]^2 q2[t] T1)/L + (
3 \[Pi]^4 q1[t]^2 q2[t] T1)/(4 L^3) + (3 \[Pi]^4 q2[t]^3 T1)/(
8 L^3) -
2 A L p Derivative[1][\[CurlyPhi]][t] q2[t] Derivative[
1][\[Theta]][t] +
2 A L p Derivative[1][q1][t] (
Derivative[1][\[CurlyPhi]][t] + Derivative[1][\[Theta]][t]) -
A L p q2[t] (
Derivative[1][\[CurlyPhi]][t]^2 +
Derivative[1][\[Theta]][t]^2) + ((2 A L^2 p)/\[Pi] +
A L p q1[t]) (\[Theta]^\[Prime]\[Prime])[t];
eqnq1 = (A Eo \[Pi]^2 q1[t])/L + A L p (q1^\[Prime]\[Prime])[t] -
A L p (\[CurlyPhi]^\[Prime]\[Prime])[t] q2[t] - (
3 A Eo \[Pi]^4 q1[t] q2[t]^2)/(4 L^3) + (15 \[Pi]^4 q1[t]^3 T1)/(
8 L^3) + (3 \[Pi]^4 q1[t] q2[t]^2 T1)/(4 L^3) -
Derivative[1][\[CurlyPhi]][
t] ((4 A L^2 p)/\[Pi] + 2 A L p q1[t]) Derivative[1][\[Theta]][
t] - 2 A L p Derivative[1][q2][t] (
Derivative[1][\[CurlyPhi]][t] + Derivative[1][\[Theta]][t]) - ((
2 A L^2 p)/\[Pi] + A L p q1[t]) (
Derivative[1][\[CurlyPhi]][t]^2 + Derivative[1][\[Theta]][t]^2) -
A L p q2[t] (\[Theta]^\[Prime]\[Prime])[t];
eqnpsi = 2 L^2 MP (\[CurlyPhi]^\[Prime]\[Prime])[t] +
5/6 A L^3 p (\[CurlyPhi]^\[Prime]\[Prime])[t] + (
4 A L^2 p (\[CurlyPhi]^\[Prime]\[Prime])[t] q1[t])/\[Pi] +
A L p (\[CurlyPhi]^\[Prime]\[Prime])[t] q1[t]^2 + (
4 A L^2 p Derivative[1][\[CurlyPhi]][t] Derivative[1][q1][
t])/\[Pi] +
2 A L p Derivative[1][\[CurlyPhi]][t] q1[t] Derivative[1][q1][t] -
A L p (q1^\[Prime]\[Prime])[t] q2[t] +
A L p (\[CurlyPhi]^\[Prime]\[Prime])[t] q2[t]^2 +
2 A L p Derivative[1][\[CurlyPhi]][t] q2[t] Derivative[1][q2][
t] + (2 A L^2 p (q2^\[Prime]\[Prime])[t])/\[Pi] +
A L p q1[t] (q2^\[Prime]\[Prime])[t] +
1/2 MM (\[CurlyPhi]^\[Prime]\[Prime])[t] rm^2 +
MP (\[CurlyPhi]^\[Prime]\[Prime])[t] rp^2 +
1/2 A L p (\[CurlyPhi]^\[Prime]\[Prime])[t] rt^2 + (
4 A L^2 p Derivative[1][q1][t] Derivative[1][\[Theta]][
t])/\[Pi] +
2 A L p q1[t] Derivative[1][q1][t] Derivative[1][\[Theta]][t] +
2 A L p q2[t] Derivative[1][q2][t] Derivative[1][\[Theta]][t] +
2 L^2 MP (\[Theta]^\[Prime]\[Prime])[t] +
5/6 A L^3 p (\[Theta]^\[Prime]\[Prime])[t] + (
4 A L^2 p q1[t] (\[Theta]^\[Prime]\[Prime])[t])/\[Pi] +
A L p q1[t]^2 (\[Theta]^\[Prime]\[Prime])[t] +
A L p q2[t]^2 (\[Theta]^\[Prime]\[Prime])[t] +
1/2 MM rm^2 (\[Theta]^\[Prime]\[Prime])[t] +
MP rp^2 (\[Theta]^\[Prime]\[Prime])[t] +
1/2 A L p rt^2 (\[Theta]^\[Prime]\[Prime])[t] + (
L M2 mu R[t] Sin[\[CurlyPhi][t]])/(L^2 + R[t]^2 -
2 L R[t] Cos[\[CurlyPhi][t]])^(3/2) - (
L M1 mu R[t] Sin[\[CurlyPhi][t]])/(L^2 + R[t]^2 +
2 L R[t] Cos[\[CurlyPhi][t]])^(3/2) + \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]
\*FractionBox[\(A\ \((\(-2\) + 4\ i)\)\
\*SuperscriptBox[\(L\), \(2\)]\ mu\ p\ R[
t]\ Sin[\[CurlyPhi][t]]\), \(4\
\*SuperscriptBox[\(n\), \(2\)]\
\*SuperscriptBox[\((
\*FractionBox[\(
\*SuperscriptBox[\((\(-1\) + 2\ i)\), \(2\)]\
\*SuperscriptBox[\(L\), \(2\)]\), \(4\
\*SuperscriptBox[\(n\), \(2\)]\)] + \
\*SuperscriptBox[\(R[t]\), \(2\)] -
\*FractionBox[\(\((\(-2\) + 4\ i)\)\ L\ R[
t]\ Cos[\[CurlyPhi][t]]\), \(2\ n\)])\), \(3/
2\)]\)]\) - \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]
\*FractionBox[\(A\ \((\(-2\) + 4\ i)\)\
\*SuperscriptBox[\(L\), \(2\)]\ mu\ p\ R[
t]\ Sin[\[CurlyPhi][t]]\), \(4\
\*SuperscriptBox[\(n\), \(2\)]\
\*SuperscriptBox[\((
\*FractionBox[\(
\*SuperscriptBox[\((\(-1\) + 2\ i)\), \(2\)]\
\*SuperscriptBox[\(L\), \(2\)]\), \(4\
\*SuperscriptBox[\(n\), \(2\)]\)] + \
\*SuperscriptBox[\(R[t]\), \(2\)] +
\*FractionBox[\(\((\(-2\) + 4\ i)\)\ L\ R[
t]\ Cos[\[CurlyPhi][t]]\), \(2\ n\)])\), \(3/2\)]\)]\);
system1 =
NDSolve[{eqnq2 == 0, eqnq1 == 0, eqnpsi == 0, q2[0] == 0.01,
Derivative[1][q2][0] == 0.01, q1[0] == 0.01,
Derivative[1][q1][0] ==
0.01, \[CurlyPhi][0] == -0.9, \[CurlyPhi]'[0] == 0.01}, { q2,
q1, \[CurlyPhi]}, {t, 0, 54908.9}, MaxSteps -> Infinity];
Plot[Evaluate[ \[CurlyPhi][t] /. system1], {t, 0, 54908.9},
Frame -> True, LabelStyle -> Directive[12],
FrameTicks -> {{All,
None}, {All, {{0, "0"}, {10981.8, "2"}, {21963.6, "4"}, {32945.4,
"6"}, {43927.1, "8"}, {54908.9, "10"}}}},
FrameLabel -> {{"Angular Displacement (rad)", None}, {"time (s)",
"Number of Orbits"}}]
when I run it , there’s an error " stating try to give initial conditions for both values and derivatives of the functions.
How do I go about doing so ? Thank a lot. Appreciate your effort in even readin this. Cheers.
-Aaron Aw -