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Mathematica Differential Equation

aarouroni — Sun, 05 Jan 2014 01:21:52 +0800

Hi guys...I'm having trouble with my Mathematica code on ordinary differential equations.
What I have programmed here is 4 equations with 8 initial conditions and there's an error , but
when I started off earlier with just 3 equations and 6 initial conditions, there's not a problem at all and a graph
was generated.

Hope u guys can aid me on this, truly appreciate your time and effort in guiding me here. Thanks.

Dude in distress.

Best regards,
Aaron Aw

PS- Just copy the code below entirely beginning from {A,L....right to the bottom and paste it in ur mathematica and it will be alright, no adjustments need to be made :

{A, L, p, mu, MM, MP, RM, RP, RT1, RT2, h} = {62.83*10^-6, 10000, 970,
3.9877848*10^14, 5000, 1000, 0.5, 0.5, 4.47207*10^-3,
4.47207*10^-3, 1};

eqnphi = -((
A (-1 + 2 i) L^2 mu p Sin[\[CurlyPhi][t]] Cos[\[Alpha][t]] R[t])/(
2 N^2 (((-1 + 2 i)^2 L^2)/(
4 N^2) - ((-1 + 2 i) L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[
t])/ N + R[t]^2)^(3/2))) + (
A (-1 + 2 i) L^2 mu p Sin[\[CurlyPhi][t]] Cos[\[Alpha][t]] R[t])/(
2 N^2 (((-1 + 2 i)^2 L^2)/(
4 N^2) + ((-1 + 2 i) L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[
t])/ N + R[t]^2)^(3/2)) - (
L mu Sin[\[CurlyPhi][t]] Cos[\[Alpha][t]] R[t] MP)/(L^2 -
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] + R[t]^2)^(3/2) + (
L mu Sin[\[CurlyPhi][t]] Cos[\[Alpha][t]] R[t] MP)/(L^2 +
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] + R[t]^2)^(3/2) +
0.5 A L^3 p Sin[2 \[Alpha][t]] Derivative[1][\[Alpha]][
t] Derivative[1][\[Theta]][t] +
2 L^2 Sin[2 \[Alpha][t]] MP Derivative[1][\[Alpha]][t] Derivative[
1][\[Theta]][t] +
0.5 A L^3 p Sin[2 \[Alpha][t]] Derivative[1][\[Alpha]][
t] Derivative[1][\[CurlyPhi]][t] +

2 L^2 Sin[2 \[Alpha][t]] MP Derivative[1][\[Alpha]][t] Derivative[
1][\[CurlyPhi]][t] - 2 L^2 (\[Theta]^\[Prime]\[Prime])[t] -
0.25 A L^3 p (\[Theta]^\[Prime]\[Prime])[t] -
0.25 A L^3 p Cos[\[Alpha][t]]^2 (\[Theta]^\[Prime]\[Prime])[t] +
0.25 A L^3 p Sin[\[Alpha][t]]^2 (\[Theta]^\[Prime]\[Prime])[t] -
L^2 MP (\[Theta]^\[Prime]\[Prime])[t] -
L^2 Cos[\[Alpha][t]]^2 MP (\[Theta]^\[Prime]\[Prime])[t] -
L^2 Sin[\[Alpha][t]]^2 MP (\[Theta]^\[Prime]\[Prime])[t] -
0.5 MM RM^2 (\[Theta]^\[Prime]\[Prime])[t] -
MP RP^2 (\[Theta]^\[Prime]\[Prime])[t] -
0.5 L \[Pi] p RT1^4 (\[Theta]^\[Prime]\[Prime])[t] +
0.5 L \[Pi] p RT2^4 (\[Theta]^\[Prime]\[Prime])[t] -
2 L^2 (\[CurlyPhi]^\[Prime]\[Prime])[t] -
0.25 A L^3 p (\[CurlyPhi]^\[Prime]\[Prime])[t] -
0.25 A L^3 p Cos[\[Alpha][t]]^2 (\[CurlyPhi]^\[Prime]\[Prime])[t] +
0.25 A L^3 p Sin[\[Alpha][t]]^2 (\[CurlyPhi]^\[Prime]\[Prime])[t] -
L^2 MP (\[CurlyPhi]^\[Prime]\[Prime])[t] -
L^2 Cos[\[Alpha][t]]^2 MP (\[CurlyPhi]^\[Prime]\[Prime])[t] +
L^2 Sin[\[Alpha][t]]^2 MP (\[CurlyPhi]^\[Prime]\[Prime])[t] -
0.5 MM RM^2 (\[CurlyPhi]^\[Prime]\[Prime])[t] -
MP RP^2 (\[CurlyPhi]^\[Prime]\[Prime])[t] -
0.5 L \[Pi] p RT1^4 (\[CurlyPhi]^\[Prime]\[Prime])[t] +
0.5 L \[Pi] p RT2^4 (\[CurlyPhi]^\[Prime]\[Prime])[t];

eqntheta =
0.5 L^2 Sin[2 \[Alpha][t]] (A L p + 4 MP) Derivative[1][\[Alpha]][
t] (Derivative[1][\[Theta]][t] +
Derivative[1][\[CurlyPhi]][t]) -
2 (MM + 2 (A L p + MP)) R[t] Derivative[1][\[Theta]][t] R'[
t] - (MM + 2 (A L p + MP)) R[t]^2 (\[Theta]^\[Prime]\[Prime])[
t] - 0.25 (2 MM RM^2 + 4 MP (2 L^2 Cos[\[Alpha][t]]^2 + RP^2) +
L (L (8 + A L p + A L p Cos[\[Alpha][t]]) + 2 \[Pi] p RT1^4 -
2 \[Pi] p RT2^4)) ((\[CurlyPhi]^\[Prime]\[Prime])[
t] + (\[Theta]^\[Prime]\[Prime])[t]);

eqnR = 0.5 (L^2 - 2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(
1/2) (L^2 + 2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(
1/2) (2 L^4 -
L^2 (-2 + 2 Cos[\[Alpha][t]] +
Cos[2 (\[Alpha][t] - \[CurlyPhi][t])] +
2 Cos[2 \[CurlyPhi][t]] +
Cos[2 (\[Alpha][t] + \[CurlyPhi][t])]) R[t]^2 +
2 R[t]^4) (R[
t]^2 (-A L p mu ((-(((-1 + 2 i) L^2 Cos[\[CurlyPhi][
t]] Cos[\[Alpha][t]])/ N) + 2 R[t])/(
2 N^2 (((-1 + 2 i)^2 L^2)/(
4 N^2) - ((-1 + 2 i) L Cos[\[Alpha][t]] Cos[\[CurlyPhi][
t]] R[t])/ N + R[t]^2)^(3/2))) -
A L p mu ((((-1 + 2 i) L^2 Cos[\[CurlyPhi][t]] Cos[\[Alpha][
t]])/ N + 2 R[t])/(
2 N^2 (((-1 + 2 i)^2 L^2)/(
4 N^2) + ((-1 + 2 i) L Cos[\[Alpha][t]] Cos[\[CurlyPhi][
t]] R[t])/ N + R[t]^2)^(3/2))) +
2 A L p (R[t] Derivative[1][\[Theta]][t]^2 - R''[t])) -
MM (mu - R[t]^3 Derivative[1][\[Theta]][t]^2 + R[t]^2 R''[t])) -
(MP R[t]^2 (-2 R[
t]^5 (L^2 - 2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(
1/2) (L^2 + 2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(1/2) Derivative[1][\[Theta]][t]^2 +
L^2 R[t] (-0.25 mu (-2 + 2 Cos[2 \[Alpha][t]] +
Cos[2 (\[Alpha][t] - \[CurlyPhi][t])] +
2 Cos[2 \[CurlyPhi][t]] +
Cos[2 (\[Alpha][t] + \[CurlyPhi][t])]) ((L^2 -
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(
1/2) + (L^2 +
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(1/2)) -
2 L^2 (L^2 -
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(
1/2) (L^2 +
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(1/2) Derivative[1][\[Theta]][t]^2) +
R[t]^3 (mu ((L^2 -
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(
1/2) + (L^2 +
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(1/2)) +
L^2 (-2 + 2 Cos[2 \[Alpha][t]] +
Cos[2 (\[Alpha][t] + \[CurlyPhi][t])]) (L^2 -
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(
1/2) (L^2 +
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(1/2) Derivative[1][\[Theta]][t]^2) +
2 R[
t]^4 (L^2 - 2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(
1/2) (L^2 + 2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(1/2) R''[t] +
L^3 (mu Cos[\[Alpha][t]] Cos[\[CurlyPhi][
t]] ((L^2 -
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(
1/2) - (L^2 +
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(1/2)) +
2 L (L^2 - 2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(
1/2) (L^2 +
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(1/2) R''[t]) +
L R[t]^2 (mu Cos[\[Alpha][t]] Cos[\[CurlyPhi][
t]] (-(L^2 -
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(
1/2) + (L^2 +
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(1/2)) -
L (-2 + 2 Cos[2 \[Alpha][t]] +
Cos[2 (\[Alpha][t] - \[CurlyPhi][t])] +
2 Cos[\[CurlyPhi][t]] +
Cos[2 (\[Alpha][t] + \[CurlyPhi][t])]) (L^2 -
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(
1/2) (L^2 +
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(1/2) R''[t])))/(R[
t]^2 ((L^2 - 2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(
3/2) (L^2 + 2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] +
R[t]^2)^(3/2)));

eqnalpha = -((
A (-1 + 2 i) L^2 mu p Cos[\[CurlyPhi][t]] Sin[\[Alpha][t]] R[t])/(
2 N^2 (((-1 + 2 i)^2 L^2)/(
4 N^2) - ((-1 + 2 i) L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[
t])/ N + R[t]^2)^(3/2))) + (
A (-1 + 2 i) L^2 mu p Cos[\[CurlyPhi][t]] Sin[\[Alpha][t]] R[t])/(
2 N^2 (((-1 + 2 i)^2 L^2)/(
4 N^2) + ((-1 + 2 i) L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[
t])/ N + R[t]^2)^(3/2)) - (
L mu Cos[\[CurlyPhi][t]] Sin[\[Alpha][t]] R[t] MP)/(L^2 -
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] + R[t]^2)^(3/2) +
( L mu Cos[\[CurlyPhi][t]] Sin[\[Alpha][t]] R[t] MP)/(L^2 +
2 L Cos[\[Alpha][t]] Cos[\[CurlyPhi][t]] R[t] + R[t]^2)^(3/2) +
2.5 L^2 Sin[2 \[Alpha][t]] Derivative[1][\[Alpha]][t]^2 -
0.25 A L^3 p Sin[2 \[Alpha][t]] Derivative[1][\[Alpha]][t]^2 -
L^2 MP Sin[2 \[Alpha][t]] Derivative[1][\[Alpha]][t]^2 -
0.25 A L^3 p Sin[2 \[Alpha][t]] Derivative[1][\[Theta]][t]^2 -
-L^2 MP Sin[2 \[Alpha][t]] Derivative[1][\[Theta]][t]^2 -
0.5 A L^3 p Sin[2 \[Alpha][t]] Derivative[1][\[Theta]][
t] Derivative[1][\[CurlyPhi]][t] -
2 L^2 Sin[2 \[Alpha][t]] MP Derivative[1][\[Theta]][t] Derivative[
1][\[CurlyPhi]][t] -
0.25 A L^3 p Sin[2 \[Alpha][t]] Derivative[1][\[CurlyPhi]][t]^2 -
L^2 MP Sin[2 \[Alpha][t]] Derivative[1][\[CurlyPhi]][t]^2 -
3 h^2 (\[Alpha]^\[Prime]\[Prime])[t] -
4.5 L^2 (\[Alpha]^\[Prime]\[Prime])[t] -
0.25 A L^3 p (\[Alpha]^\[Prime]\[Prime])[t] -
2.5 Cos[\[Alpha][t]]^2 (\[Alpha]^\[Prime]\[Prime])[t] +
0.25 A L^3 p Cos[\[Alpha][t]]^2 (\[Alpha]^\[Prime]\[Prime])[t] +
2.5 Sin[\[Alpha][t]]^2 (\[Alpha]^\[Prime]\[Prime])[t] -
0.25 A L^3 p Sin[\[Alpha][t]]^2 (\[Alpha]^\[Prime]\[Prime])[t] -
L^2 MP (\[Alpha]^\[Prime]\[Prime])[t] +
L^2 Cos[\[Alpha][t]]^2 MP (\[Alpha]^\[Prime]\[Prime])[t] -
L^2 Sin[\[Alpha][t]]^2 MP (\[Alpha]^\[Prime]\[Prime])[t] -
0.25 MM RM^2 (\[Alpha]^\[Prime]\[Prime])[t] -
0.5 MP RP^2 (\[Alpha]^\[Prime]\[Prime])[t] -
0.5 L \[Pi] p RT1^4 (\[Alpha]^\[Prime]\[Prime])[t] +
0.5 L \[Pi] p RT2^4 (\[Alpha]^\[Prime]\[Prime])[t];

system1 =
NDSolve[{eqnphi == 0, eqntheta == 0, eqnR == 0,
eqnalpha == 0, \[CurlyPhi][0] == -0.9,
Derivative[1][\[CurlyPhi]][0] == 0, \[Theta][0] == 0,
Derivative[1][\[Theta]][0] == 0.00114, R[0] == 6728000,
R'[0] == 0, \[Alpha][0] == -0.01,
Derivative[1][\[Alpha]][0] == 0}, {\[CurlyPhi], \[Theta],
R, \[Alpha]}, {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"}}]

Mathematica Differential Equation

aarouroni — Sun, 05 Jan 2014 01:20:09 +0800

Mathematica Differential Equation

aarouroni — Sun, 05 Jan 2014 01:15:25 +0800

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