# Mathematica Problem for ODE's

Posted 10 years ago
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 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 -
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Posted 10 years ago
 Is your equation elliptic?  Or better, is it an evolution equation?There is a huge difference in numerically solving an evolution equation and a nonevolution equation. NDSolve solves evolution equations. The first step is to create the simplest possible version of the kind of equation you are trying to solve and  then to try solving that.
Posted 10 years ago
 Here's an example from the documentations = NDSolve[{y''[x] + Sin[y[x]] y[x] == 0, y[0] == 1, y'[0] == 0},   y, {x, 0, 30}]http://reference.wolfram.com/mathematica/ref/NDSolve.html