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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 160762, 4129]*) (*NotebookOutlinePosition[ 165460, 4242]*) (* CellTagsIndexPosition[ 164447, 4218]*) (*WindowFrame->Normal*) Notebook[{ Cell["Differential Operators and Harmonic Oscillators", "Title"], Cell["\<\ Operator Calculus to Solve Heterogeneous Second-Order Linear ODEs\ \ \>", "Subtitle"], Cell[TextData[{ "Does ", StyleBox["Mathematica", FontSlant->"Italic"], " apply the Fourier Transform/Derivative Rule Automagically?" }], "Text", CellTags->"mmtag:22:implementing_rules_for_linear_operators"], Cell[CellGroupData[{ Cell[BoxData[ \(FourierTransform[D[f[x], {x, 1}], x, k]\)], "Input", CellTags->"mmtag:22:FourierTransform[]__of_derivatives"], Cell[BoxData[ \(\(-\[ImaginaryI]\)\ k\ FourierTransform[f[x], x, k]\)], "Output"] }, Open ]], Cell["It does as of version 5.0! .", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[FourierTransform[D[f[x], {x, i}], x, k], {i, 24}] // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", TagBox[GridBox[{ {\(\(-\[ImaginaryI]\)\ k\ FourierTransform[f[x], x, k]\)}, {\(\(-k\^2\)\ FourierTransform[f[x], x, k]\)}, {\(\[ImaginaryI]\ k\^3\ FourierTransform[f[x], x, k]\)}, {\(k\^4\ FourierTransform[f[x], x, k]\)}, {\(\(-\[ImaginaryI]\)\ k\^5\ FourierTransform[f[x], x, k]\)}, {\(\(-k\^6\)\ FourierTransform[f[x], x, k]\)}, {\(\[ImaginaryI]\ k\^7\ FourierTransform[f[x], x, k]\)}, {\(k\^8\ FourierTransform[f[x], x, k]\)}, {\(\(-\[ImaginaryI]\)\ k\^9\ FourierTransform[f[x], x, k]\)}, {\(\(-k\^10\)\ FourierTransform[f[x], x, k]\)}, {\(\[ImaginaryI]\ k\^11\ FourierTransform[f[x], x, k]\)}, {\(k\^12\ FourierTransform[f[x], x, k]\)}, {\(\(-\[ImaginaryI]\)\ k\^13\ FourierTransform[f[x], x, k]\)}, {\(\(-k\^14\)\ FourierTransform[f[x], x, k]\)}, {\(\[ImaginaryI]\ k\^15\ FourierTransform[f[x], x, k]\)}, {\(k\^16\ FourierTransform[f[x], x, k]\)}, {\(\(-\[ImaginaryI]\)\ k\^17\ FourierTransform[f[x], x, k]\)}, {\(\(-k\^18\)\ FourierTransform[f[x], x, k]\)}, {\(\[ImaginaryI]\ k\^19\ FourierTransform[f[x], x, k]\)}, {\(k\^20\ FourierTransform[f[x], x, k]\)}, {\(\(-\[ImaginaryI]\)\ k\^21\ FourierTransform[f[x], x, k]\)}, {\(\(-k\^22\)\ FourierTransform[f[x], x, k]\)}, {\(\[ImaginaryI]\ k\^23\ FourierTransform[f[x], x, k]\)}, {\(k\^24\ FourierTransform[f[x], x, k]\)} }, RowSpacings->1, ColumnAlignments->{Left}], Column], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "Does ", StyleBox["Mathematica", FontSlant->"Italic"], " apply the rules according to the Fourier Transform being a linear \ operator?" }], "Text", CellTags->{ "mmtag:22:FourierTransform[]__implementing_rules_as_linear_operator", "mmtag:22:implementing_rules_for_linear_operators__FourierTransform[]"}], Cell[CellGroupData[{ Cell[BoxData[ \(FourierTransform[a\ f[x]\ + \ b\ g[x], x, k]\)], "Input"], Cell[BoxData[ \(FourierTransform[a\ f[x] + b\ g[x], x, k]\)], "Output"] }, Open ]], Cell["\<\ Apparently not--so we make some rules that can be applied. It may \ be instructive to see how to do this.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\(ConstantRule\ = \ FourierTransform[a_\ \ f_[y_], x_, k_] \[RuleDelayed] \ a\ FourierTransform[f[y], x, k]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(DistributeRule = \ \ \[IndentingNewLine]FourierTransform[Plus[expr_], x_, \ k_]\ \[RuleDelayed] Distribute[FourierTransform[expr, x, k], Plus]\)}], "Input"], Cell[BoxData[ \(FourierTransform[a_\ f_[y_], x_, k_] \[RuleDelayed] a\ FourierTransform[f[y], x, k]\)], "Output"], Cell[BoxData[ \(FourierTransform[expr_, x_, k_] \[RuleDelayed] Distribute[FourierTransform[expr, x, k], Plus]\)], "Output"] }, Open ]], Cell["Try them out:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(temp\ = \(FourierTransform[a\ f[x]\ + \ b\ g[x]\ + \ d\ p[x], x, k] /. DistributeRule\) /. ConstantRule\)], "Input"], Cell[BoxData[ \(a\ FourierTransform[f[x], x, k] + b\ FourierTransform[g[x], x, k] + d\ FourierTransform[p[x], x, k]\)], "Output"] }, Open ]], Cell["\<\ Let's Fourier Transform the left-hand -side of a second-order ODE:\ \ \>", "Text", CellTags->{ "mmtag:22:FourierTransform[]__to_solve_damped_forced_harmonic_oscillator", "mmtag:22:damped_forced_harmonic_operator__solution_using_\ FourierTransform[]"}], Cell[CellGroupData[{ Cell[BoxData[ \(ODE2nd\ = \ Mass\ D[y[t], {t, 2}]\ + \ Viscosity\ D[y[t], t]\ + \ SpringK\ y\ [t]\)], "Input"], Cell[BoxData[ RowBox[{\(SpringK\ y[t]\), "+", RowBox[{"Viscosity", " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", RowBox[{"Mass", " ", RowBox[{ SuperscriptBox["y", "\[Prime]\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FrrODE2nd\ = \(FourierTransform[ODE2nd, t, \[Omega]] /. DistributeRule\) /. ConstantRule\)], "Input"], Cell[BoxData[ \(SpringK\ FourierTransform[y[t], t, \[Omega]] - \[ImaginaryI]\ Viscosity\ \[Omega]\ FourierTransform[ y[t], t, \[Omega]] - Mass\ \[Omega]\^2\ FourierTransform[y[t], t, \[Omega]]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(rhs\ = \ FourierTransform[Cos[\ \[Omega]0\ \ t]\ , t, \[Omega]]\)], "Input", CellTags->"mmtag:22:DiracDelta[]__as_FourierTransform[]_of_Cos[]"], Cell[BoxData[ \(\@\(\[Pi]\/2\)\ DiracDelta[\[Omega] - \[Omega]0] + \@\(\[Pi]\/2\)\ \ DiracDelta[\[Omega] + \[Omega]0]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ftsol\ = \ Solve[FrrODE2nd\ \[Equal] rhs, FourierTransform[y[t], t, \[Omega]]]\)], "Input"], Cell[BoxData[ \({{FourierTransform[y[t], t, \[Omega]] \[Rule] \(\@\(2\ \[Pi]\)\ DiracDelta[\[Omega] - \ \[Omega]0] + \@\(2\ \[Pi]\)\ DiracDelta[\[Omega] + \[Omega]0]\)\/\(2\ \ \((SpringK - \[ImaginaryI]\ Viscosity\ \[Omega] - Mass\ \ \[Omega]\^2)\)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(InverseFourierTransform[ FourierTransform[y[t], t, \[Omega]] /. Flatten[ftsol], \[Omega], t, Assumptions\ \[Rule] \ \[Omega]0 > \ 0]\)], "Input", CellTags-> "mmtag:22:InverseFourierTransform[]__in_solution_to_harmonic_oscillator"], Cell[BoxData[ \(\(\((SpringK - Mass\ \[Omega]0\^2)\)\ Cos[t\ \[Omega]0] + Viscosity\ \ \[Omega]0\ Sin[t\ \[Omega]0]\)\/\(SpringK\^2 - 2\ Mass\ SpringK\ \[Omega]0\^2 \ + Viscosity\^2\ \[Omega]0\^2 + Mass\^2\ \[Omega]0\^4\)\)], "Output"] }, Open ]], Cell["\<\ Otherwise, the complete solution (i.e., the inhomogeneous plus the \ homogeneous parts) can be calculated by the DSolve function\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[ y[t] /. \(DSolve[{Mass\ D[y[t], {t, 2}]\ + \ Damper\ D[y[t], t]\ + \ SpringK\ y[t]\ \[Equal] \ Cos[\[Omega]\_o\ t], \ y[0] == 1, \(y'\)[0] == 0}, y[t], t]\)[\([0]\)], Assumptions -> \ \[Omega]0\ > \ 0\ && \ \ Mass\ > \ 0\ && \ Damper\ > \ 0\ && \ SpringK\ > \ 0]\)], "Input", CellTags->"mmtag:22:damped_forced_harmonic_operator__general_solution"], Cell[BoxData[ \({\((\[ExponentialE]\^\(-\(\(\((Damper + \@\(Damper\^2 - 4\ Mass\ \ SpringK\))\)\ t\)\/\(2\ Mass\)\)\)\ \((SpringK\ \((\((\(-1\) + SpringK)\)\ \((Damper\ \((\(-1\) + \ \[ExponentialE]\^\(\(\@\(Damper\^2 - 4\ Mass\ SpringK\)\ t\)\/Mass\))\) + \ \((1 + \[ExponentialE]\^\(\(\@\(Damper\^2 - 4\ Mass\ SpringK\)\ \ t\)\/Mass\))\)\ \@\(Damper\^2 - 4\ Mass\ SpringK\))\) + 2\ \[ExponentialE]\^\(\(\((Damper + \@\(Damper\^2 - 4\ \ Mass\ SpringK\))\)\ t\)\/\(2\ Mass\)\)\ \@\(Damper\^2 - 4\ Mass\ SpringK\)\ \ Cos[t\ \[Omega]\_o])\) + 2\ Damper\ \[ExponentialE]\^\(\(\((Damper + \@\(Damper\^2 - 4\ \ Mass\ SpringK\))\)\ t\)\/\(2\ Mass\)\)\ \@\(Damper\^2 - 4\ Mass\ SpringK\)\ \ Sin[t\ \[Omega]\_o]\ \[Omega]\_o + \((Damper\^3\ \((\(-1\) + \ \[ExponentialE]\^\(\(\@\(Damper\^2 - 4\ Mass\ SpringK\)\ t\)\/Mass\))\) - Damper\ \((\(-1\) + \[ExponentialE]\^\(\(\@\(Damper\^2 \ - 4\ Mass\ SpringK\)\ t\)\/Mass\))\)\ Mass\ \((1 + 2\ SpringK)\) + Damper\^2\ \((1 + \[ExponentialE]\^\(\(\@\(Damper\^2 - \ 4\ Mass\ SpringK\)\ t\)\/Mass\))\)\ \@\(Damper\^2 - 4\ Mass\ SpringK\) - \((1 \ + \[ExponentialE]\^\(\(\@\(Damper\^2 - 4\ Mass\ SpringK\)\ t\)\/Mass\))\)\ \ Mass\ \((\(-1\) + 2\ SpringK)\)\ \@\(Damper\^2 - 4\ Mass\ SpringK\) - 2\ \[ExponentialE]\^\(\(\((Damper + \@\(Damper\^2 - 4\ \ Mass\ SpringK\))\)\ t\)\/\(2\ Mass\)\)\ Mass\ \@\(Damper\^2 - 4\ Mass\ \ SpringK\)\ Cos[t\ \[Omega]\_o])\)\ \[Omega]\_o\%2 + Mass\^2\ \((Damper\ \((\(-1\) + \ \[ExponentialE]\^\(\(\@\(Damper\^2 - 4\ Mass\ SpringK\)\ t\)\/Mass\))\) + \ \((1 + \[ExponentialE]\^\(\(\@\(Damper\^2 - 4\ Mass\ SpringK\)\ \ t\)\/Mass\))\)\ \@\(Damper\^2 - 4\ Mass\ SpringK\))\)\ \ \[Omega]\_o\%4)\))\)/\((2\ \@\(Damper\^2 - 4\ Mass\ SpringK\)\ \((SpringK\^2 \ + \((Damper\^2 - 2\ Mass\ SpringK)\)\ \[Omega]\_o\%2 + Mass\^2\ \[Omega]\_o\%4)\))\)}\)], "Output", CellTags->"mmtag:22:damped_forced_harmonic_operator__general_solution"] }, Open ]], Cell["\<\ Functionals: An Introduction to Variational Calculus by way of \ Variation of Parameters\ \>", "Subtitle"], Cell[CellGroupData[{ Cell[TextData[{ "An early application of introductory calculus involves finding extreme \ values of a function ", StyleBox["y", FontSlant->"Italic"], " = ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "): values of the independent variable ", StyleBox["x", FontSlant->"Italic"], " are sought that maximize or minimize ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "). In the calculus of variations, a common problem involves finding an \ unknown ", StyleBox["function y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") that minimizes a different function in which ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") appears inside an integral. Here we pursue an answer to a problem of \ this type." }], "Section", CellTags->{ "mmtag:22:functionals", "mmtag:22:calculus_of_variations__introduction"}], Cell[TextData[{ "The \"quickest\" path of running up-hill can be estimated with a model of \ how fast one runs up hill. Given a starting point ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "=0) = 0 and an ending point ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "=1) = 1, the question is to find, among all possible choices of path ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") that connects the end and starting point, which is the one that takes \ the least time if there is a hill ", StyleBox["h", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") that has to be \"run-up.\"\nA reasonable model is that running speed is \ proportional to Cos(climbing-angle)--that is the velocity is 1 for running on \ a flat surface and goes to zero as the climbing-angle goes to ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Pi]\/2\)\(\ \)\(.\)\)\)]], "\n\nIn this case, the time can be represented as an integral of the \ unknown ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") (this is worked out in the accompanying lecture notes)\n", StyleBox["T", FontSlant->"Italic"], "[", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")] = ", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_\(x = 0\)\%\(x = 1\)\), RowBox[{ SqrtBox[ RowBox[{"1", " ", "+", " ", SuperscriptBox[ RowBox[{"(", FractionBox[ StyleBox["dy", FontSlant->"Italic"], StyleBox["dx", FontSlant->"Italic"]], ")"}], "2"], " ", "+", SuperscriptBox[ RowBox[{ RowBox[{"(", FractionBox[ StyleBox["dh", FontSlant->"Italic"], StyleBox["dx", FontSlant->"Italic"]], ")"}], " "}], "2"]}]], " ", \(\[DifferentialD]x\)}]}], TraditionalForm]]], "\nSuppose that the \"hill\" has the shape ", StyleBox["h", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") = ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], ".\n\nHere, the \"real\" problem of finding the optimal path from the \ infinite set of all paths connecting ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "=0) = 0 to", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "=1) = 1 will be replaced by looking at all second-order polynomials: ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") = ", StyleBox["a", FontSlant->"Italic"], " + ", StyleBox["bx", FontSlant->"Italic"], " + ", StyleBox["c", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^2\)]] }], "Text", CellTags->"mmtag:22:optimal_path__example"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(h\ = \ x^2;\)\), "\[IndentingNewLine]", \(\(YGeneral\ = \ a\ + \ b\ x\ + \ c\ x\^2;\)\)}], "Input"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(YGeneral\\)\\\" is similar to existing \ symbol \\\"\\!\\(General\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"] }, Open ]], Cell["The general path must satisfy the boundary conditions:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(YSatisifyingBCs\ = \ YGeneral /. \((Solve[{\((YGeneral /. x \[Rule] 0)\) \[Equal] 0, \((YGeneral /. x \[Rule] 1)\) \[Equal] 1}, {a, c}] // Flatten)\)\)], "Input"], Cell[BoxData[ \(b\ x + \((1 - b)\)\ x\^2\)], "Output"] }, Open ]], Cell["\<\ There is one remaining free variable, it can be determined by \ minimizing the integral\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TimeIntegral\ = \ Integrate[ Sqrt[1\ + \ \((D[YSatisifyingBCs, x])\)^2\ + \ \((D[h, x])\)^2], x]\)], "Input"], Cell[BoxData[ \(\((\(-\(\(\((\(-1\) + b)\)\ b\)\/\(4\ \((2 - 2\ b + b\^2)\)\)\)\) + x\/2)\)\ \@\(1 + b\^2\ \((1 - 2\ x)\)\^2 + 4\ b\ \((1 - 2\ x)\)\ \ x + 8\ x\^2\) + \(\((1 - b + b\^2)\)\ Log[2\ \((\@\(1 + b\^2\ \((1 - 2\ \ x)\)\^2 + 4\ b\ \((1 - 2\ x)\)\ x + 8\ x\^2\) + \(b + 4\ x - 4\ b\ x + b\^2\ \ \((\(-1\) + 2\ x)\)\)\/\@\(2 - 2\ b + b\^2\))\)]\)\/\(2\ \((2 - 2\ b + \ b\^2)\)\^\(3/2\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Time\ = \ Simplify[\((TimeIntegral /. x \[Rule] 1)\)\ - \ \((TimeIntegral /. x \[Rule] 0)\)]\)], "Input"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(Time\\)\\\" is similar to existing symbol \\\ \"\\!\\(Times\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ \(1\/4\ \((\(\((\(-1\) + b)\)\ b\ \@\(1 + b\^2\)\)\/\(2 - 2\ b + b\^2\) + \ \(\@\(9 - 4\ b + b\^2\)\ \((4 - 3\ b + b\^2)\)\)\/\(2 - 2\ b + b\^2\) - \(2\ \ \((1 - b + b\^2)\)\ Log[2\ \((\@\(1 + b\^2\) - \(\((\(-1\) + b)\)\ b\)\/\@\(2 \ - 2\ b + b\^2\))\)]\)\/\((2 - 2\ b + b\^2)\)\^\(3/2\) + \(2\ \((1 - b + b\^2)\ \)\ Log[2\ \((\@\(9 - 4\ b + b\^2\) + \(4 - 3\ b + b\^2\)\/\@\(2 - 2\ b + \ b\^2\))\)]\)\/\((2 - 2\ b + b\^2)\)\^\(3/2\))\)\)], 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The solution is obtained from a boundary-value problem which we \ do not take up at this point, but it is interesting to see the exact solution \ and compare it with the approximate one we obtained above. The closed-form \ expression for the function that minimizes the climbing time is:\ \>", "Text"], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(YExactSolution\ = \ \((2 x\ \@\(1 + 4\ \ x\^2\) + \ ArcSinh[2\ x])\)\/\((2\ \@5 + ArcSinh[2])\);\)\)\)], "Input"], Cell["and a series approimation to this function is:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Series[YExactSolution, {x, 0, 6}] // Normal\) // N\)], "Input"], Cell[BoxData[ \(0.6761586459535304`\ x + 0.4507724306356869`\ x\^3 - 0.27046345838141217`\ x\^5\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Now plot the exact solution and compare it to our calculated \ approximation...\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(ExactSolution = \ Plot[YExactSolution, {x, 0, 1}, PlotStyle \[Rule] {Thickness[0.005], Hue[1]}]\)], "Input"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(ExactSolution\\)\\\" is similar to 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