(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 48342, 1451]*) (*NotebookOutlinePosition[ 49421, 1486]*) (* CellTagsIndexPosition[ 49377, 1482]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Use the package IntroToSymmetry.m to work out the point group of \ of the linear 2rd order ODE\n\nXtt+X=0.\n\nThe function \ SolveDeterminingEquations will only find the groups corresponding to \ translational invariance in time and dilational invariance in X. ", FontSize->14, FontWeight->"Plain"], StyleBox["Mathematica", FontSize->14, FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" built-in functions are used to simplify the determining \ equations and find the remaining groups that depend in Sines and Cosines of \ time. For further information the user should see the 1976 paper by C. E. \ Wulfman and B. G. Wybourne entitled \"The Lie group of Newton's and \ Lagrange's equations for the harmonic oscillator\" in J. Physics A: Math. \ Gen., Vol. 9, No. 4.", FontSize->14, FontWeight->"Plain"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(Off[General::spell]\)], "Input"], Cell["First read in the package.", "Text"], Cell["Needs[\"SymmetryAnalysis`IntroToSymmetry`\"]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell["Enter the input equation as a string, ie, in quotes.", "Text"], Cell["\<\ inputequation= \"D[x[t],t,t]+x[t]\";\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ The function x[t] is a solution of the equation and this constraint \ must be applied to the invariance condition. Replace the second derivative \ wherever it appears in the invariance condition.\ \>", "Text"], Cell["\<\ rulesarray= {\"D[x[t],t,t]->-x[t]\"};\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[StyleBox["Enter the list of independent variables.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(\(independentvariables = {"\"};\)\)], "Input"], Cell[TextData[StyleBox["Enter the list of dependent variables.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(\(dependentvariables = {"\"};\)\)], "Input"], Cell[TextData[StyleBox["Enter the list of function and constant names that \ must be preserved when the equation is expressed in terms of generic (x1,y1) \ variables.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(\(frozennames = {"\<\>"};\)\)], "Input"], Cell[TextData[StyleBox["Enter the maximum derivative order of the equation.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(\(p = 2;\)\)], "Input"], Cell[TextData[StyleBox["The maximum derivative order that the \ infinitesimalsare assumed to depend on is specified by the input parameter r. \ This parameter is only nonzero when the user is looking for Lie contact \ groups or Lie-Backlund groups. For the usual case where one is searching for \ point groups one sets r=0.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(\(r = 0;\)\)], "Input"], Cell[TextData[StyleBox["When searching for Lie-Backlund groups (r=1 or \ greater) one can, without loss of generality, leave the independent variables \ untransformed. The corresponding infinitesimals (the xse's) are set to zero \ by setting xseon=0. If one is searching for point groups then set xseon=1. \ The choice xseon=1 is also an option when looking for Lie-Backlund groups and \ this can be useful when looking for contact symmetries.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(\(xseon = 1;\)\)], "Input"], Cell[TextData[StyleBox["When searching for Lie-Backlund groups, it is \ necessary to differentiate the input equation r times producing derivatives \ of order p+r. These higher order differential consequences are appended to \ the set of rules applied to the invariance condition. This process is carried \ out automatically when internalrules=1. For point groups the equation or \ equation system is the only rule or set of rules needed and one sets \ internalrules=0.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(\(internalrules = 0;\)\)], "Input"], Cell[TextData[StyleBox["Now work out the determining equations of the Lie \ point group that leaves the equation invariant. The output is available as a \ table of strings called zdeterminingequations.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell["\<\ Timing[FindDeterminingEquations[ independentvariables,dependentvariables,frozennames,p,r,xseon,inputequation,\ rulesarray,internalrules]]\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \("FindDeterminingEquations has finished executing. You can look at the \ output in the table zdeterminingequations. Each entry in this table is a \ determining equation in string format expressed in terms of z-variables. \ Rules for converting between z-variables and conventional variables are \ contained in the table ztableofrules. To view the determining equations in \ terms of conventional variables use the command \ ToExpression[zdeterminingequations]/.ztableofrules. There are two other items \ the user may wish to look at; the equation converted to generic \ (x1,x2,...,y1,y2,...) variables is designated equationgenericvariables and \ the various derivatives of the equation that appear in the invariance \ condition can be viewed in the table invarconditiontable. Rules for \ converting between z-variables and generic variables are contained in the \ table ztableofrulesxy."\)], "Print"], Cell[BoxData[ \({0.30000000000000004`\ Second, Null}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(equationgenericvariables\)], "Input"], Cell[BoxData[ \("D[y1[x1],x1,x1]+y1[x1]"\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["invarconditiontable", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \({0, 1, 0, 1}\)], "Output"] }, Open ]], Cell["\<\ Here are the determining equations in a readable form as \ expressions rather than strings.\ \>", "Text"], Cell[CellGroupData[{ Cell["zdeteqns=ToExpression[zdeterminingequations]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{\(eta1[z1, z2]\), "-", RowBox[{"z2", " ", RowBox[{ SuperscriptBox["eta1", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(z1, z2\), "]"}]}], "+", RowBox[{"2", " ", "z2", " ", RowBox[{ SuperscriptBox["xse1", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(z1, z2\), "]"}]}], "+", RowBox[{ SuperscriptBox["eta1", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(z1, z2\), "]"}]}], "==", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"3", " ", "z2", " ", RowBox[{ SuperscriptBox["xse1", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(z1, z2\), "]"}]}], "+", RowBox[{"2", " ", RowBox[{ SuperscriptBox["eta1", TagBox[\((1, 1)\), Derivative], MultilineFunction->None], "[", \(z1, z2\), "]"}]}], "-", RowBox[{ SuperscriptBox["xse1", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(z1, z2\), "]"}]}], "==", "0"}], ",", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["eta1", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(z1, z2\), "]"}], "-", RowBox[{"2", " ", RowBox[{ SuperscriptBox["xse1", TagBox[\((1, 1)\), Derivative], MultilineFunction->None], "[", \(z1, z2\), "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{"-", RowBox[{ SuperscriptBox["xse1", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(z1, z2\), "]"}]}], "==", "0"}]}], "}"}]], "Output"] }, Open ]], Cell["\<\ Here is the correspondence between z-variables and conventional \ variables.\ \>", "Text"], Cell[CellGroupData[{ Cell["ztableofrules", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \({z1 \[Rule] t, z2 \[Rule] x[t]}\)], "Output"] }, Open ]], Cell["\<\ Now solve the determining equations in terms of multivariable \ polynomials of some selected order. The Mathematica function Solve uses \ Gaussian elimination to solve a large number of linear equations for the \ polynomial coefficients. The time roughly follows time/timeref=((number of equations)/(number of equationsref))^n where the exponent is between 2.4 and 2.7. The Mathematica function Timing \ outputs the time required for the SolveDeterminingEquations function to \ execute.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Timing[SolveDeterminingEquations[ independentvariables,dependentvariables,r, xseon,zdeterminingequations,order=3]]\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ InterpretationBox["\<\"The number of unknown polynomial coefficients = \ \\!\\(20\\)\"\>", StringForm[ "The number of unknown polynomial coefficients = ``", 20], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox["\<\"The number of equations for the polynomial \ coefficients = \\!\\(25\\)\"\>", StringForm[ "The number of equations for the polynomial coefficients = ``", 25], Editable->False]], "Print"], Cell[BoxData[ \("SolveDeterminingEquations has finished executing. You can look at the \ output in the tables xsefunctions and etafunctions. Each entry in these \ tables is an infinitesimal function in string format expressed in terms of \ z-variables and the group parameters. The output can also be viewed with the \ group parameters stripped away by looking at the table infinitesimalgroups. \ In either case you may wish to convert the z-variables to conventional \ variables using the table ztableofrules. Keep in mind that this function \ only finds solutions of the determining equations that are of algebraic form. \ The determining equations may admit solutions that involve transcendental \ functions and/or integrals. Note that arbitrary functions may appear in the \ infinitesimals and that these can be detected by running the package function \ SolveDeterminingEquations for several polynomial orders. If terms of ever \ increasing order appear, then an arbitrary function is indicated."\)], "Print"], Cell[BoxData[ \({0.1333333333333333`\ Second, Null}\)], "Output"] }, Open ]], Cell["\<\ Here are the infinitesimal transformation functions for the \ independent variables.\ \>", "Text"], Cell[CellGroupData[{ Cell["xsefunctions", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \({"xse1[z1_, z2_]=a10"}\)], "Output"] }, Open ]], Cell["and for the dependent variables.", "Text"], Cell[CellGroupData[{ Cell["etafunctions", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \({"eta1[z1_, z2_]=b14*z2"}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Express the xse functions in terms of x,y variables.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(ToExpression[xsefunctions] /. ztableofrules\)], "Input"], Cell[BoxData[ \({a10}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Express the eta function in terms of these \ variables.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(ToExpression[etafunctions] /. ztableofrules\)], "Input"], Cell[BoxData[ \({b14\ x[t]}\)], "Output"] }, Open ]], Cell["Here is a list of the various infinitesimal groups.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(infinitesimalgroupsxy = infinitesimalgroups /. {z1 \[Rule] t, z2 \[Rule] x}\)], "Input"], Cell[BoxData[ \({{{1}, {0}}, {{0}, {x}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ColumnForm[infinitesimalgroupsxy]\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\({{1}, {0}}\)}, {\({{0}, {x}}\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ {{{1}, {0}}, {{0}, {x}}}], Editable->False]], "Output"] }, Open ]], Cell[TextData[StyleBox["The final result indicates that the equation is \ invariant under two groups. The parameter a10 corresponds to a translation \ group in time and the parameter b14 represents a dilation in y. Now generate \ the commutator table.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(MakeCommutatorTable[independentvariables, dependentvariables, infinitesimalgroupsxy]\)], "Input"], Cell[BoxData[ \("MakeCommutatorTable has finished executing. You can look at the output \ in the table commutatortable. To present the output in the most readable form \ you may want view it as a matrix using MatrixForm[commutatortable]. \ Occasionally the entries in the commutatortable will have terms that cancel. \ To get rid of these terms use the function Simplify before viewing the \ table."\)], "Print"] }, Open ]], Cell[TextData[StyleBox["Put the commutator table in readable form.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[commutatortable]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{"(", "\[NoBreak]", GridBox[{ {"0"}, {"0"} }], "\[NoBreak]", ")"}], RowBox[{"(", "\[NoBreak]", GridBox[{ {"0"}, {"0"} }], "\[NoBreak]", ")"}]}, { RowBox[{"(", "\[NoBreak]", GridBox[{ {"0"}, {"0"} }], "\[NoBreak]", ")"}], RowBox[{"(", "\[NoBreak]", GridBox[{ {"0"}, {"0"} }], "\[NoBreak]", ")"}]} }], "\[NoBreak]", ")"}], MatrixForm[ {{{{0}, {0}}, {{0}, {0}}}, {{{0}, {0}}, {{0}, { 0}}}}]]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["This is an Abelian Lie algebra. The equation admits more groups \ than this and we need to use ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" built-in functions to simplify the determining equations and try \ various ansatz.", FontWeight->"Plain"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[StyleBox["Try a solution of the determining equations of the \ following form.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(xse1[t_, x_] := fa[x]*Cos[t]\)], "Input"], Cell[BoxData[ \(eta1[t_, x_] := ga[x]*Sin[t]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(determiningequationsxta = zdeteqns /. {z1 \[Rule] t, z2 \[Rule] x}\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{\(\(-2\)\ x\ fa[x]\ Sin[t]\), "-", RowBox[{"x", " ", \(Sin[t]\), " ", RowBox[{ SuperscriptBox["ga", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{\(Cos[t]\ fa[x]\), "+", RowBox[{"3", " ", "x", " ", \(Cos[t]\), " ", RowBox[{ SuperscriptBox["fa", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "+", RowBox[{"2", " ", \(Cos[t]\), " ", RowBox[{ SuperscriptBox["ga", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"2", " ", \(Sin[t]\), " ", RowBox[{ SuperscriptBox["fa", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "+", RowBox[{\(Sin[t]\), " ", RowBox[{ SuperscriptBox["ga", "\[Prime]\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{\(-Cos[t]\), " ", RowBox[{ SuperscriptBox["fa", "\[Prime]\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "==", "0"}]}], "}"}]], "Output"] }, Open ]], Cell[TextData[StyleBox["This system is solved by (fb[x], gb[x])=(-x, 1+x^2), \ (x, 1-x^2). ", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[StyleBox["Now try a solution of the determining equations of \ the form.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(xse1[t_, x_] := fb[x]*Sin[t]\)], "Input"], Cell[BoxData[ \(eta1[t_, x_] := gb[x]*Cos[t]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(determiningequationsxtb = zdeteqns /. {z1 \[Rule] t, z2 \[Rule] x}\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{\(2\ x\ Cos[t]\ fb[x]\), "-", RowBox[{"x", " ", \(Cos[t]\), " ", RowBox[{ SuperscriptBox["gb", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{\(fb[x]\ Sin[t]\), "+", RowBox[{"3", " ", "x", " ", \(Sin[t]\), " ", RowBox[{ SuperscriptBox["fb", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "-", RowBox[{"2", " ", \(Sin[t]\), " ", RowBox[{ SuperscriptBox["gb", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{ RowBox[{\(-2\), " ", \(Cos[t]\), " ", RowBox[{ SuperscriptBox["fb", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "+", RowBox[{\(Cos[t]\), " ", RowBox[{ SuperscriptBox["gb", "\[Prime]\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{\(-Sin[t]\), " ", RowBox[{ SuperscriptBox["fb", "\[Prime]\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "==", "0"}]}], "}"}]], "Output"] }, Open ]], Cell[TextData[StyleBox["This system is solved by (fb[x], gb[x])=(x, 1+x^2), \ (-x, 1-x^2). Note the sign change compared to the first case.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[StyleBox["Next try a solution of the determining equations of \ the form.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(xse1[t_, x_] := fc[x]*Sin[2*t]\)], "Input"], Cell[BoxData[ \(eta1[t_, x_] := gc[x]*Cos[2*t]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(determiningequationsxtc = zdeteqns /. {z1 \[Rule] t, z2 \[Rule] x}\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{\(4\ x\ Cos[2\ t]\ fc[x]\), "-", \(3\ Cos[2\ t]\ gc[x]\), "-", RowBox[{"x", " ", \(Cos[2\ t]\), " ", RowBox[{ SuperscriptBox["gc", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{\(4\ fc[x]\ Sin[2\ t]\), "+", RowBox[{"3", " ", "x", " ", \(Sin[2\ t]\), " ", RowBox[{ SuperscriptBox["fc", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "-", RowBox[{"4", " ", \(Sin[2\ t]\), " ", RowBox[{ SuperscriptBox["gc", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{ RowBox[{\(-4\), " ", \(Cos[2\ t]\), " ", RowBox[{ SuperscriptBox["fc", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "+", RowBox[{\(Cos[2\ t]\), " ", RowBox[{ SuperscriptBox["gc", "\[Prime]\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{\(-Sin[2\ t]\), " ", RowBox[{ SuperscriptBox["fc", "\[Prime]\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "==", "0"}]}], "}"}]], "Output"] }, Open ]], Cell[TextData[StyleBox["This system is solved by fc[x]=1, gc[x]=x.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[StyleBox["And finally try a solution of the determining \ equations of the form.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(xse1[t_, x_] := fd[x]*Cos[2*t]\)], "Input"], Cell[BoxData[ \(eta1[t_, x_] := gd[x]*Sin[2*t]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(determiningequationsxtd = zdeteqns /. {z1 \[Rule] t, z2 \[Rule] x}\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{\(\(-4\)\ x\ fd[x]\ Sin[2\ t]\), "-", \(3\ gd[x]\ Sin[2\ t]\), "-", RowBox[{"x", " ", \(Sin[2\ t]\), " ", RowBox[{ SuperscriptBox["gd", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{\(4\ Cos[2\ t]\ fd[x]\), "+", RowBox[{"3", " ", "x", " ", \(Cos[2\ t]\), " ", RowBox[{ SuperscriptBox["fd", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "+", RowBox[{"4", " ", \(Cos[2\ t]\), " ", RowBox[{ SuperscriptBox["gd", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"4", " ", \(Sin[2\ t]\), " ", RowBox[{ SuperscriptBox["fd", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "+", RowBox[{\(Sin[2\ t]\), " ", RowBox[{ SuperscriptBox["gd", "\[Prime]\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{\(-Cos[2\ t]\), " ", RowBox[{ SuperscriptBox["fd", "\[Prime]\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "==", "0"}]}], "}"}]], "Output"] }, Open ]], Cell[TextData[StyleBox["This system is solved by fd[x]=1, gd[x]=-x. Finally \ the full set of groups is", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(infinitesimalgroupsxyall\)\(=\)\({{{1}, {0}}, \ {{0}, {x}}, \ \ {{\(-x\)*Cos[t]}, {\((1 + x^2)\)*Sin[t]}}, \ {{x*Cos[t]}, {\((1 - x^2)\)* Sin[t]}}, \ {{x*Sin[t]}, {\((1 + x^2)\)*Cos[t]}}, \ {{\(-x\)* Sin[t]}, {\((1 - x^2)\)*Cos[t]}}, \ {{Sin[2*t]}, {x* Cos[2*t]}}, \ {{Cos[2*t]}, {\(-x\)* Sin[2*t]}}}\)\(\ \)\)\)], "Input"], Cell[BoxData[ \({{{1}, {0}}, {{0}, {x}}, {{\(-x\)\ Cos[t]}, {\((1 + x\^2)\)\ Sin[ t]}}, {{x\ Cos[t]}, {\((1 - x\^2)\)\ Sin[t]}}, {{x\ Sin[ t]}, {\((1 + x\^2)\)\ Cos[t]}}, {{\(-x\)\ Sin[ t]}, {\((1 - x\^2)\)\ Cos[t]}}, {{Sin[ 2\ t]}, {x\ Cos[2\ t]}}, {{Cos[ 2\ t]}, {\(-x\)\ Sin[2\ t]}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[%]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{"(", "\[NoBreak]", GridBox[{ {"1"} }], "\[NoBreak]", ")"}], RowBox[{"(", "\[NoBreak]", GridBox[{ {"0"} }], "\[NoBreak]", ")"}]}, { RowBox[{"(", "\[NoBreak]", GridBox[{ {"0"} }], "\[NoBreak]", ")"}], RowBox[{"(", "\[NoBreak]", GridBox[{ {"x"} }], "\[NoBreak]", ")"}]}, { RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-x\)\ Cos[t]\)} }], "\[NoBreak]", ")"}], RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\((1 + x\^2)\)\ Sin[t]\)} }], "\[NoBreak]", ")"}]}, { RowBox[{"(", "\[NoBreak]", GridBox[{ {\(x\ Cos[t]\)} }], "\[NoBreak]", ")"}], RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\((1 - x\^2)\)\ Sin[t]\)} }], "\[NoBreak]", ")"}]}, { RowBox[{"(", "\[NoBreak]", GridBox[{ {\(x\ Sin[t]\)} }], "\[NoBreak]", ")"}], RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\((1 + x\^2)\)\ Cos[t]\)} }], "\[NoBreak]", ")"}]}, { RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-x\)\ Sin[t]\)} }], "\[NoBreak]", ")"}], RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\((1 - x\^2)\)\ Cos[t]\)} }], "\[NoBreak]", ")"}]}, { RowBox[{"(", "\[NoBreak]", GridBox[{ {\(Sin[2\ t]\)} }], "\[NoBreak]", ")"}], RowBox[{"(", "\[NoBreak]", GridBox[{ {\(x\ Cos[2\ t]\)} }], "\[NoBreak]", ")"}]}, { RowBox[{"(", "\[NoBreak]", GridBox[{ {\(Cos[2\ t]\)} }], "\[NoBreak]", ")"}], RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(-x\)\ Sin[2\ t]\)} }], "\[NoBreak]", ")"}]} }], "\[NoBreak]", ")"}], MatrixForm[ {{{1}, {0}}, {{0}, {x}}, {{ Times[ -1, x, Cos[ t]]}, { Times[ Plus[ 1, Power[ x, 2]], Sin[ t]]}}, {{ Times[ x, Cos[ t]]}, { Times[ Plus[ 1, Times[ -1, Power[ x, 2]]], Sin[ t]]}}, {{ Times[ x, Sin[ t]]}, { Times[ Plus[ 1, Power[ x, 2]], Cos[ t]]}}, {{ Times[ -1, x, Sin[ t]]}, { Times[ Plus[ 1, Times[ -1, Power[ x, 2]]], Cos[ t]]}}, {{ Sin[ Times[ 2, t]]}, { Times[ x, Cos[ Times[ 2, t]]]}}, {{ Cos[ Times[ 2, t]]}, { Times[ -1, x, Sin[ Times[ 2, t]]]}}}]]], "Output"] }, Open ]], Cell[TextData[StyleBox["The harmonic oscillator admits an 8-dimensional Lie \ algebra.", FontWeight->"Plain"]], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(MakeCommutatorTable[independentvariables, dependentvariables, infinitesimalgroupsxyall]\)], "Input"], Cell[BoxData[ \("MakeCommutatorTable has finished executing. You can look at the output \ in the table commutatortable. 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