(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 232350, 5622] NotebookOptionsPosition[ 207671, 5092] NotebookOutlinePosition[ 224267, 5390] CellTagsIndexPosition[ 224224, 5387] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["The Principle of Least Action", "Title", CellChangeTimes->{{3.5006705747374506`*^9, 3.500670584976754*^9}, { 3.5006712637088437`*^9, 3.500671265205888*^9}}, TextJustification->1.], Cell["Jason Gross, December 7, 2010", "Subtitle", CellChangeTimes->{{3.5006712660239124`*^9, 3.500671284020445*^9}}, FontSize->10], Cell[CellGroupData[{ Cell["Introduction", "Section", CellChangeTimes->{{3.5006711405361977`*^9, 3.500671150592496*^9}}], Cell[TextData[{ "Recall that we defined the ", StyleBox["Lagrangian", FontSlant->"Italic"], " to be the kinetic energy less potential energy, ", Cell[BoxData[ FormBox[ RowBox[{"L", "=", RowBox[{"K", "-", "U"}]}], TraditionalForm]], CellChangeTimes->{ 3.5006541640755157`*^9, {3.5006542325275416`*^9, 3.500654238071706*^9}}], ", at a point. The action is then defined to be the integral of the \ Lagrangian along the path," }], "Text", CellChangeTimes->{{3.500670589548889*^9, 3.5006706389823523`*^9}, { 3.5006706715093155`*^9, 3.500670767850167*^9}}, TextJustification->1.], Cell[BoxData[ FormBox[ RowBox[{"S", "\[LongEqual]", RowBox[{ SubsuperscriptBox["\[Integral]", SubscriptBox["t", "0"], SubscriptBox["t", "1"]], RowBox[{"L", RowBox[{"\[DifferentialD]", "t"}]}]}], "\[LongEqual]", RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", SubscriptBox["t", "0"], SubscriptBox["t", "1"]], "K"}], "-", RowBox[{"U", RowBox[{"\[DifferentialD]", "t"}]}]}]}], TraditionalForm]], "DisplayFormula", CellChangeTimes->{{3.5006708476535287`*^9, 3.5006708559487743`*^9}, { 3.5006713260626893`*^9, 3.500671332004865*^9}}, TextAlignment->Center, TextJustification->1.], Cell["\<\ It is (remarkably!) true that, in any physical system, the path an object \ actually takes minimizes the action. It can be shown that the extrema of \ action occur at\ \>", "Text", CellChangeTimes->{{3.500670900769101*^9, 3.500670995401902*^9}}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FractionBox[ RowBox[{"\[PartialD]", "L"}], RowBox[{"\[PartialD]", "q"}], MultilineFunction->None], "-", TagBox[ RowBox[{ FractionBox["\[DifferentialD]", RowBox[{"\[DifferentialD]", "t"}]], FractionBox[ RowBox[{"\[PartialD]", "L"}], RowBox[{"\[PartialD]", OverscriptBox["q", "."]}], MultilineFunction->None]}], D]}], "\[LongEqual]", "0"}], TraditionalForm]], "DisplayFormula", TextAlignment->Center], Cell["\<\ This is called the Euler equation, or the Euler-Lagrange Equation.\ \>", "Text", CellChangeTimes->{{3.500670900769101*^9, 3.500670995401902*^9}, { 3.500673679282343*^9, 3.5006736885386167`*^9}}], Cell[CellGroupData[{ Cell["Derivation", "Subsection", CellChangeTimes->{{3.5006763059450893`*^9, 3.5006763145333433`*^9}}], Cell["\<\ Courtesy of Scott Hughes\[CloseCurlyQuote]s Lecture notes for 8.033. (Most \ of this is copied almost verbatim from that.)\ \>", "Text", CellChangeTimes->{{3.5006717491212115`*^9, 3.5006717937725334`*^9}, { 3.5006721187191515`*^9, 3.500672127592414*^9}}], Cell[TextData[{ "Suppose we have a function ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", RowBox[{"x", ",", RowBox[{ OverscriptBox["x", "."], ";", "t"}]}], ")"}], TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], " of a variable ", Cell[BoxData[ FormBox["x", TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], " and its derivative ", Cell[BoxData[ FormBox[ RowBox[{ OverscriptBox["x", "."], "\[LongEqual]", RowBox[{ RowBox[{"\[DifferentialD]", "x"}], "/", RowBox[{"\[DifferentialD]", "t"}]}]}], TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], ". 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We consider the limits of integration to be fixed. \ That is, ", Cell[BoxData[ FormBox[ RowBox[{"x", "(", SubscriptBox["t", "1"], ")"}], TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], " will be the same for any ", Cell[BoxData[ FormBox["x", TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], " we care about, as will ", Cell[BoxData[ FormBox[ RowBox[{"x", "(", SubscriptBox["t", "2"], ")"}], TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], "." }], "Text", CellChangeTimes->{{3.500671369759983*^9, 3.5006713850514355`*^9}, { 3.500671426645666*^9, 3.500671496201725*^9}, {3.500671539809016*^9, 3.5006717309726744`*^9}, {3.5006717985466747`*^9, 3.5006718669707*^9}, { 3.5006718976136065`*^9, 3.5006718986796384`*^9}, {3.50067215313717*^9, 3.500672154587213*^9}}], Cell[TextData[{ "Imagine we have some ", Cell[BoxData[ FormBox[ RowBox[{"x", "(", "t", ")"}], TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], " for which ", Cell[BoxData[ FormBox["J", TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], " is at an extremum, and imagine that we have a function which parametrizes \ how far our current path is from our choice of ", Cell[BoxData[ FormBox["x", TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], ":" }], "Text", CellChangeTimes->{{3.5006718997056684`*^9, 3.500671911209009*^9}, { 3.5006721550152254`*^9, 3.5006721554812393`*^9}, 3.5006953333305945`*^9}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"x", "(", RowBox[{"t", ";", "\[Alpha]"}], ")"}], "=", RowBox[{ RowBox[{"x", "(", "t", ")"}], "+", RowBox[{"\[Alpha]", " ", RowBox[{"A", "(", "t", ")"}]}]}]}], TraditionalForm]], "DisplayFormula", CellChangeTimes->{{3.5006708476535287`*^9, 3.5006708559487743`*^9}, { 3.5006713260626893`*^9, 3.500671332004865*^9}, {3.500671513757245*^9, 3.5006715594855986`*^9}, {3.500671872658868*^9, 3.500671887000293*^9}, { 3.5006721559292526`*^9, 3.500672156408267*^9}}, TextAlignment->Center, TextJustification->1.], Cell[TextData[{ "The function ", Cell[BoxData[ FormBox["A", TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], " is totally arbitrary, except that we require it to vanish at the \ endpoints: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"A", "(", SubscriptBox["t", "0"], ")"}], "=", RowBox[{ RowBox[{"A", "(", SubscriptBox["t", "1"], ")"}], "=", "0"}]}], TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], ". The parameter ", Cell[BoxData[ FormBox["\[Alpha]", TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], " allows us to control how the variation ", Cell[BoxData[ FormBox[ RowBox[{"A", "(", "t", ")"}], TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], " enters into our path ", Cell[BoxData[ FormBox[ RowBox[{"x", "(", RowBox[{"t", ";", "\[Alpha]"}], ")"}], TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], "." }], "Text", CellChangeTimes->{{3.500671369759983*^9, 3.5006713850514355`*^9}, { 3.500671426645666*^9, 3.500671496201725*^9}, {3.500671539809016*^9, 3.5006717309726744`*^9}, {3.5006717985466747`*^9, 3.5006718669707*^9}, { 3.5006719158081455`*^9, 3.5006720457879925`*^9}, {3.5006721598763695`*^9, 3.5006721598763695`*^9}}], Cell[TextData[{ "The \[OpenCurlyDoubleQuote]correct\[CloseCurlyDoubleQuote] path ", Cell[BoxData[ FormBox[ RowBox[{"x", "(", "t", ")"}], TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], " is unknown; our goal is to figure out how to construct it, or to figure \ out how ", Cell[BoxData[ FormBox["f", TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], " behaves when we are on it." }], "Text", CellChangeTimes->{{3.500671369759983*^9, 3.5006713850514355`*^9}, { 3.500671426645666*^9, 3.500671496201725*^9}, {3.500671539809016*^9, 3.5006717309726744`*^9}, {3.5006717985466747`*^9, 3.5006718669707*^9}, { 3.5006719158081455`*^9, 3.500672116011071*^9}, {3.500672160462387*^9, 3.5006721698736653`*^9}, 3.5006953359166713`*^9}], Cell[TextData[{ "Our basic idea is to ask how does the integral ", Cell[BoxData[ FormBox["J", TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], " behave when we are in the vicinity of the extremum. We know that ordinary \ functions are flat --- have zero first derivative --- when we are at an \ extremum. 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However, this doesn\[CloseCurlyQuote]t teach us anything useful, sine we \ don\[CloseCurlyQuote]t know the path ", Cell[BoxData[ FormBox[ RowBox[{"x", "(", "t", ")"}], TraditionalForm]], CellChangeTimes->{ 3.500670658189921*^9, {3.5006707933409214`*^9, 3.5006708043192463`*^9}, { 3.500671040435235*^9, 3.5006710482484665`*^9}, 3.500671100846023*^9, { 3.500671413501278*^9, 3.5006714205264854`*^9}}, TextJustification->1.], " that corresponds to the extremum." }], "Text", CellChangeTimes->{{3.5006721716217175`*^9, 3.500672176736869*^9}, { 3.5006722139379697`*^9, 3.5006722921302843`*^9}, {3.5006723350535545`*^9, 3.5006723904011927`*^9}}], Cell[TextData[{ "But we also know We know that ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ RowBox[{ FractionBox[ RowBox[{"\[PartialD]", "J"}], RowBox[{"\[PartialD]", "\[Alpha]"}], MultilineFunction->None], "\[RightBracketingBar]"}], RowBox[{"\[Alpha]", "=", "0"}]], "=", "0"}], TraditionalForm]], CellChangeTimes->{{3.5006708476535287`*^9, 3.5006708559487743`*^9}, { 3.5006713260626893`*^9, 3.500671332004865*^9}, {3.500671513757245*^9, 3.5006715594855986`*^9}, {3.5006721520241375`*^9, 3.5006721524491496`*^9}, {3.5006722975874453`*^9, 3.500672326280295*^9}, 3.500672692848145*^9}, TextAlignment->Center, TextJustification->1.], " since it\[CloseCurlyQuote]s an extremum. 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UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.5006967329170213`*^9, {3.5006970295007997`*^9, 3.500697080490309*^9}, { 3.5006971311868095`*^9, 3.5006971368149757`*^9}, 3.5007549454965773`*^9, { 3.50075497935958*^9, 3.500754994052015*^9}}] }, Open ]] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Problem: Bead on a Ring", "Section", CellChangeTimes->{{3.5006734704251604`*^9, 3.500673531351964*^9}, 3.500678571330143*^9, {3.5006786379991164`*^9, 3.500678639358156*^9}, { 3.5006910355213833`*^9, 3.5006910420225754`*^9}}], Cell["From 8.033 Quiz #2", "Text", CellChangeTimes->{{3.5006911555499363`*^9, 3.5006911653902273`*^9}}], Cell[CellGroupData[{ Cell["Problem", "Subsection", CellChangeTimes->{{3.500654268050593*^9, 3.5006542794169292`*^9}}, TextJustification->1.], Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 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