(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 6664018, 112115] NotebookOptionsPosition[ 6657283, 111927] NotebookOutlinePosition[ 6657865, 111948] CellTagsIndexPosition[ 6657822, 111945] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ RowBox[{ RowBox[{"Quit", "[", "]"}], ";"}]], "Input", CellChangeTimes->{{3.7184061759904623`*^9, 3.718406177362875*^9}}], Cell["\<\ El lagrangiano, los impulsos generalizados y las velocidades escritas en t\ \[EAcute]rminos de los impulsos:\ \>", "Subsection", CellChangeTimes->{{3.7182031573634*^9, 3.7182032095466003`*^9}}, FontSize->18], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"L", "[", RowBox[{ "\[Theta]_", ",", " ", "\[Theta]p_", ",", " ", "\[Psi]p_", ",", " ", "\[Phi]p_"}], "]"}], ":=", " ", RowBox[{ RowBox[{ FractionBox["I3", "2"], SuperscriptBox[ RowBox[{"(", RowBox[{"\[Psi]p", " ", "+", " ", RowBox[{"\[Phi]p", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ")"}], "2"]}], "+", " ", RowBox[{ FractionBox["I1", "2"], RowBox[{"(", RowBox[{ SuperscriptBox["\[Theta]p", "2"], "+", " ", RowBox[{ SuperscriptBox["\[Phi]p", "2"], SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]}], ")"}]}], " ", "-", " ", RowBox[{"mgl", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"sol", " ", "=", " ", RowBox[{"Collect", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"Solve", "[", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"p\[Theta]", " ", "\[Equal]", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"L", "[", RowBox[{ "\[Theta]", ",", " ", "\[Theta]p", ",", " ", "\[Psi]p", ",", " ", "\[Phi]p"}], "]"}], ",", " ", "\[Theta]p"}], "]"}]}], ",", " ", "\[IndentingNewLine]", RowBox[{"p\[Psi]", " ", "\[Equal]", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"L", "[", RowBox[{ "\[Theta]", ",", " ", "\[Theta]p", ",", " ", "\[Psi]p", ",", " ", "\[Phi]p"}], "]"}], ",", " ", "\[Psi]p"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"p\[Phi]", " ", "\[Equal]", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"L", "[", RowBox[{ "\[Theta]", ",", " ", "\[Theta]p", ",", " ", "\[Psi]p", ",", " ", "\[Phi]p"}], "]"}], ",", " ", "\[Phi]p"}], "]"}]}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"\[Theta]p", ",", " ", "\[Psi]p", ",", " ", "\[Phi]p"}], "}"}]}], "]"}], "\[IndentingNewLine]", ",", " ", RowBox[{"{", RowBox[{"I1", ",", " ", "I3"}], "}"}], ",", " ", "Simplify"}], "]"}]}]}], "Input", CellChangeTimes->{{3.7182022107722*^9, 3.7182025176172*^9}, { 3.7182152137338*^9, 3.7182152230832*^9}}], Cell[TextData[{ "Uso \[Theta]p, \[Phi]p y \[Psi]p para denotar ", Cell[BoxData[ FormBox[ OverscriptBox["\[Theta]", "."], TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox[ OverscriptBox["\[Phi]", "."], TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox[ OverscriptBox["\[Psi]", "."], TraditionalForm]], FormatType->"TraditionalForm"], ". 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Lo vamos a multiplicar por I1.\n\nA la constante \ mgl I1 la llamamos \[Alpha], y a la relaci\[OAcute]n ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["I", "1"], "/", SubscriptBox["I", "3"]}], TraditionalForm]], FormatType->"TraditionalForm"], " la llamamos \[Beta]. 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Ignorando la \ constante aditiva ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"\[Beta]", " ", SuperscriptBox[ SubscriptBox["p", "\[Psi]"], "2"]}], "2"], TraditionalForm]]], ", podemos entonces definir un hamiltoniano efectivo para \[Theta] y ", Cell[BoxData[ FormBox[ SubscriptBox["p", "\[Theta]"], TraditionalForm]], FormatType->"TraditionalForm"], ", que depende de un \[UAcute]nico par\[AAcute]metro, \[Alpha], y de los \ valores de las constantes de movimiento ", Cell[BoxData[ FormBox[ SubscriptBox["p", "\[Psi]"], TraditionalForm]], FormatType->"TraditionalForm"], " y ", Cell[BoxData[ FormBox[ SubscriptBox["p", "\[Phi]"], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Subsubsection", CellChangeTimes->{{3.7182036877212*^9, 3.718203795014*^9}, { 3.7182330715116*^9, 3.7182331030504*^9}, {3.7182331471004*^9, 3.7182331671404*^9}}, FontSize->16], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"H\[Theta]", "[", RowBox[{ "\[Theta]_", ",", " ", "p\[Theta]_", ",", " ", "p\[Psi]_", ",", " ", "p\[Phi]_", ",", " ", "\[Alpha]_"}], "]"}], ":=", " ", RowBox[{ FractionBox[ SuperscriptBox["p\[Theta]", "2"], "2"], "+", FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"p\[Phi]", "-", RowBox[{"p\[Psi]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ")"}], "2"], " ", SuperscriptBox[ RowBox[{"Csc", "[", "\[Theta]", "]"}], "2"]}], "2"], "+", RowBox[{"\[Alpha]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}]}], ";"}]], "Input", CellChangeTimes->{{3.7182037197768*^9, 3.7182037399158*^9}, { 3.7182038046692*^9, 3.7182038125568*^9}, {3.7182040430874*^9, 3.7182040440268*^9}, {3.7182146857622004`*^9, 3.7182146899314003`*^9}}], Cell[TextData[{ "Veamos el caso particular en que ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "\[Psi]"], "=", RowBox[{ SubscriptBox["p", "\[Phi]"], "=", "p"}]}], TraditionalForm]]], "." }], "Subsection", CellChangeTimes->{{3.7182039868884*^9, 3.7182040281482*^9}, { 3.7182331101602*^9, 3.7182331219074*^9}}, FontSize->18], Cell[BoxData[ RowBox[{"Collect", "[", RowBox[{ RowBox[{"H\[Theta]", "[", RowBox[{ "\[Theta]", ",", " ", "p\[Theta]", ",", " ", "p", ",", " ", "p", ",", " ", "\[Alpha]"}], "]"}], ",", " ", "p", ",", " ", "Simplify"}], "]"}]], "Input", CellChangeTimes->{{3.7182040689736*^9, 3.7182041958574*^9}, { 3.7182042329684*^9, 3.7182043436604*^9}}], Cell[BoxData[ RowBox[{"TrigReduce", "[", SuperscriptBox[ RowBox[{"Tan", "[", FractionBox["\[Theta]", "2"], "]"}], "2"], "]"}]], "Input", CellChangeTimes->{{3.7182042071644*^9, 3.7182042222602*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"H\[Theta]1", "[", RowBox[{ "\[Theta]_", ",", " ", "p\[Theta]_", ",", " ", "p_", ",", " ", "\[Alpha]_"}], "]"}], ":=", " ", RowBox[{ FractionBox[ SuperscriptBox["p\[Theta]", "2"], "2"], "+", RowBox[{"\[Alpha]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], "+", RowBox[{ FractionBox["1", "2"], " ", SuperscriptBox["p", "2"], " ", FractionBox[ RowBox[{"1", "-", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], RowBox[{"1", "+", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]]}]}]}]], "Input", CellChangeTimes->{{3.7182040317986*^9, 3.7182040566172*^9}, { 3.718204372241*^9, 3.7182043786591997`*^9}}], Cell[TextData[{ "Es un sistema conservativo, de modo que el punto en el espacio de fases ", Cell[BoxData[ FormBox[ RowBox[{"\[Theta]", "-", SubscriptBox["p", "\[Theta]"]}], TraditionalForm]], FormatType->"TraditionalForm"], " se mueve a lo largo de curvas de H\[Theta]1 constante.\nDividamos por la \ constante ", Cell[BoxData[ FormBox["p", TraditionalForm]]], " y redefinamos las coordenadas en el espacio de fases como \[Theta] y P = ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "\[Theta]"], "/", "p"}], TraditionalForm]], FormatType->"TraditionalForm"], ". 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Para ", Cell[BoxData[ FormBox["A", TraditionalForm]], FormatType->"TraditionalForm"], " mayores que este valor cr\[IAcute]tico, ", Cell[BoxData[ FormBox[ SubscriptBox["V", "ef"], TraditionalForm]]], " tiene un m\[AAcute]ximo en \[Theta] = 0 y dos m\[IAcute]nimos, \ sim\[EAcute]tricos respecto de \[Theta] = 0. Los m\[IAcute]nimos aparecen en ", Cell[BoxData[ FormBox[ RowBox[{"\[Theta]", " ", "=", " ", RowBox[{"\[ImplicitPlus]", "0"}]}], TraditionalForm]]], " y se mueven hacia \[PlusMinus]\[Pi] a medida que aumenta ", Cell[BoxData[ FormBox["A", TraditionalForm]]], ", lo que tiene sentido, porque aumentar ", Cell[BoxData[ FormBox["A", TraditionalForm]]], " equivale a aumentar ", Cell[BoxData[ FormBox[ RowBox[{"g", "."}], TraditionalForm]]], "\n\nVe\[AAcute]moslo. Para calcular la derivada del Vef respecto de \ \[Theta], definimos x = Cos[\[Theta]], derivamos respecto de x y, siguiendo \ la regla de la cadena, luego multiplicamos por -", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"1", "-", SuperscriptBox["x", "2"]}]], TraditionalForm]]], ". 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La \ soluci\[OAcute]n ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Cos", "[", "\[Theta]", "]"}], " ", "=", " ", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"1", "/", SqrtBox["A"]}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " es realizable si \n\n", Cell[BoxData[ FormBox[ RowBox[{"\t", StyleBox[ RowBox[{"A", ">", " ", RowBox[{"1", "/", "4"}]}], FontColor->GrayLevel[0]]}], TraditionalForm]]], ".\n\nEn tal caso, para ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Cos", "[", "\[Theta]", "]"}], " ", "=", " ", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"1", "/", SqrtBox["A"]}]}]}], TraditionalForm]]], "existen dos \[AAcute]ngulos posibles, sim\[EAcute]tricos respecto del cero, \ como es f\[AAcute]cil convencerse a trav\[EAcute]s de la siguiente figura:" }], "Subsubsection", CellChangeTimes->{{3.7182071697116003`*^9, 3.7182071875458*^9}, { 3.7182073365822*^9, 3.7182074088831997`*^9}, {3.718207440192*^9, 3.718207458087*^9}, {3.7182077830916*^9, 3.718207974954*^9}, { 3.7182080740996*^9, 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"\nResumiendo: ", Cell[BoxData[ FormBox[ SubscriptBox["V", "ef"], TraditionalForm]]], " tiene un s\[OAcute]lo extremo para \[Theta] real si ", Cell[BoxData[ FormBox[ RowBox[{"A", " ", "<", " ", RowBox[{"1", "/", "2"}]}], TraditionalForm]]], ". Es decir, para valores suficientemente peque\[NTilde]os de ", Cell[BoxData[ FormBox["g", TraditionalForm]]], " el \[UAcute]nico punto de equilibrio es \[Theta] = 0. Vemos por la forma \ del potencial que en ese caso todas las \[OAcute]rbitas en \[Theta] son \ acotadas y giran alrededor del origen." }], "Subsubsection", CellChangeTimes->{{3.7182071697116003`*^9, 3.7182071875458*^9}, { 3.7182073365822*^9, 3.7182074088831997`*^9}, {3.718207440192*^9, 3.718207458087*^9}, {3.7182077830916*^9, 3.718207974954*^9}, { 3.7182080740996*^9, 3.7182082431706*^9}, {3.7182084060222*^9, 3.7182084148536*^9}, {3.7182338809488*^9, 3.7182339482422*^9}, { 3.7182840768936*^9, 3.7182842093574*^9}, {3.7182842563526*^9, 3.7182843360257998`*^9}, 3.7182845207623997`*^9}, FontSize->16], Cell[BoxData[ GraphicsBox[GraphicsComplexBox[CompressedData[" 1:eJxs3Xm8pUdVL/zuhD4MV7iAgHidgIuReQogkz4FJECYp4RRFLmGy4yIcEEQ ZRKQCASJMiPKICBIxAAmsp+rCALhAglJgHQg6dCBJCfh9JT0lO73cJ79Xfut 3/b8059fr1WrVtWup2rVqlWrbvm7L3js7x2xadOmw0dv2vTTf9c+d+Zzz3vT lnHT9Df+5zHHv+6fdm0u3P7m9Pv+yucPzuCJf8+sp68G3lp4U/+HPkT5wnP5 QT9YeK5f6/m3JB63nva880973oGZ9sDaA2tPT18NvLWwhgR9iPJDyA/6wSH0 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{None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImageSize->300, Method->{"DefaultBoundaryStyle" -> Automatic, "ScalingFunctions" -> None}, PlotLabel->FormBox["\"Curva separatriz\"", TraditionalForm], PlotRange-> NCache[{{-Pi, Pi}, {-2.049887960952952, 2.049887960952952}}, {{-3.141592653589793, 3.141592653589793}, {-2.049887960952952, 2.049887960952952}}], PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{{3.7184871416766*^9, 3.7184871608504*^9}, 3.7185040814522*^9}, CellID->297322959] }, Open ]], Cell[CellGroupData[{ Cell["\<\ El movimiento de la peonza en el espacio: integraci\[OAcute]n \ num\[EAcute]rica de las ecuaciones\ \>", "Section", CellFrame->{{0, 0}, {0, 3}}, CellChangeTimes->{{3.7182149812469997`*^9, 3.718214993808*^9}, { 3.7182153559268*^9, 3.7182153691470003`*^9}, {3.7182341544356003`*^9, 3.7182341559702*^9}}], Cell["Partimos del hamiltoniano original", "Subsubsection", CellChangeTimes->{{3.718215060467*^9, 3.7182150655702*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"H", "[", RowBox[{ "\[Theta]_", ",", " ", "p\[Theta]_", ",", " ", "p\[Psi]_", ",", " ", "p\[Phi]_"}], "]"}], ":=", " ", RowBox[{ FractionBox[ SuperscriptBox["p\[Theta]", "2"], RowBox[{"2", " ", "I1"}]], "+", FractionBox[ SuperscriptBox["p\[Psi]", "2"], RowBox[{"2", " ", "I3"}]], "+", FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"p\[Phi]", "-", RowBox[{"p\[Psi]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ")"}], "2"], " "}], RowBox[{"2", " ", "I1", " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]], "+", RowBox[{"mgl", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}]}], ";"}]], "Input", CellChangeTimes->{{3.7182154873964*^9, 3.7182154914906*^9}}], Cell[TextData[{ "Ya sabemos que ", Cell[BoxData[ FormBox[ SubscriptBox["p", "\[Psi]"], TraditionalForm]]], " y ", Cell[BoxData[ FormBox[ SubscriptBox["p", "\[Phi]"], TraditionalForm]]], " son constantes. Las ecuaciones de movimiento relevantes son las que dan \ las velocidades en t\[EAcute]rminos de los impulsos\n\n\t", Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"\[Theta]p", "\[Rule]", FractionBox["p\[Theta]", "I1"]}], ",", RowBox[{"\[Psi]p", "\[Rule]", RowBox[{ FractionBox["p\[Psi]", "I3"], "-", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"p\[Phi]", "-", RowBox[{"p\[Psi]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ")"}], " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], RowBox[{"I1", " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]]}]}], ",", RowBox[{"\[Phi]p", "\[Rule]", FractionBox[ RowBox[{"(", RowBox[{"p\[Phi]", "-", RowBox[{"p\[Psi]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ")"}], RowBox[{"I1", " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]]}]}], "}"}]], CellChangeTimes->{{3.718202467818*^9, 3.7182025197622004`*^9}, 3.7182152250772*^9}, FontSize->18, FontColor->GrayLevel[0]], "\n\ny la que da ", Cell[BoxData[ FormBox[ SubscriptBox[ OverscriptBox["p", "."], "\[Theta]"], TraditionalForm]]], ", que se obtiene derivando H respecto de \[Theta] y multiplicando \[Times] \ -1." }], "Subsubsection", CellChangeTimes->{{3.7182150739192*^9, 3.718215177784*^9}, { 3.7182152551963997`*^9, 3.7182153454894*^9}, {3.7182153894742002`*^9, 3.71821547649*^9}, {3.7182155099302*^9, 3.7182155420248003`*^9}, { 3.7182157294472*^9, 3.7182157392188*^9}, 3.7182173824248*^9, { 3.7182341896540003`*^9, 3.718234189764*^9}, {3.7182342291654*^9, 3.7182342322047997`*^9}}, FontSize->16], Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"-", RowBox[{"D", "[", RowBox[{ RowBox[{"H", "[", RowBox[{ "\[Theta]", ",", " ", "p\[Theta]", ",", " ", "p\[Psi]", ",", " ", "p\[Phi]"}], "]"}], ",", " ", "\[Theta]"}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.7182155440694*^9, 3.7182155979048*^9}}], Cell[TextData[{ "Es decir\n\n", StyleBox["\t ", FontColor->GrayLevel[0]], Cell[BoxData[ RowBox[{"{", RowBox[{"p\[Theta]p", "\[Rule]", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"p\[Psi]", " ", RowBox[{"(", RowBox[{"p\[Phi]", "-", RowBox[{"p\[Psi]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ")"}], " "}], RowBox[{"I1", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]]}], "+", FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"p\[Phi]", "-", RowBox[{"p\[Psi]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ")"}], "2"], " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], RowBox[{"I1", " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]], "+", RowBox[{"mgl", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}]}], "}"}]], CellChangeTimes->{3.7182155981544*^9}], "\n" }], "Subsubsection", CellChangeTimes->{{3.718215641942*^9, 3.7182157028236*^9}, { 3.7182157595062*^9, 3.7182157755324*^9}, 3.7182173808004*^9}, FontSize->16], Cell[TextData[{ "Caso ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "\[Phi]"], " ", "=", " ", RowBox[{ SubscriptBox["p", "\[Psi]"], "=", " ", "p"}]}], TraditionalForm]]], ". " }], "Subsection", CellFrame->{{0, 0}, {0, 0.5}}, CellChangeTimes->{{3.7182156224546003`*^9, 3.7182156285727997`*^9}, { 3.7182157796538*^9, 3.718215821755*^9}}, FontSize->16], Cell[TextData[{ "Para fijar el problema usaremos ", Cell[BoxData[ FormBox[ SubscriptBox["I", "3"], TraditionalForm]]], " = 2 ", Cell[BoxData[ FormBox[ SubscriptBox["I", "1"], TraditionalForm]]], ", como si la peonza fuera un disco atravesado por un eje sin masa. Dado ", Cell[BoxData[ FormBox["p", TraditionalForm]]], ", podemos definir una velocidad angular caracter\[IAcute]stica del \ problema, \n\n\t", Cell[BoxData[ FormBox[ RowBox[{"\[Omega]", " ", "=", " ", RowBox[{"p", "/", SubscriptBox["I", "3"]}]}], TraditionalForm]], FontColor->GrayLevel[0]], StyleBox[". ", FontColor->GrayLevel[0]], "\n\nEsto nos permite introducir una variable temporal adimensional \n\n", StyleBox["\t\[Tau] = \[Omega] t. ", FontColor->GrayLevel[0]], "\n\t\nUna variaci\[OAcute]n de \[Tau] en una unidad, corresponde, a grandes \ rasgos, a una vuelta de la peonza sobre su eje.\n\nVolviendo a las ecs. de \ movimiento, con ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "\[Psi]"], "=", " ", RowBox[{ SubscriptBox["p", "\[Phi]"], "=", " ", "p"}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["I", "3"], TraditionalForm]]], " = 2 ", Cell[BoxData[ FormBox[ SubscriptBox["I", "1"], TraditionalForm]]], ", y cambiando a la variable \[Tau] (de modo que ahora \[Theta]p es d\ \[Theta]/d\[Tau], etc.), queda:" }], "Subsubsection", CellChangeTimes->{{3.7182158258733997`*^9, 3.7182162225234003`*^9}, { 3.7182163227138*^9, 3.718216377295*^9}, {3.7182164128482*^9, 3.7182165165984*^9}, {3.7182343013442*^9, 3.7182343753946*^9}}, FontSize->16], Cell[BoxData[ RowBox[{"TrigReduce", "[", RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"\[Theta]p", "\[Rule]", RowBox[{ FractionBox["I3", "p"], RowBox[{"(", FractionBox["p\[Theta]", "I1"], ")"}]}]}], ",", RowBox[{"\[Psi]p", "\[Rule]", RowBox[{ RowBox[{"(", FractionBox["I3", "p"], ")"}], RowBox[{"(", RowBox[{ FractionBox["p\[Psi]", "I3"], "-", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"p\[Phi]", "-", RowBox[{"p\[Psi]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ")"}], " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], RowBox[{"I1", " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]]}], ")"}]}]}], ",", RowBox[{"\[Phi]p", "\[Rule]", RowBox[{ RowBox[{"(", FractionBox["I3", "p"], ")"}], RowBox[{"(", FractionBox[ RowBox[{"(", RowBox[{"p\[Phi]", "-", RowBox[{"p\[Psi]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ")"}], RowBox[{"I1", " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]], ")"}]}]}]}], "}"}], " ", "/.", " ", RowBox[{"p\[Phi]", "\[Rule]", " ", "p\[Psi]"}]}], " ", "/.", " ", RowBox[{"I1", "\[Rule]", " ", FractionBox["I3", "2"]}]}], "/.", " ", RowBox[{"p\[Psi]", "\[Rule]", " ", "p"}]}], " ", "]"}], "]"}]], "Input", CellChangeTimes->{{3.7182165217328*^9, 3.7182166495102*^9}, { 3.7182166891956*^9, 3.718216741318*^9}, {3.7182168307882*^9, 3.7182168748012*^9}}], Cell[BoxData[ RowBox[{"Collect", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", FractionBox["I3", "p"], ")"}], RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"p\[Psi]", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "p\[Phi]"}], "+", RowBox[{"p\[Psi]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ")"}], " ", RowBox[{"Csc", "[", "\[Theta]", "]"}]}], "I1"], "+", FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"p\[Phi]", "-", RowBox[{"p\[Psi]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], ")"}], "2"], " ", RowBox[{"Cot", "[", "\[Theta]", "]"}], " ", SuperscriptBox[ RowBox[{"Csc", "[", "\[Theta]", "]"}], "2"]}], "I1"], "+", RowBox[{"mgl", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}], ")"}]}], " ", "/.", " ", RowBox[{"p\[Phi]", "\[Rule]", " ", "p\[Psi]"}]}], " ", "/.", " ", RowBox[{"I1", "\[Rule]", " ", FractionBox["I3", "2"]}]}], "/.", " ", RowBox[{"p\[Psi]", "\[Rule]", " ", "p"}]}], " ", ",", " ", "mgl", ",", " ", "FullSimplify"}], "]"}]], "Input", CellChangeTimes->{{3.7182167647552*^9, 3.7182167920067997`*^9}, { 3.7182168875982*^9, 3.7182169774709997`*^9}}], Cell[TextData[{ "Es decir\n\n\t", Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"\[Theta]p", "\[Rule]", RowBox[{"2", FractionBox["p\[Theta]", "p"]}]}], ",", RowBox[{"\[Psi]p", "\[Rule]", FractionBox[ RowBox[{"1", "-", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], RowBox[{"1", "+", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]]}], ",", RowBox[{"\[Phi]p", "\[Rule]", FractionBox["2", RowBox[{"1", "+", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]]}], ",", " ", RowBox[{"p\[Theta]p", " ", "\[Rule]", " ", RowBox[{ RowBox[{ RowBox[{"-", "8"}], " ", "p", " ", SuperscriptBox[ RowBox[{"Csc", "[", "\[Theta]", "]"}], "3"], " ", SuperscriptBox[ RowBox[{"Sin", "[", FractionBox["\[Theta]", "2"], "]"}], "4"]}], "+", FractionBox[ RowBox[{"I3", " ", "mgl", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}], "p"]}]}]}], "}"}]], CellChangeTimes->{{3.718202467818*^9, 3.7182025197622004`*^9}, 3.7182152250772*^9}, FontColor->GrayLevel[0]], "\n\nYa casi estamos. As\[IAcute] como adimensionalizamos el tiempo, podemos \ adimensionalizar los impulsos, dividi\[EAcute]ndolos por el impulso caracter\ \[IAcute]stico ", Cell[BoxData[ FormBox["p", TraditionalForm]]], ".\n\nEntonces, ahora escribiremos ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ SubscriptBox["p", "\[Theta]"], "p"], "\[Rule]", " ", "P"}], TraditionalForm]]], " y ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["p\[Theta]p", "p"], "\[Rule]", " ", "Pp"}], TraditionalForm]]], ". Al dividir por ", Cell[BoxData[ FormBox["p", TraditionalForm]]], " en el t\[EAcute]rmino asociado al potencial gravitatorio aparece la \ constante\n\n\t", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["I", "3"], RowBox[{"mgl", " ", "/", " ", SuperscriptBox["p", "2"]}]}], TraditionalForm]], FontColor->GrayLevel[0]], "\n\t\nque, seg\[UAcute]n nuestras primeras definiciones, es igual a ", Cell[BoxData[ FormBox[ RowBox[{"2", "A"}], TraditionalForm]]], ", el par\[AAcute]metro que mide la importancia relativa de la gravedad \ respecto del esp\[IAcute]n.\n\nFinalmente, las ecuaciones de movimiento \ adimensionalizadas son:\n\n\t", Cell[BoxData[ StyleBox[ RowBox[{"{", RowBox[{ RowBox[{"\[Theta]p", "\[Rule]", RowBox[{"2", "P"}]}], ",", RowBox[{"\[Psi]p", "\[Rule]", FractionBox[ RowBox[{"1", "-", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], RowBox[{"1", "+", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]]}], ",", RowBox[{"\[Phi]p", "\[Rule]", FractionBox["2", RowBox[{"1", "+", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]]}], ",", " ", RowBox[{"Pp", " ", "\[Rule]", " ", RowBox[{ RowBox[{ RowBox[{"-", "8"}], " ", SuperscriptBox[ RowBox[{"Csc", "[", "\[Theta]", "]"}], "3"], " ", SuperscriptBox[ RowBox[{"Sin", "[", FractionBox["\[Theta]", "2"], "]"}], "4"]}], "+", RowBox[{"2", "A", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}]}]}], "}"}], FontWeight->"Bold"]], CellChangeTimes->{{3.718202467818*^9, 3.7182025197622004`*^9}, 3.7182152250772*^9}, FontColor->GrayLevel[0]], "\n\t\nIntroduciendo algunas simplificaciones para las funciones trigonom\ \[EAcute]tricas resultan:" }], "Subsubsection", CellChangeTimes->{{3.7182158258733997`*^9, 3.7182162225234003`*^9}, { 3.7182163227138*^9, 3.718216377295*^9}, {3.7182164128482*^9, 3.7182165165984*^9}, {3.7182166564904003`*^9, 3.7182166589406*^9}, { 3.7182170226247997`*^9, 3.7182170279982*^9}, {3.7182170879856*^9, 3.718217475035*^9}, {3.7182176100002003`*^9, 3.7182176172592*^9}, { 3.7182176526328*^9, 3.7182176529136*^9}, {3.7182176930478*^9, 3.7182179091762*^9}, {3.718217997903*^9, 3.718218239134*^9}, { 3.7182186079144*^9, 3.7182186363496*^9}, {3.7182189022958*^9, 3.7182189693*^9}, {3.7182190225436*^9, 3.718219028145*^9}, { 3.7182193473761997`*^9, 3.7182193570355997`*^9}, {3.7182344321134*^9, 3.7182344428552*^9}, {3.7182344793932*^9, 3.7182345030222*^9}, { 3.718285052854*^9, 3.7182850872148*^9}}, FontSize->16], Cell[BoxData[ RowBox[{"FullSimplify", "[", StyleBox[ RowBox[{"{", RowBox[{ RowBox[{"\[Theta]p", "\[Rule]", RowBox[{"2", "P"}]}], ",", RowBox[{"\[Psi]p", "\[Rule]", FractionBox[ RowBox[{"1", "-", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], RowBox[{"1", "+", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]]}], ",", RowBox[{"\[Phi]p", "\[Rule]", FractionBox["2", RowBox[{"1", "+", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]]}], ",", " ", RowBox[{"Pp", " ", "\[Rule]", " ", RowBox[{ RowBox[{ RowBox[{"-", "8"}], " ", SuperscriptBox[ RowBox[{"Csc", "[", "\[Theta]", "]"}], "3"], " ", SuperscriptBox[ RowBox[{"Sin", "[", FractionBox["\[Theta]", "2"], "]"}], "4"]}], "+", RowBox[{"2", "A", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}]}]}], "}"}], FontWeight->"Bold"], StyleBox["]", FontWeight->"Bold"]}]], "Input", CellChangeTimes->{{3.718285091007*^9, 3.7182850939816*^9}}], Cell[TextData[{ "El verdedero problema aqu\[IAcute] se reduce a integrar la ecuaci\[OAcute]n \ para P\[Theta] y \[Theta]. Vamos a hacerlo num\[EAcute]ricamente.\n\nYa \ tenemos las ecs. diferenciales. Las condiciones iniciales para ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "\[Phi]"], "=", " ", RowBox[{ SubscriptBox["p", "\[Psi]"], "=", " ", "p"}]}], TraditionalForm]]], " est\[AAcute]n contenidas en el par\[AAcute]metro ", Cell[BoxData[ FormBox["A", TraditionalForm]]], ". \n\nComo \[Phi] y \[Psi] son c\[IAcute]clicas, podemos fijar sus valores \ iniciales arbitrariamente. Uno estar\[IAcute]a inclinado a usar \[Phi] = \ \[Psi] = 0, pero ese punto corresponde a un punto singular de los \ \[AAcute]ngulos de Euler, de modo que conviene elegir \[Phi] igual a \ cualquier otro valor.\n\nLas condiciones iniciales para \[Theta] y p\[Theta] \ deben considerarse con mayor cuidado. Al analizar los retratos de fase vimos \ que \[Theta] puede hacer dos cosas bien diferentes: rodear el origen o rodear \ (si los hay) alguno de los puntos de equilibrio que aparecen cuando ", Cell[BoxData[ FormBox["A", TraditionalForm]]], " es mayor que 1/4. En cualquier caso, algo es seguro, para todas las \ trayectorias hay puntos de retorno, en los cuales ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "\[Theta]"], " ", "=", " ", "0"}], TraditionalForm]]], ". De manera que podemos tomar \[EAcute]sa como condici\[OAcute]n inicial \ para ", Cell[BoxData[ FormBox[ SubscriptBox["p", "\[Theta]"], TraditionalForm]]], ".\n\nSi estamos en el caso ", Cell[BoxData[ FormBox[ RowBox[{"A", "<", " ", RowBox[{"1", "/", "4"}]}], TraditionalForm]]], ", para distintos valores iniciales de \[Theta] no obtendremos resultados \ cualitativamente diferentes. Siempre son \[OAcute]rbitas alrededor del \ origen. \n\nSi estamos en el caso ", Cell[BoxData[ FormBox[ RowBox[{"A", ">", " ", RowBox[{"1", "/", "4"}]}], TraditionalForm]]], ", la forma de las \[OAcute]rbitas depender\[AAcute] de si el valor inicial \ de \[Theta] est\[AAcute] dentro o fuera de la curva separatriz. \n\n\t", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{"|", "\[Theta]", "|", RowBox[{"<", " ", SubscriptBox["\[Theta]", "+"]}]}], "=", " ", RowBox[{ RowBox[{"ArcCos", "[", RowBox[{ FractionBox["1", RowBox[{"2", "A"}]], "-", "1"}], "]"}], "\[Rule]"}]}], FontColor->GrayLevel[0]], TraditionalForm]]], StyleBox[" dentro de la separatriz", FontColor->GrayLevel[0]] }], "Subsection", CellChangeTimes->{{3.7182158258733997`*^9, 3.7182162225234003`*^9}, { 3.7182163227138*^9, 3.718216377295*^9}, {3.7182164128482*^9, 3.7182165165984*^9}, {3.7182166564904003`*^9, 3.7182166589406*^9}, { 3.7182170226247997`*^9, 3.7182170279982*^9}, {3.7182170879856*^9, 3.718217475035*^9}, {3.7182176100002003`*^9, 3.7182176172592*^9}, { 3.7182176526328*^9, 3.7182176529136*^9}, {3.7182176930478*^9, 3.7182179091762*^9}, {3.718217997903*^9, 3.718218239134*^9}, { 3.7182186079144*^9, 3.7182186363496*^9}, {3.7182189022958*^9, 3.7182189693*^9}, {3.7182190225436*^9, 3.718219028145*^9}, { 3.7182193473761997`*^9, 3.7182193570355997`*^9}, {3.7182344321134*^9, 3.7182344428552*^9}, {3.7182344793932*^9, 3.7182345030222*^9}, { 3.718285052854*^9, 3.7182851061184*^9}, {3.7182851398606*^9, 3.7182851469835997`*^9}}, FontSize->16], Cell["\<\ Probamos primero como marcha el asunto integrando un caso particular: \ \>", "Subsection", CellFrame->{{0, 0}, {0, 0.5}}, CellChangeTimes->{{3.7182199258504*^9, 3.7182199532942*^9}, { 3.718220118278*^9, 3.7182201241476*^9}, {3.7182345787416*^9, 3.7182346014726*^9}}, FontSize->18], Cell["\<\ (Ustedes cambien el valor de Apr y el de \[Theta]0. Prueben con Apr mayor o \ menor que 1/4, y -en el primer caso- con \[Theta]0 mayor o menor que \ \[Theta]mas)\ \>", "Text", CellChangeTimes->{{3.718220126863*^9, 3.7182201922525997`*^9}, 3.7182852022574*^9, {3.7184872502682*^9, 3.7184872532488003`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"Apr", " ", "=", " ", RowBox[{"Rationalize", "[", "0.33", "]"}]}], ";"}], " ", RowBox[{"(*", RowBox[{ "uso", " ", "Rationalize", " ", "para", " ", "generar", " ", "un", " ", RowBox[{"n\[UAcute]mero", " ", "'"}], RowBox[{"exacto", "'"}]}], "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{ "Quiero", " ", "tener", " ", "a", " ", "la", " ", "vista", " ", "el", " ", "\[AAcute]ngulo", " ", "que", " ", "separa", " ", "las", " ", "condiciones", " ", "iniciales", " ", "que", " ", "para", " ", "A"}], " ", ">", " ", RowBox[{ RowBox[{"1", "/", "4"}], " ", "caen", " ", "de", " ", "un", " ", "lado", " ", "o", " ", "del", " ", "otro", " ", "de", " 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