(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 13.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 6193215, 102766] NotebookOptionsPosition[ 6178538, 102520] NotebookOutlinePosition[ 6179074, 102539] CellTagsIndexPosition[ 6179031, 102536] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["3.029 Spring 2022\[LineSeparator]Additional Reading Week 01: 01/31 - \ 02/04", "Subtitle", CellChangeTimes->{{3.8525512993398438`*^9, 3.8525513206118402`*^9}, { 3.852652054138073*^9, 3.8526520591301193`*^9}, {3.852840289958778*^9, 3.852840345460693*^9}},ExpressionUUID->"895e2cbe-e1e7-4314-9198-\ d042d761ec59"], Cell[CellGroupData[{ Cell["Euler\[CloseCurlyQuote]s Equations of Rigid Bodies", "Chapter", CellChangeTimes->{{3.852551340964005*^9, 3.852551346980482*^9}, 3.8526520737398577`*^9, {3.852840413215933*^9, 3.8528404399249563`*^9}},ExpressionUUID->"064dca75-ae72-43ad-8bb6-\ e9a63ab98e3f"], Cell[TextData[{ "Remember this quote from the ", Cell[BoxData[ FormBox[ SuperscriptBox["1", "st"], TraditionalForm]],ExpressionUUID-> "03b56107-b51f-46a9-b7db-1a5fdcc115c5"], " lecture?\[LineSeparator]\[OpenCurlyDoubleQuote]Undergraduate [physics] \ students come in expecting that the hardest thing they\[CloseCurlyQuote]ll \ have to learn will be relativity or quantum mechanics. \ \[LineSeparator]Actually those are the most ", StyleBox["novel ", FontSlant->"Italic"], "topics. The hardest thing that an undergraduate physics student must learn \ is the classical dynamics of spinning tops\[CloseCurlyDoubleQuote]" }], "Item", CellChangeTimes->{ 3.852840461251688*^9, {3.8528404939807167`*^9, 3.852840494863771*^9}, { 3.8528405674065*^9, 3.852840602838396*^9}},ExpressionUUID->"c2b5cb2f-2674-4137-a2e7-\ ea9d7c8c36d0"], Cell["\<\ This (optional) reading will introduce the relevant concepts and equations in \ modeling rigid bodies such as spinning tops\ \>", "Item", CellChangeTimes->{ 3.852840461251688*^9, {3.8528404939807167`*^9, 3.852840494863771*^9}, { 3.8528405674065*^9, 3.852840650081538*^9}, {3.852840683425118*^9, 3.852840683426093*^9}},ExpressionUUID->"dce008cd-16f1-45e1-b12d-\ 6ad36a6e66bc"], Cell[CellGroupData[{ Cell["\<\ The exposition here follows that in the excellent book \ \[OpenCurlyDoubleQuote]Structure and Interpretation of Classical Mechanics\ \[CloseCurlyDoubleQuote] by Prof. Sussman and Wisdom\ \>", "Item", CellChangeTimes->{ 3.852840461251688*^9, {3.8528404939807167`*^9, 3.852840494863771*^9}, { 3.8528405674065*^9, 3.852840650081538*^9}, {3.852840684282563*^9, 3.8528407314640617`*^9}},ExpressionUUID->"4e9b9956-3753-4803-8a80-\ a86d69497ff7"], Cell["\<\ Prof. Sussman and Wisdom teach an advanced class on these topics, 12.650J. If \ you find this reading to be of interest, you might enjoy taking that class in \ the future.\ \>", "Subitem", CellChangeTimes->{ 3.852840461251688*^9, {3.8528404939807167`*^9, 3.852840494863771*^9}, { 3.8528405674065*^9, 3.852840650081538*^9}, {3.852840684282563*^9, 3.8528408038845778`*^9}, {3.8528408613320227`*^9, 3.852840861621523*^9}},ExpressionUUID->"51562415-927d-4f7b-a9b7-\ b21d7741968d"] }, Open ]], Cell[CellGroupData[{ Cell["Rotation Kinematics", "Section", CellChangeTimes->{{3.852840455604553*^9, 3.852840460004809*^9}, { 3.853149169795454*^9, 3.853149172627393*^9}, {3.853150239592525*^9, 3.85315024595256*^9}, {3.853150336672606*^9, 3.8531503400887823`*^9}, 3.853150402098156*^9},ExpressionUUID->"2d98b11d-7963-43be-9bbd-\ e977465201ae"], Cell[CellGroupData[{ Cell["We start by examining the kinetic energy of a rigid body", "Item", CellChangeTimes->{ 3.852840461251688*^9, {3.8528404939807167`*^9, 3.852840494863771*^9}, { 3.8528405674065*^9, 3.852840650081538*^9}, 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"448a405d-aa04-499c-a7e2-32a3a632cc4c"], ", and velocities ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ SubscriptBox[ OverscriptBox[ OverscriptBox[ StyleBox["r", "TI"], "\[RightVector]"], "."], "\[Alpha]"], TraditionalForm], "errors" -> {}, "input" -> "\\dot{\\vec{r}}_{\\alpha}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "d20543f3-08ab-4182-8982-b1e78a3c481a"], " with fixed positional constraints between them." }], "Subitem", CellChangeTimes->{ 3.852840461251688*^9, {3.8528404939807167`*^9, 3.852840494863771*^9}, { 3.8528405674065*^9, 3.852840650081538*^9}, {3.852840684282563*^9, 3.8528408095161*^9}, {3.852840894115223*^9, 3.852840931505925*^9}, { 3.853149132555407*^9, 3.85314915713923*^9}, {3.853149191523726*^9, 3.853149253243885*^9}, {3.853149292492627*^9, 3.853149328655552*^9}},ExpressionUUID->"652dc8ea-6163-48e8-956c-\ 3bd06d44fe14"], Cell["The kinetic energy is the simply given by", "Subitem", CellChangeTimes->{ 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"errors" -> {}, "input" -> "\\sum_{\\alpha}m_{\\alpha}\\dot{\\vec{\\xi}}_{\\alpha} =0", "state" -> "Boxes"|>, "TeXAssistantTemplate"]}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.8531493955569363`*^9, 3.853149413359498*^9}, 3.853149722746519*^9, {3.853149784878642*^9, 3.85314978549498*^9}, { 3.853149906552137*^9, 3.853149909000956*^9}, {3.853150007824274*^9, 3.853150039842774*^9}},ExpressionUUID->"f54eeb4d-6ce2-42bd-8dc3-\ 39a1a1bc7294"], Cell["and thus drop the second term to give", "Subitem", CellChangeTimes->{{3.853150042283321*^9, 3.8531500518066587`*^9}},ExpressionUUID->"eaab4bac-c389-4f14-951d-\ f27d4d12432a"], Cell[BoxData[ RowBox[{"\t", TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["T", "TI"], "\[LongEqual]", UnderscriptBox["\[Sum]", "\[Alpha]", LimitsPositioning -> True], FractionBox["1", "2"], SubscriptBox[ StyleBox["m", "TI"], "\[Alpha]"], OverscriptBox[ OverscriptBox[ StyleBox["X", "TI"], "\[RightVector]"], "."], "\[CenterDot]", OverscriptBox[ OverscriptBox[ 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CellChangeTimes->{{3.853150120844096*^9, 3.8531501566341753`*^9}},ExpressionUUID->"2fb63d3e-5e90-4431-aa59-\ 7ade851d0d69"], Cell["kinetic energy of motion of the total center of mass", "Subitem", CellChangeTimes->{{3.853150120844096*^9, 3.853150157508768*^9}},ExpressionUUID->"5474c1c8-a749-4957-a33c-\ 94ae544b0854"], Cell["kinetic energy of rotation about the center of mass", "Subitem", CellChangeTimes->{{3.853150120844096*^9, 3.853150167951516*^9}},ExpressionUUID->"efdcd453-6836-4855-a600-\ ece5d2ef32f1"] }, Open ]], Cell["\<\ Finally, as we saw with the double pendulum, it\[CloseCurlyQuote]s convenient \ to express the velocities in terms of generalized \ coordinates.\[LineSeparator]To do this, we introduce the following formula:\ \>", "Item", CellChangeTimes->{{3.853150449293395*^9, 3.853150517305606*^9}},ExpressionUUID->"4fb85223-213b-4897-bb47-\ 6aaab65be34a"], Cell[BoxData[ RowBox[{"\t", TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SubscriptBox[ OverscriptBox[ 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the angular velocity vector, and is related to the rotation matrix \ acting on the rigid body\[LineSeparator](Note: a detailed derivation is \ outside the scope of this reading - but can be found ", ButtonBox["here", BaseStyle->"Hyperlink", ButtonData->{ URL["https://mitpress.mit.edu/sites/default/files/titles/content/sicm_\ edition_2/chapter002.html#h1-8"], None}, ButtonNote-> "https://mitpress.mit.edu/sites/default/files/titles/content/sicm_edition_\ 2/chapter002.html#h1-8"], ")" }], "Subitem", CellChangeTimes->{{3.853150598525981*^9, 3.853150630249825*^9}, { 3.853150664363915*^9, 3.853150717876709*^9}, 3.853150931753601*^9},ExpressionUUID->"872628cf-f38b-4c59-aa3b-\ a7f7de14fb04"] }, Open ]], Cell[CellGroupData[{ Cell["Moments of Inertia", "Section", CellChangeTimes->{{3.8531503451928377`*^9, 3.8531503477368097`*^9}, { 3.853150404873208*^9, 3.853150407689146*^9}},ExpressionUUID->"1159aead-4f3d-4857-8755-\ 1cc45f0d9410"], Cell["We can now express the rotational kinetic energy as", 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our degrees of freedom:", "Item", CellChangeTimes->{{3.853190896321587*^9, 3.85319090704088*^9}, { 3.853190981322287*^9, 3.853190983890236*^9}, {3.853191018930233*^9, 3.853191037550688*^9}},ExpressionUUID->"57a05f2a-bd13-4399-a913-\ 9171631acb5e"], Cell["We have the three Euler angles", "Subitem", CellChangeTimes->{{3.853190896321587*^9, 3.85319090704088*^9}, { 3.853190981322287*^9, 3.853190983890236*^9}, {3.853191018930233*^9, 3.853191058450088*^9}},ExpressionUUID->"9f664b04-e1e4-4e26-8541-\ b8523d5a831b"], Cell["and their three derivatives", "Subitem", CellChangeTimes->{{3.853190896321587*^9, 3.85319090704088*^9}, { 3.853190981322287*^9, 3.853190983890236*^9}, {3.853191018930233*^9, 3.8531910729848223`*^9}},ExpressionUUID->"3db338c8-00a2-4533-aa9d-\ 3c75f17f4f7d"] }, Open ]], Cell[CellGroupData[{ Cell["Now, let\[CloseCurlyQuote]s count our constants of motion (i.e. \ constraints)", "Item", CellChangeTimes->{{3.853190896321587*^9, 3.85319090704088*^9}, { 3.853190981322287*^9, 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