(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. 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[ 5817, 178]*) (*NotebookOutlinePosition[ 6516, 202]*) (* CellTagsIndexPosition[ 6472, 198]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Mathematica Lab Exercise #6", "Subtitle"], Cell[CellGroupData[{ Cell["Variational Calculus & Lagrange's Equations of Motion", "Subsection"], Cell[TextData[{ "Lagrange's equations can be derived by assuming that the action is \ minimized on a physical trajectory. The action is:\n\n ", StyleBox["S=", FontSize->16], Cell[BoxData[ \(TraditionalForm\`\[Integral]\(\(L\)\((\)\(\(\(\(x, x, t\)\()\)\)\& . \) \[DifferentialD]t\)\)\)], FontSize->14], ",\n \n", "where L is the Lagrangian. Anytime there is a conservative potential, V, \ L=T-V, where T is the kinetic energy. We will use T=", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "m", Cell[BoxData[ \(TraditionalForm\`v\^2\)]], " and study several simple systems. We will see in class that if we vary \ the function x(t) over all possible functions, with the endpoints fixed, the \ action is extremized by a function that satisfies:\n\n ", StyleBox[" ", FontSize->16], Cell[BoxData[ \(TraditionalForm\`\[DifferentialD]L\/\[DifferentialD]x\)], FontSize->16], StyleBox["- ", FontSize->16], Cell[BoxData[ \(TraditionalForm\`d\/dt\)], FontSize->16], Cell[BoxData[ \(TraditionalForm\`\[DifferentialD]L\/\[DifferentialD]v\)], FontSize->16], StyleBox["=0,\n \n", FontSize->14], "where v=", Cell[BoxData[ \(TraditionalForm\`\(x\& . \)\)]], ".\n " }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[TextData[{ "Problem 1) ", "As a first exercise, assume V=0 and derive the equation of motion. Solve \ it and then compute the action between time ", Cell[BoxData[ \(TraditionalForm\`t\_1\)]], "and ", Cell[BoxData[ \(TraditionalForm\`t\_2\)]], ". To calculate the action you must first determine x( ", Cell[BoxData[ \(TraditionalForm\`t\_1\)]], ") and x(", Cell[BoxData[ \(TraditionalForm\`t\_2\)]], "). Convince yourself that this completely specifies x(t) if x(t) satisfies \ the equation of motion, and compute the action. Study what happens when you \ change x(t) slightly, but you must keep x( ", Cell[BoxData[ \(TraditionalForm\`t\_1\)]], ") and x(", Cell[BoxData[ \(TraditionalForm\`t\_2\)]], ") fixed, so you can only vary the function between the initial and final \ times. For example, you can add \[Alpha] sin(\[Pi]", Cell[BoxData[ \(TraditionalForm\`t\/\(\(t\_\(\(2\)\(-\)\)\) t\_1\)\)]], ") to the solution, and recompute the action. Try to vary the function and \ get a lower action." }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[BoxData[ \(Clear["\"]\)], "Input"], Cell[TextData[{ "Problem 2) Next consider V=", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "k", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], ". Find the equation of motion, solve it, find the specific solution for a \ given x( ", Cell[BoxData[ \(TraditionalForm\`t\_1\)]], ") and x(", Cell[BoxData[ \(TraditionalForm\`t\_2\)]], ") and then compute the action. Again, vary x(t) and try to find a lower \ action. What happens if you use the solution to the first problem for x(t)." }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[BoxData[ \(Clear["\"]\)], "Input"], Cell["\<\ Problem 3) Now think about a bead on a ring of radius R. Write the \ velocity in terms of r and \[Theta], and use this to write the Lagrangian for \ a \"free\" particle on a ring. Determine the equation of motion and solve it. \ Next suspend the ring in a gravitational field and write V in terms of r and \ \[Theta]. Given L=T-V, determine the equation of motion. Think about how you \ might test whether the solution to this equation of motion minimizes the \ action.\ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[BoxData[ \(Clear["\"]\)], "Input"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.2 for Macintosh", ScreenRectangle->{{0, 1280}, {0, 998}}, WindowSize->{721, 764}, WindowMargins->{{16, Automatic}, {Automatic, 73}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 47, 0, 65, "Subtitle"], Cell[CellGroupData[{ Cell[1848, 57, 75, 0, 46, "Subsection"], Cell[1926, 59, 1410, 41, 257, "Text"], Cell[3339, 102, 1142, 31, 125, "Text"], Cell[4484, 135, 54, 1, 27, "Input"], Cell[4541, 138, 584, 18, 86, "Text"], Cell[5128, 158, 54, 1, 27, "Input"], Cell[5185, 161, 547, 10, 102, "Text"], Cell[5735, 173, 54, 1, 27, "Input"] }, Open ]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)