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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[ 81144, 1960]*) (*NotebookOutlinePosition[ 110542, 3027]*) (* CellTagsIndexPosition[ 110498, 3023]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox[ "A brief summary of important laws in physics written as quaternions"]], "Subtitle", CellMargins->{{1, 48}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times New Roman"], Cell[TextData[StyleBox["doug "]], "Subsubtitle", CellMargins->{{1, 48}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->16], Cell[TextData[StyleBox[ "Classical physics\n Newton's 2nd law in an inertial reference frame, \ Cartesian coordinates\n Newton's 2nd law in a rotating reference frame\n \ Newton's 2nd law in an inertial reference frame, polar coordinates, for a \ central force\n The wave equation\n The simple harmonic oscillator\n \ A test if a force in conservative\n \nSpecial relativity\n The \ invariant interval squared between two events\n Transformations between \ inertial reference frames for a boost along the x axis\n \n\ Electromagnetism\n The electric and magnetic fields\n Electrostatics\n \ Magnetostatics\n The gauge-invariant Maxwell equations\n The Maxwell \ equations in the Lorenz gauge\n The Maxwell equations in the light gauge \ (QED?)\n The Lorentz force\n \nQuantum mechanics\n A complete inner \ product space\n The wave function\n The Schr\[ODoubleDot]dinger \ equation\n The Klein-Gordon equation\n Time reversal\n \nGravity\n \ The gravitational field\n The Maxwell gravitational wave equations\n \ The Lorentz gravitational force\n"]], "Text", CellMargins->{{1, 48}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[TextData[StyleBox[ "The following laws of physics are generated by quaternion operators acting \ on the appropriate quaternion-valued functions. 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\ 2\ \[Gamma]\ \[Beta]\ t\ x\ + \ \[Gamma]\ x\^2\)\/\(t\^2 + \ x\^2\), \(\(-\[Gamma]\)\ \[Beta] \((t\^2 - \ x\^2)\)\)\/\(t\^2 + \ x\^2\), 0, 0]\n.q[t, x, 0, 0])\).{1, 0, 0, 0}]\n == {\((t - x\ \[Beta])\)\ \[Gamma], \((x - t\ \[Beta])\)\ \[Gamma], 0, 0}\)], "Input", CellMargins->{{1, 48}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ \(L = \((\(\[Gamma]\ t\^2\ - \ 2\ \[Gamma]\ \[Beta]\ t\ x\ + \ \[Gamma]\ x\^2\)\/\(t\^2 + \ x\^2\), \(\(-\[Gamma]\)\ \[Beta] \((t\^2 - \ x\^2)\)\)\/\(t\^2 + \ x\^2\), 0, 0)\)\), \(q\ -> \ q\^\[Prime]\ = \ Lq\t\), \(\((t, x, 0, 0)\) -> \((t\^\[Prime], x\^\[Prime], 0, 0)\)\n \t\t\t\t\t\t\t\t\t\t\t = \((\[Gamma]\ t - \[Gamma]\ x\ \[Beta]\ , \(-\[Gamma]\)\ \[Beta]\ t + \[Gamma]\ x\ , 0, 0)\)\)}], "Input", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["In general for a boost along the x axis:", "Text", CellMargins->{{1, 48}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[BoxData[ \({t', x', y', z'} == \n Simplify[1\/\(t\^2 + \ x\^2 + \ y\^2\ + \ z\^2\)\n q[ \[Gamma]\ t\^2 - 2\ \[Gamma]\ \[Beta]\ t\ x\ + \ \[Gamma]\ x\^2\ + \ y\^2\ + \ z\^2, \ \(-\ \[Gamma]\)\ \[Beta] \((t\^2 - \ x\^2)\), \n t\ y\ - \ x\ z\ + \ \[Gamma]\ y \((\(-t\)\ + \ \[Beta]\ x)\)\ + \ \[Gamma]\ z \((\(-\[Beta]\)\ t\ + \ x)\), \n x\ y\ + \ t\ z\ + \ \[Gamma]\ y \((\[Beta]\ t\ - \ x)\)\ \ + \ \[Gamma]\ z \((\(-t\)\ + \ \[Beta]\ x)\)].\n q[t, x, y, z].{1, 0, 0, 0}]\n == {\((t - x\ \[Beta])\)\ \[Gamma], \((x - t\ \[Beta])\)\ \[Gamma], y, z}\)], "Input", CellMargins->{{1, 48}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ \(L = 1\/\(t\^2 + \ x\^2 + \ y\^2\ + \ z\^2\)\), \(\((\[Gamma]\ t\^2 - 2\ \[Gamma]\ \[Beta]\ t\ x\ + \ \[Gamma]\ x\^2 + \ y\^2 + \ z\^2, \(-\ \[Gamma]\)\ \[Beta] \((t\^2 - \ x\^2)\), \n t\ y - x\ z + \ \[Gamma]\ y \((\(-t\)\ + \ \[Beta]\ x)\)\ + \ \[Gamma]\ z \((\(-\[Beta]\)\ t\ + \ x)\), \n x\ y + \ t\ z + \ \[Gamma]\ y \((\[Beta]\ t\ - \ x)\) + \ \[Gamma]\ z \((\(-t\)\ + \ \[Beta]\ x)\))\)\), \(q\ -> \ q\^\[Prime]\ = \ Lq\t\), \(\((t, x, 0, 0)\) -> \((t\^\[Prime], x\^\[Prime], 0, 0)\)\n \t\t\t\t\t\t\t\t\t\t\t = \((\[Gamma]\ t - 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