Raj Chaklashiya Honors Contract Report 105B.nb

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     RowBox[{"Aw", ",", "As", ",", "z"}], "]"}], ",", " ", 
    "\"\<\\nso the new dimensionless vector potential is\\na = \>\"", ",", 
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     RowBox[{"aw", ",", "as"}], "]"}]}], "]"}], 
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   RowBox[{"r_", ",", "P_", ",", "t_"}], "]"}], ";"}], "\[IndentingNewLine]", 
 RowBox[{"Print", "[", 
  "\"\<\\nWe will assume no transverse variation (Az = 0) so that we can \
assume Px and Py to be constants. Here we will declare these constants to be \
0. Additionally, the signal field is taken to be circularly \
polarized.\\n\>\"", "]"}], "\[IndentingNewLine]", 
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   RowBox[{
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  RowBox[{
  "\"\<We also introduce the dimensionless vector \
potentials\\n\!\(\*SuperscriptBox[\(\[Mu]\), \(2\)]\) = \>\"", ",", 
   RowBox[{"\[Mu]square", "[", "z", "]"}], ",", "\"\<\\n\[Psi] = \>\"", ",", 
   RowBox[{"\[Psi]", "[", 
    RowBox[{"z", ",", "t"}], "]"}], ",", "\"\<\\nand \[Gamma] = \>\"", ",", 
   RowBox[{"\[Gamma]", "/.", "\[Gamma]rule"}], ",", " ", 
   "\"\<\\n Here, \[Psi] represents the phase of the electron and \[Gamma] \
represents the energy of the electron.\>\""}], "]"}], "\[IndentingNewLine]", 
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  RowBox[{
   RowBox[{"H1", "[", 
    RowBox[{"E0_", ",", "t_", ",", "z_"}], "]"}], "=", 
   RowBox[{
    RowBox[{"-", "m"}], " ", "c", " ", 
    RowBox[{"Sqrt", "[", 
     RowBox[{
      RowBox[{"-", 
       SuperscriptBox[
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       RowBox[{"Cos", "[", 
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         RowBox[{"z", ",", "t"}], "]"}], "]"}]}]}], "]"}]}]}], 
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 RowBox[{"Print", "[", 
  RowBox[{
  "\"\<Now with these substitutions and the assumption that there is no \
transverse variation, the new Hamiltonian\\n H1 = \>\"", ",", 
   RowBox[{"H1Hold", "[", 
    RowBox[{"E0", ",", "t", ",", "z"}], "]"}], ",", 
   "\"\<\\n which is now much easier to solve, as it is a one-dimensional \
Hamiltonian of relativistic motion.\\n\>\""}], "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"\[Psi]rule", "=", 
   RowBox[{"\[Psi]", "\[Rule]", " ", 
    RowBox[{"\[Psi]", "[", 
     RowBox[{"z", ",", "t"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", 
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      RowBox[{"\[Psi]", "[", 
       RowBox[{"z", ",", "t"}], "]"}], "]"}]}]}]}], 
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 RowBox[{
  RowBox[{
   RowBox[{"H1", "[", 
    RowBox[{"E0_", ",", "t_", ",", "z_"}], "]"}], "=", 
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    RowBox[{"-", "m"}], " ", "c", " ", 
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       RowBox[{"Cos", "[", 
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  ";"}], "\[IndentingNewLine]", 
 RowBox[{"Print", "[", 
  RowBox[{
  "\"\<To convert this Hamiltonian to one in terms of the canonical \
coordinates \[Psi] and \[Gamma], which are the new canonical position and \
momentum, respectively, we use the generating function\\nG1 = \>\"", " ", 
   RowBox[{"G1", "[", 
    RowBox[{"\[Gamma]", ",", "\[Psi]"}], "]"}]}], 
  "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"G1", "[", 
    RowBox[{"\[Gamma]_", ",", "\[Psi]_"}], "]"}], "=", 
   RowBox[{
    RowBox[{"E0", " ", "t"}], "-", 
    RowBox[{
     RowBox[{"(", " ", 
      RowBox[{"1", "-", 
       RowBox[{"m", " ", 
        FractionBox[
         SuperscriptBox["c", "2"], "\[Omega]s"]}]}], ")"}], "\[Gamma]", " ", 
     RowBox[{"\[Psi]", "[", 
      RowBox[{"z", ",", "t"}], "]"}]}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"E0rule", "=", 
    RowBox[{"E0", "\[Rule]", " ", 
     RowBox[{"m", " ", 
      SuperscriptBox["c", "2"], " ", "\[Gamma]"}]}]}], ";"}], 
  "\[IndentingNewLine]"}], "\[IndentingNewLine]", 
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  RowBox[{
   RowBox[{"DG1z", "[", 
    RowBox[{"\[Gamma]_", ",", "\[Psi]_", ",", "z_"}], "]"}], "=", 
   RowBox[{
    RowBox[{"D", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"G1", "[", 
         RowBox[{"\[Gamma]", ",", "\[Psi]"}], "]"}], "/.", "E0rule"}], "/.", 
       "trule"}], ",", "z"}], "]"}], " ", "//", "FullSimplify"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"H2", "[", 
    RowBox[{"\[Gamma]_", ",", "\[Psi]_", ",", "z_"}], "]"}], " ", "=", 
   RowBox[{
    RowBox[{
     FractionBox["\[Omega]s", 
      RowBox[{"m", " ", 
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        RowBox[{"\[Gamma]", ",", "\[Psi]", ",", "z"}], "]"}]}], ")"}]}], "//",
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 RowBox[{"Print", "[", 
  RowBox[{
  "\"\<When we take the partial derivative of the generating function with \
respect to z, we get\\n G1'[z] = \>\"", ",", 
   RowBox[{"DG1z", "[", 
    RowBox[{"\[Gamma]", ",", "\[Psi]", ",", "z"}], "]"}], ",", " ", 
   "\"\<\\n and we find that when we add this quantity to H1 and multiply by \
the constant \!\(\*FractionBox[\(\[Omega]s\), \(m\\\ \*SuperscriptBox[\(c\), \
\(2\)]\)]\), we get\\n H2 = \>\"", ",", 
   RowBox[{"H2", "[", 
    RowBox[{"\[Gamma]", ",", "\[Psi]", ",", "z"}], "]"}], ",", " ", 
   "\"\<\\n which is the new Hamiltonian in terms of \[Gamma] (the new \
canonical momentum), \[Psi] (the new canonical coordinate), and z (the new \
independent variable). It is more convenient for us to use these new \
coordinates because the beam velocity about matches the bunching wave for \
when \[Psi] is almost constant.\\n\>\""}], "]"}], "\[IndentingNewLine]", 
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  RowBox[{
   RowBox[{"H3", "[", 
    RowBox[{"\[Gamma]_", ",", "\[Psi]_", ",", "z_"}], "]"}], " ", "=", " ", 
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    RowBox[{"\[Gamma]", " ", 
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       RowBox[{"2", " ", 
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        RowBox[{"as", "[", "z", "]"}], 
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  RowBox[{"\[Delta]ksrule", "=", 
   RowBox[{"\[Delta]ks", "\[Rule]", " ", "0"}]}], 
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    RowBox[{
     RowBox[{
      RowBox[{"\[Gamma]", " ", 
       RowBox[{"(", 
        RowBox[{
         RowBox[{"kw", "[", "z", "]"}], "+", "\[Delta]ks"}], ")"}]}], "+", 
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  ";"}], "\[IndentingNewLine]", 
 RowBox[{"Print", "[", 
  RowBox[{
  "\"\<The new Hamiltonian H3 is obtained from making the approximations that \
kw >> ks, \!\(\*SuperscriptBox[\(\[Gamma]\), \(\(2\)\(\\\ \)\)]\)>> \
\!\(\*SuperscriptBox[\(\[Mu]\), \(2\)]\), \!\(\*SuperscriptBox[\(\[Gamma]\), \
\(\(2\)\(\\\ \)\)]\)>> aw as, and aw >> as. Thus, \\nH3 = \>\"", " ", 
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    RowBox[{"\[Gamma]", ",", "\[Psi]", ",", "z"}], "]"}]}], 
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  RowBox[{
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 RowBox[{
  RowBox[{"\[Gamma]rDz", "=", 
   RowBox[{
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      RowBox[{
       RowBox[{"\[Gamma]r", "[", "z", "]"}], ",", "z"}], "]"}], "]"}], 
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     FractionBox[
      RowBox[{"-", 
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     RowBox[{"Sin", "[", "\[Psi]r", "]"}]}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{"Print", "[", 
  RowBox[{
  "\"\<The Synchronous Energy \[Gamma]r and Synchronous Phase \[Psi]r are \
defined such that\\n\[Gamma]r[z] = \>\"", ",", 
   RowBox[{"\[Gamma]r", "[", "z", "]"}], ",", " ", "\"\<\\n and \\n \>\"", 
   ",", " ", "\[Gamma]rDz", ",", " ", 
   "\"\<\\n The synchronous energy is where \[Gamma] - \[Gamma]r and \[Psi] - \
\[Psi]r change very little throughout the motion, such that we can treat them \
like constants of motion. Electrons will thus exhibit stable oscillations \
about such values.\>\""}], "]"}], "\[IndentingNewLine]", 
 RowBox[{"Print", "[", 
  "\"\<For motion about the Synchronous Energy, we use H4 = H3 + \[Psi] D[\
\[Gamma]r[z],z], and we will examine this motion in the next section.\>\"", 
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   "\[InvisibleSpace]", "\<\"\\n with P being the momentum of the electron, m \
being the electron mass, c being the speed of light, and A being the vector \
potential. Here, the conjugate momenta are Px, Py, and Pz, and the conjugate \
positions are x, y, and z. t is the independent variable. H0 is equal to E0, \
the energy of the system.\\n\"\>"}],
  SequenceForm[
  "The original Hamiltonian is\nH0 = E0 = ", $CellContext`c ($CellContext`c^2 \
$CellContext`m^2 + (((-$CellContext`Ax)/$CellContext`c) $CellContext`e + \
$CellContext`Px)^2 + (((-$CellContext`Ay)/$CellContext`c) $CellContext`e + \
$CellContext`Py)^2 + (((-$CellContext`Az)/$CellContext`c) $CellContext`e + \
$CellContext`Pz)^2)^Rational[1, 2], 
   "\n with P being the momentum of the electron, m being the electron mass, \
c being the speed of light, and A being the vector potential. Here, the \
conjugate momenta are Px, Py, and Pz, and the conjugate positions are x, y, \
and z. t is the independent variable. H0 is equal to E0, the energy of the \
system.\n"],
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Cell[BoxData[
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  RowBox[{"\<\"The first modified Hamiltonian,\\nH1 = -Pz = \"\>", 
   "\[InvisibleSpace]", 
   RowBox[{"-", 
    FractionBox[
     RowBox[{
      RowBox[{"Az", " ", "e"}], "+", 
      SqrtBox[
       RowBox[{
        RowBox[{
         RowBox[{"-", 
          SuperscriptBox["Ax", "2"]}], " ", 
         SuperscriptBox["e", "2"]}], "+", 
        SuperscriptBox["E0", "2"], "+", 
        RowBox[{"2", " ", "Ax", " ", "c", " ", "e", " ", "Px"}], "-", 
        RowBox[{
         SuperscriptBox["c", "2"], " ", 
         RowBox[{"(", 
          RowBox[{
           RowBox[{
            SuperscriptBox["c", "2"], " ", 
            SuperscriptBox["m", "2"]}], "+", 
           SuperscriptBox["Px", "2"]}], ")"}]}], "-", 
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{
           RowBox[{"Ay", " ", "e"}], "-", 
           RowBox[{"c", " ", "Py"}]}], ")"}], "2"]}]]}], "c"]}], 
   "\[InvisibleSpace]", "\<\"\\nuses -E0 as the conjugate momentum instead of \
Pz and z as the independent variable instead of t. The new momenta are Px, \
Py, and -E0, and the new coordinates are x, y, and t. H1 is equal to -Pz now.\
\\n\"\>"}],
  SequenceForm[
  "The first modified Hamiltonian,\nH1 = -Pz = ", ((-1)/$CellContext`c) \
($CellContext`Az $CellContext`e + ((-$CellContext`Ax^2) $CellContext`e^2 + \
$CellContext`E0^2 + (((
          2 $CellContext`Ax) $CellContext`c) $CellContext`e) $CellContext`Px - \
$CellContext`c^2 ($CellContext`c^2 $CellContext`m^2 + $CellContext`Px^2) - \
($CellContext`Ay $CellContext`e - $CellContext`c $CellContext`Py)^2)^
     Rational[1, 2]), 
   "\nuses -E0 as the conjugate momentum instead of Pz and z as the \
independent variable instead of t. The new momenta are Px, Py, and -E0, and \
the new coordinates are x, y, and t. H1 is equal to -Pz now.\n"],
  Editable->False]], "Print",
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Cell[BoxData[
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  RowBox[{"\<\"The vector potential can be written as a sum of the vector \
potential of the wiggler field,\\nAw[z] = \"\>", "\[InvisibleSpace]", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{"-", 
       RowBox[{"Aw", "[", "z", "]"}]}], " ", 
      RowBox[{"Cos", "[", 
       RowBox[{
        SubsuperscriptBox["\[Integral]", "0", "z"], 
        RowBox[{
         RowBox[{"kw", "[", "z1", "]"}], 
         RowBox[{"\[DifferentialD]", "z1"}]}]}], "]"}]}], ",", 
     RowBox[{
      RowBox[{"-", 
       RowBox[{"Aw", "[", "z", "]"}]}], " ", 
      RowBox[{"Sin", "[", 
       RowBox[{
        SubsuperscriptBox["\[Integral]", "0", "z"], 
        RowBox[{
         RowBox[{"kw", "[", "z1", "]"}], 
         RowBox[{"\[DifferentialD]", "z1"}]}]}], "]"}]}], ",", "0"}], "}"}], 
   "\[InvisibleSpace]", "\<\"\\n and the vector potential of the signal \
field,\\nAs[z] = \"\>", "\[InvisibleSpace]", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{"As", "[", "z", "]"}], " ", 
      RowBox[{"Cos", "[", 
       RowBox[{
        RowBox[{"t", " ", "\[Omega]s"}], "-", 
        RowBox[{
         SubsuperscriptBox["\[Integral]", "0", "z"], 
         RowBox[{
          RowBox[{"ks", "[", "z1", "]"}], 
          RowBox[{"\[DifferentialD]", "z1"}]}]}]}], "]"}]}], ",", 
     RowBox[{
      RowBox[{"As", "[", "z", "]"}], " ", 
      RowBox[{"Sin", "[", 
       RowBox[{
        RowBox[{"t", " ", "\[Omega]s"}], "-", 
        RowBox[{
         SubsuperscriptBox["\[Integral]", "0", "z"], 
         RowBox[{
          RowBox[{"ks", "[", "z1", "]"}], 
          RowBox[{"\[DifferentialD]", "z1"}]}]}]}], "]"}]}], ",", "0"}], 
    "}"}], "\[InvisibleSpace]", "\<\"\\nto produce\\nA = \"\>", 
   "\[InvisibleSpace]", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"As", "[", "z", "]"}], " ", 
       RowBox[{"Cos", "[", 
        RowBox[{
         RowBox[{"t", " ", "\[Omega]s"}], "-", 
         RowBox[{
          SubsuperscriptBox["\[Integral]", "0", "z"], 
          RowBox[{
           RowBox[{"ks", "[", "z1", "]"}], 
           RowBox[{"\[DifferentialD]", "z1"}]}]}]}], "]"}]}], "-", 
      RowBox[{
       RowBox[{"Aw", "[", "z", "]"}], " ", 
       RowBox[{"Cos", "[", 
        RowBox[{
         SubsuperscriptBox["\[Integral]", "0", "z"], 
         RowBox[{
          RowBox[{"kw", "[", "z1", "]"}], 
          RowBox[{"\[DifferentialD]", "z1"}]}]}], "]"}]}]}], ",", 
     RowBox[{
      RowBox[{
       RowBox[{"As", "[", "z", "]"}], " ", 
       RowBox[{"Sin", "[", 
        RowBox[{
         RowBox[{"t", " ", "\[Omega]s"}], "-", 
         RowBox[{
          SubsuperscriptBox["\[Integral]", "0", "z"], 
          RowBox[{
           RowBox[{"ks", "[", "z1", "]"}], 
           RowBox[{"\[DifferentialD]", "z1"}]}]}]}], "]"}]}], "-", 
      RowBox[{
       RowBox[{"Aw", "[", "z", "]"}], " ", 
       RowBox[{"Sin", "[", 
        RowBox[{
         SubsuperscriptBox["\[Integral]", "0", "z"], 
         RowBox[{
          RowBox[{"kw", "[", "z1", "]"}], 
          RowBox[{"\[DifferentialD]", "z1"}]}]}], "]"}]}]}], ",", "0"}], 
    "}"}]}],
  SequenceForm[
  "The vector potential can be written as a sum of the vector potential of \
the wiggler field,\nAw[z] = ", {(-$CellContext`Aw[$CellContext`z]) Cos[
      Integrate[
       $CellContext`kw[$CellContext`z1], {$CellContext`z1, 
        0, $CellContext`z}]], (-$CellContext`Aw[$CellContext`z]) Sin[
      Integrate[
       $CellContext`kw[$CellContext`z1], {$CellContext`z1, 
        0, $CellContext`z}]], 0}, 
   "\n and the vector potential of the signal field,\nAs[z] = ", \
{$CellContext`As[$CellContext`z] 
    Cos[$CellContext`t $CellContext`\[Omega]s - Integrate[
       $CellContext`ks[$CellContext`z1], {$CellContext`z1, 
        0, $CellContext`z}]], $CellContext`As[$CellContext`z] 
    Sin[$CellContext`t $CellContext`\[Omega]s - Integrate[
       $CellContext`ks[$CellContext`z1], {$CellContext`z1, 
        0, $CellContext`z}]], 0}, 
   "\nto produce\nA = ", {$CellContext`As[$CellContext`z] 
     Cos[$CellContext`t $CellContext`\[Omega]s - Integrate[
        $CellContext`ks[$CellContext`z1], {$CellContext`z1, 
         0, $CellContext`z}]] - $CellContext`Aw[$CellContext`z] Cos[
      Integrate[
       $CellContext`kw[$CellContext`z1], {$CellContext`z1, 
        0, $CellContext`z}]], $CellContext`As[$CellContext`z] 
     Sin[$CellContext`t $CellContext`\[Omega]s - Integrate[
        $CellContext`ks[$CellContext`z1], {$CellContext`z1, 
         0, $CellContext`z}]] - $CellContext`Aw[$CellContext`z] Sin[
      Integrate[
       $CellContext`kw[$CellContext`z1], {$CellContext`z1, 
        0, $CellContext`z}]], 0}],
  Editable->False]], "Print",
 CellChangeTimes->{{3.6902295893911552`*^9, 3.690229616318773*^9}, 
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Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"\\nThese two vector potentials can be converted to \
dimensionless vector potentials by the following equivalence:\\nA = \"\>", 
   "\[InvisibleSpace]", 
   RowBox[{"{", 
    RowBox[{
     FractionBox[
      RowBox[{
       SuperscriptBox["c", "2"], " ", "m", " ", 
       RowBox[{"aw", "[", "z", "]"}]}], "e"], ",", 
     FractionBox[
      RowBox[{
       SuperscriptBox["c", "2"], " ", "m", " ", 
       RowBox[{"as", "[", "z", "]"}]}], "e"], ",", "0"}], "}"}], 
   "\[InvisibleSpace]", "\<\"\\nso the new dimensionless vector potential \
is\\na = \"\>", "\[InvisibleSpace]", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"aw", "[", "z", "]"}], ",", 
     RowBox[{"as", "[", "z", "]"}], ",", "0"}], "}"}]}],
  SequenceForm[
  "\nThese two vector potentials can be converted to dimensionless vector \
potentials by the following equivalence:\nA = ", \
{(($CellContext`c^2/$CellContext`e) $CellContext`m) \
$CellContext`aw[$CellContext`z], (($CellContext`c^2/$CellContext`e) \
$CellContext`m) $CellContext`as[$CellContext`z], 0}, 
   "\nso the new dimensionless vector potential is\na = ", {
    $CellContext`aw[$CellContext`z], 
    $CellContext`as[$CellContext`z], 0}],
  Editable->False]], "Print",
 CellChangeTimes->{{3.6902295893911552`*^9, 3.690229616318773*^9}, 
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   3.6902313482769785`*^9}, 3.6902361213174877`*^9, 3.6902366613790736`*^9, 
   3.7378301453459253`*^9}],

Cell[BoxData["\<\"\\nWe will assume no transverse variation (Az = 0) so that \
we can assume Px and Py to be constants. Here we will declare these constants \
to be 0. Additionally, the signal field is taken to be circularly \
polarized.\\n\"\>"], "Print",
 CellChangeTimes->{{3.6902295893911552`*^9, 3.690229616318773*^9}, 
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Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"We also introduce the dimensionless vector potentials\\n\\!\\(\
\\*SuperscriptBox[\\(\[Mu]\\), \\(2\\)]\\) = \"\>", "\[InvisibleSpace]", 
   RowBox[{"1", "+", 
    SuperscriptBox[
     RowBox[{"as", "[", "z", "]"}], "2"], "+", 
    SuperscriptBox[
     RowBox[{"aw", "[", "z", "]"}], "2"]}], 
   "\[InvisibleSpace]", "\<\"\\n\[Psi] = \"\>", "\[InvisibleSpace]", 
   RowBox[{
    RowBox[{
     RowBox[{"-", "t"}], " ", "\[Omega]s"}], "+", 
    RowBox[{
     SubsuperscriptBox["\[Integral]", "0", "z"], 
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        RowBox[{"kw", "[", "z1", "]"}], "+", 
        RowBox[{"kz", "[", "z1", "]"}]}], ")"}], 
      RowBox[{"\[DifferentialD]", "z1"}]}]}]}], 
   "\[InvisibleSpace]", "\<\"\\nand \[Gamma] = \"\>", "\[InvisibleSpace]", 
   FractionBox["E0", 
    RowBox[{
     SuperscriptBox["c", "2"], " ", "m"}]], 
   "\[InvisibleSpace]", "\<\"\\n Here, \[Psi] represents the phase of the \
electron and \[Gamma] represents the energy of the electron.\"\>"}],
  SequenceForm[
  "We also introduce the dimensionless vector potentials\n\
\!\(\*SuperscriptBox[\(\[Mu]\), \(2\)]\) = ", 
   1 + $CellContext`as[$CellContext`z]^2 + $CellContext`aw[$CellContext`z]^2, 
   "\n\[Psi] = ", (-$CellContext`t) $CellContext`\[Omega]s + 
   Integrate[$CellContext`kw[$CellContext`z1] + \
$CellContext`kz[$CellContext`z1], {$CellContext`z1, 0, $CellContext`z}], 
   "\nand \[Gamma] = ", $CellContext`c^(-2) ($CellContext`E0/$CellContext`m), 
   "\n Here, \[Psi] represents the phase of the electron and \[Gamma] \
represents the energy of the electron."],
  Editable->False]], "Print",
 CellChangeTimes->{{3.6902295893911552`*^9, 3.690229616318773*^9}, 
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Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Now with these substitutions and the assumption that there is \
no transverse variation, the new Hamiltonian\\n H1 = \"\>", 
   "\[InvisibleSpace]", 
   RowBox[{
    RowBox[{"-", "c"}], " ", "m", " ", 
    SqrtBox[
     RowBox[{
      RowBox[{"2", " ", "as", " ", "aw", " ", 
       RowBox[{"Cos", "[", 
        TagBox[
         RowBox[{"\[Psi]", "[", 
          RowBox[{"z", ",", "t"}], "]"}],
         HoldForm], "]"}]}], "+", 
      SuperscriptBox[
       RowBox[{"\[Gamma]", "[", "z", "]"}], "2"], "-", 
      SuperscriptBox[
       RowBox[{"\[Mu]", "[", "z", "]"}], "2"]}]]}], 
   "\[InvisibleSpace]", "\<\"\\n which is now much easier to solve, as it is \
a one-dimensional Hamiltonian of relativistic motion.\\n\"\>"}],
  SequenceForm[
  "Now with these substitutions and the assumption that there is no \
transverse variation, the new Hamiltonian\n H1 = ", ((-$CellContext`c) \
$CellContext`m) (((2 $CellContext`as) $CellContext`aw) Cos[
        HoldForm[
         $CellContext`\[Psi][$CellContext`z, $CellContext`t]]] + \
$CellContext`\[Gamma][$CellContext`z]^2 - \
$CellContext`\[Mu][$CellContext`z]^2)^Rational[1, 2], 
   "\n which is now much easier to solve, as it is a one-dimensional \
Hamiltonian of relativistic motion.\n"],
  Editable->False]], "Print",
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Cell[BoxData[
 RowBox[{"\<\"To convert this Hamiltonian to one in terms of the canonical \
coordinates \[Psi] and \[Gamma], which are the new canonical position and \
momentum, respectively, we use the generating function\\nG1 = \"\>", " ", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{
     RowBox[{"-", "\[Gamma]"}], " ", 
     RowBox[{"(", 
      RowBox[{"1", "-", 
       FractionBox[
        RowBox[{
         SuperscriptBox["c", "2"], " ", "m"}], "\[Omega]s"]}], ")"}], " ", 
     TagBox[
      RowBox[{"\[Psi]", "[", 
       RowBox[{"z", ",", "t"}], "]"}],
      HoldForm]}], "+", 
    RowBox[{"E0", " ", 
     RowBox[{"t", "[", "z", "]"}]}]}], ")"}]}]], "Print",
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Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"When we take the partial derivative of the generating function \
with respect to z, we get\\n G1'[z] = \"\>", "\[InvisibleSpace]", 
   FractionBox[
    RowBox[{
     SuperscriptBox["c", "2"], " ", "m", " ", "\[Gamma]", " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"kw", "[", "z", "]"}], "+", 
       RowBox[{"kz", "[", "z", "]"}]}], ")"}]}], "\[Omega]s"], 
   "\[InvisibleSpace]", "\<\"\\n and we find that when we add this quantity \
to H1 and multiply by the constant \\!\\(\\*FractionBox[\\(\[Omega]s\\), \\(m\
\\\\ \\*SuperscriptBox[\\(c\\), \\(2\\)]\\)]\\), we get\\n H2 = \"\>", 
   "\[InvisibleSpace]", 
   RowBox[{
    RowBox[{"\[Gamma]", " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"kw", "[", "z", "]"}], "+", 
       RowBox[{"kz", "[", "z", "]"}]}], ")"}]}], "-", 
    FractionBox[
     RowBox[{"\[Omega]s", " ", 
      SqrtBox[
       RowBox[{
        RowBox[{"2", " ", "as", " ", "aw", " ", 
         RowBox[{"Cos", "[", 
          TagBox[
           RowBox[{"\[Psi]", "[", 
            RowBox[{"z", ",", "t"}], "]"}],
           HoldForm], "]"}]}], "+", 
        SuperscriptBox[
         RowBox[{"\[Gamma]", "[", "z", "]"}], "2"], "-", 
        SuperscriptBox[
         RowBox[{"\[Mu]", "[", "z", "]"}], "2"]}]]}], "c"]}], 
   "\[InvisibleSpace]", "\<\"\\n which is the new Hamiltonian in terms of \
\[Gamma] (the new canonical momentum), \[Psi] (the new canonical coordinate), \
and z (the new independent variable). It is more convenient for us to use \
these new coordinates because the beam velocity about matches the bunching \
wave for when \[Psi] is almost constant.\\n\"\>"}],
  SequenceForm[
  "When we take the partial derivative of the generating function with \
respect to z, we get\n G1'[z] = ", ((($CellContext`c^2 $CellContext`m) \
$CellContext`\[Gamma])/$CellContext`\[Omega]s) \
($CellContext`kw[$CellContext`z] + $CellContext`kz[$CellContext`z]), 
   "\n and we find that when we add this quantity to H1 and multiply by the \
constant \!\(\*FractionBox[\(\[Omega]s\), \(m\\ \*SuperscriptBox[\(c\), \
\(2\)]\)]\), we get\n H2 = ", $CellContext`\[Gamma] \
($CellContext`kw[$CellContext`z] + $CellContext`kz[$CellContext`z]) - \
($CellContext`c^(-1) $CellContext`\[Omega]s) (((
        2 $CellContext`as) $CellContext`aw) Cos[
        HoldForm[
         $CellContext`\[Psi][$CellContext`z, $CellContext`t]]] + \
$CellContext`\[Gamma][$CellContext`z]^2 - \
$CellContext`\[Mu][$CellContext`z]^2)^Rational[1, 2], 
   "\n which is the new Hamiltonian in terms of \[Gamma] (the new canonical \
momentum), \[Psi] (the new canonical coordinate), and z (the new independent \
variable). It is more convenient for us to use these new coordinates because \
the beam velocity about matches the bunching wave for when \[Psi] is almost \
constant.\n"],
  Editable->False]], "Print",
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  "\"\<The following plots give trajectories of the electrons in the \[Psi]-\
\[Delta]\[Gamma] plane. Each one starts with random initial values for \
\[Delta]\[Gamma] and \[Psi]. The first is for \[Psi]r = \!\(\*FractionBox[\(\
\[Pi]\), \(4\)]\), the second is for \[Psi]r = \!\(\*FractionBox[\(\[Pi]\), \
\(2\)]\), and the third is for \[Psi]r = 0. These plots are for a certain \
number of particles labeled 'num', which can be modified at the top of this \
box. Currently, num = \>\"", ",", "num", ",", 
   "\"\< As we can see, the particles tend toward infinity in the \[Delta]\
\[Gamma]-\[Psi] plane for these arbitrary initial energy values. This \
motivates us to examine what happens when we have \[Delta]\[Gamma] constant \
for all particles and examine the \[Psi]-z plane motion, which we will do in \
the next section.\>\""}], "]"}], "\[IndentingNewLine]", 
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Cell[BoxData[
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  RowBox[{"\<\"The following plots give trajectories of the electrons in the \
\[Psi]-\[Delta]\[Gamma] plane. Each one starts with random initial values for \
\[Delta]\[Gamma] and \[Psi]. The first is for \[Psi]r = \\!\\(\\*FractionBox[\
\\(\[Pi]\\), \\(4\\)]\\), the second is for \[Psi]r = \
\\!\\(\\*FractionBox[\\(\[Pi]\\), \\(2\\)]\\), and the third is for \[Psi]r = \
0. These plots are for a certain number of particles labeled 'num', which can \
be modified at the top of this box. Currently, num = \"\>", 
   "\[InvisibleSpace]", "4.`", 
   "\[InvisibleSpace]", "\<\" As we can see, the particles tend toward \
infinity in the \[Delta]\[Gamma]-\[Psi] plane for these arbitrary initial \
energy values. This motivates us to examine what happens when we have \
\[Delta]\[Gamma] constant for all particles and examine the \[Psi]-z plane \
motion, which we will do in the next section.\"\>"}],
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\[Delta]\[Gamma] plane. Each one starts with random initial values for \
\[Delta]\[Gamma] and \[Psi]. The first is for \[Psi]r = \!\(\*FractionBox[\(\
\[Pi]\), \(4\)]\), the second is for \[Psi]r = \!\(\*FractionBox[\(\[Pi]\), \
\(2\)]\), and the third is for \[Psi]r = 0. These plots are for a certain \
number of particles labeled 'num', which can be modified at the top of this \
box. Currently, num = ", 4., 
   " As we can see, the particles tend toward infinity in the \
\[Delta]\[Gamma]-\[Psi] plane for these arbitrary initial energy values. This \
motivates us to examine what happens when we have \[Delta]\[Gamma] constant \
for all particles and examine the \[Psi]-z plane motion, which we will do in \
the next section."],
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conditions.:\\n\>\""}], "]"}], "\[IndentingNewLine]", 
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    ",", "\"\< particles where \[Psi]r = 0 in the \[Delta]\[Gamma]-\[Psi] \
plane for same initial energy conditions.:\\n\>\""}], 
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    ",", "\"\< particles where \[Psi]r = -\!\(\*FractionBox[\(\[Pi]\), \(4\)]\
\) in the \[Delta]\[Gamma]-\[Psi] plane for same initial energy \
conditions.:\\n\>\""}], "]"}], "\[IndentingNewLine]", 
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   "\"\<For these above trajectories, we can see that the particles tend to \
be trapped in small regions of \[Delta]\[Gamma] and \[Psi] for small \[Psi]r \
when \[Delta]\[Gamma] is constant, particularly when \[Psi]r approaches 0. \
For \[Psi]r = \[Pi]/2, the particles are no longer trapped.\>\"", "]"}], 
  "\[IndentingNewLine]"}], "\[IndentingNewLine]", 
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  RowBox[{
  "\"\<The following is a graph of the trajectories of \>\"", ",", " ", "num",
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\(4\)]\) in the \[Psi]-z plane for same initial energy conditions.:\\n\>\""}],
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  RowBox[{
  "\"\<The following is a graph of the trajectories of \>\"", ",", " ", "num",
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\(2\)]\) in the \[Psi]-z plane for same initial energy conditions.:\\n\>\""}],
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  RowBox[{
  "\"\<The following is a graph of the trajectories of \>\"", ",", " ", "num",
    ",", "\"\< particles where \[Psi]r = 0 in the \[Psi]-z plane for same \
initial energy conditions.:\\n\>\""}], "]"}], "\[IndentingNewLine]", 
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  RowBox[{
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 RowBox[{"Print", "[", 
  RowBox[{
  "\"\<The following is a graph of the trajectories of \>\"", ",", " ", "num",
    ",", "\"\< particles where \[Psi]r = -\!\(\*FractionBox[\(\[Pi]\), \(4\)]\
\) in the \[Psi]-z plane for same initial energy conditions.:\\n\>\""}], 
  "]"}], "\[IndentingNewLine]", 
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      FractionBox["zlim", "4"]}], "}"}]}], "]"}], 
  "\[IndentingNewLine]"}], "\[IndentingNewLine]", 
 RowBox[{"Print", "[", 
  "\"\<As we can see, over z, \[Psi] for 100 particles periodically bunches \
to the same value. For positive \[Psi]r, \[Psi] tends to increase, whereas \
for negative \[Psi]r, \[Psi] tends to decrease. For 0 \[Psi]r, \[Psi] tends \
to bunch at \[Psi]=0 over periodic intervals of z. The fact that these \
particles tend to bunch at certain values of \[Psi] is exactly what we expect \
for motion about synchronous energy. The variation in \[Psi]-\[Psi]r becomes \
very small and as a result the particles bunch at periodic intervals (they \
oscillate). This results in electron motion that oscillates around certain \
values of phase over z.\>\"", "]"}]}], "Input",
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  "The Hamiltonian being evaluated is:\n\nH4[\[Psi],\[Delta]\[Gamma],z] = ", \
$CellContext`kw $CellContext`\[Gamma]r^(-1) \
$CellContext`\[Delta]\[Gamma][$CellContext`z]^2 - $CellContext`as \
$CellContext`aw $CellContext`c^(-1) $CellContext`\[Gamma]r^(-1) $CellContext`\
\[Omega]s (Cos[
      $CellContext`\[Psi][$CellContext`z]] + 
    Sin[$CellContext`\[Psi]r] $CellContext`\[Psi][$CellContext`z])],
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   Derivative[1][$CellContext`\[Psi]][$CellContext`z] == 
   2 $CellContext`kw $CellContext`\[Gamma]r^(-1) \
$CellContext`\[Delta]\[Gamma][$CellContext`z], "\n\n", 
   Derivative[
    1][$CellContext`\[Delta]\[Gamma]][$CellContext`z] == $CellContext`as \
$CellContext`aw $CellContext`c^(-1) $CellContext`\[Gamma]r^(-1) $CellContext`\
\[Omega]s (Sin[$CellContext`\[Psi]r] - Sin[
      $CellContext`\[Psi][$CellContext`z]])],
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     RowBox[{"Sin", "[", 
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\!\(\*FractionBox[\(\[Pi]\), \(4\)]\) are:\n\n", 
   Derivative[
    1][$CellContext`\[Psi]][$CellContext`z] == \
$CellContext`\[Delta]\[Gamma][$CellContext`z], "\n\n", 
   Derivative[1][$CellContext`\[Delta]\[Gamma]][$CellContext`z] == 
   2^Rational[-1, 2] - Sin[
     $CellContext`\[Psi][$CellContext`z]]],
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