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Cell[CellGroupData[{
Cell["C & Q Chaos II - Integrable Hamiltonian systems", "Title",
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 "In this notebook, we will explore integrable Hamiltonian systems. \n\nWe \
will see that they have ",
 StyleBox["stable and unstable fixed points",
  FontSlant->"Italic"],
 " connected by ",
 StyleBox["stable and unstable manifolds",
  FontSlant->"Italic"],
 ": \n - When a manifold is both stable and unstable, it is known as a ",
 StyleBox["homoclinic or heteroclinic manifold",
  FontWeight->"Bold",
  FontSlant->"Italic"],
 ". This concept will be very important later. \n - It turns out that these \
manifolds also play an important role as ",
 StyleBox["separatrices",
  FontWeight->"Bold",
  FontSlant->"Italic"],
 "  (singular: ",
 StyleBox["separatrix",
  FontSlant->"Italic"],
 "), manifolds that separate between different types of motion.\n\nWe will \
also explore the concept of ",
 StyleBox["integrable systems",
  FontSlant->"Italic"],
 ": \n  - An autonomous Hamiltonian system of a single degree of freedom \
(DOF) is always integrable, since the motions stay on surfaces constant \
energy. \n  - Interestingly, a time-dependent Hamiltonian can be viewed as an \
autonomous system with two DOFs with time being a formal coordinate. \n  - \
Integrable Hamiltonian systems can be qualitatively understood as a \
\[OpenCurlyDoubleQuote]product\[CloseCurlyDoubleQuote] of motions generated \
by 1DOF Hamiltonians.\n  - The \[OpenCurlyDoubleQuote]product\
\[CloseCurlyDoubleQuote] intuition is formalized by the notion of ",
 StyleBox["separability",
  FontWeight->"Bold",
  FontSlant->"Italic"],
 " and ",
 StyleBox["Liouville integrability",
  FontWeight->"Bold",
  FontSlant->"Italic"],
 ".\n  - Real systems demonstrate that the \[OpenCurlyDoubleQuote]product\
\[CloseCurlyDoubleQuote] notion and separability can be strongly warped and \
hidden from view. \n\nFew-degree-of-freedom systems become increasingly \
harder to analyze. We will mainly focus on autonomous 2DOF systems in this \
course:\n   - In that case, we can use the method of ",
 StyleBox["Poincar\[EAcute] surfaces of section",
  FontWeight->"Bold",
  FontSlant->"Italic"],
 ". These \[OpenCurlyDoubleQuote]cut\[CloseCurlyDoubleQuote] though the flow \
of the evolution through phase space and provide insight into the dynamics.\n \
  -  The 2DOF systems also provide the \[OpenCurlyDoubleQuote]minimal\
\[CloseCurlyDoubleQuote] example where chaos occurs. (But more about that \
next time.)"
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Cell[CellGroupData[{

Cell["Mathematical pendulum and its equilibria", "Section",
 CellChangeTimes->{{3.948785860605207*^9, 3.948785872841161*^9}, {
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Cell["\<\
(Here we finish the mathematical pendulum chapter that we did not finish in \
the first session.)\
\>", "Text",
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The r,pr trajectory shape is determined by the condition that E=Hpol and L=p\
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Plot the hypersurface defined by the E=const. L=const. condition in the r,pr,\
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