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In this notebook we will delve deeper into integrability, how it is defined \
and what is it exactly that makes us say that a system is non-integrable. We \
will then go and discuss the homoclinic tangle that is one of the few \
analytical proofs of chaoticity. \
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 " - Every 1 DOF autonomous system is integrable - the shape of its phase - \
space trajectories are simply given by constant energy surfaces .\n - If we \
just add ",
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be integrable . E . g . for ",
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- Then the pairs of coordinates fulfill equations that are not coupled and \
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 "- For bound motions, the motions are oscillations in ",
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Is this Hamiltonian separable at face value? No, but it is integrable as we \
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\[Phi] is a cyclical coordinate (the Hamiltonian is independent of it), so L \
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The r,pr trajectory shape is determined by the condition that E=Hpol and L=p\
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Recall from your analytical-mechanics course that coordinate transforms could \
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 " - When a Hamiltonian does not depend on a canonical coordinate, the \
conjugate momentum is always proportionate to an actions.\n - This was the \
case of the angular momentum ",
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to an action that we encountered was the momentum ",
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 "Now we are better equipped to understand what the perturbations to the \
pendulum. For the system to be integrable, the snapshots should look like \
cutting through a torus ",
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we know we are looking at ",
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Cell["\<\
Now let us transform to the AA coordinates of the unperturbed system (at \
\[Epsilon]=0). We get:\
\>", "Text",
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Cell[TextData[{
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still represent the Hamiltonian in normal form, which means eliminating the \
dependence on the angle variables \[Psi],\[Tau]. We could try simply guessing \
a good approximate Hamiltonian by averaging out the angle dependence. \n\nIn \
our case this yields the original Hamiltonian and we know it fails badly near \
the saddle point (unstable equilibrium). Why?\n\nThe answer is in the \
so-called ",
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 "\n - In the time-dependent system, we can understand the Poincar\[EAcute] \
surfaces of section as a discrete map. These maps have fixed points \[Gamma] \
(points that are mapped onto themselves after an iteration) that are at the \
fixed points of the isolated system when \[Epsilon]=0.\n - They also have \
stable and unstable manifolds ",
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often bifurcates into unstable and stable manifolds again. \n - However, \
since the perturbed ",
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transversal intersection is present, infinitely more have to be as well. As a \
result a \[OpenCurlyDoubleQuote]tangle\[CloseCurlyDoubleQuote] of infinitely \
folding intersections appear.\n - It was shown by Smale in the 1960s that \
this leads to \[OpenCurlyDoubleQuote]horseshoe\[CloseCurlyDoubleQuote] \
dynamics that can be proven to imply what is known as ",
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 " The distance between the manifolds ",
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 " can in fact be computed by perturbation theory, the method is known as \
Melnikov\[CloseCurlyQuote]s method. The formula for a time-dependent \
perturbation is (see Wiggins - Introduction to Applied Nonlinear Dynamical \
Systems and Chaos)\n",
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Cell[TextData[{
 "Where ",
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 " is the zeroth order Hamiltonian, ",
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unperturbed homoclinic orbit. When this function expressed as a function of \
t0 oscillates through zero, it proof of chaos. "
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Cell["\<\
Exercise: (Semester project) try to compute the Melnikov integral for some \
system.\
\>", "Subsection",
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