Double Pendulum: Javascript pendulum animation Oliver Knill, July 15, 2015. Reload the page to start with a different initial condition.
Some older javascript from 16 years ago still runs today. Also the current page runs on any platform (like your phone) and will remain so, as no libraries or plugins (like Java and Flash) are used. At the moment the flow is slightly dissipative due to the discretization. The browser integrates numerically the Hamiltonian differential equations on the 4 manifold M=Z x Z, where Z is the cylinder. This is a flow on a 3-dimensional energy surface in M. The differential equations are
x'= (3v-3cos(x-y)w)/(16-9cos2(x-y))
y'= (8w-3cos(x-y)v)/(16-9cos2(x-y))
v'=-3x'y' sin(x-y)-9sin(x)
w'= 3x'y' sin(x-y)-3sin(y)
Since volume preserving flows on 3 manifolds are expected to be chaotic in general,
it is believed that the metric entropy is positive at least for large enough energies.
A mathematical proof appears currently completely out of reach. We
just don't have the math yet to analyze this. For small energies, KAM (Kolmogorov-Arnold-Moser)
dominates and make part of the phase space integrable.
There are horse shoes for large energies but this only establishes positive topological
entropy. I had tried to apply pluri-subharmonic techniques (developed by Michael Herman) already as an undergraduate
at ETH Zürich to attack the metric entropy problem for this problem:
the manifold contains a torus which bounds a poly-disc D x D. Now extend this analytically.
Since entropy depends in a pluri-subharmonic manner on the larger space using complex parameters (z,w),
one can try to estimate the functional at the origin (0,0) of the polydisk, where it does not
correspond to a mechanical problem any more but where we would get the estimation job done.
Such attempts failed even for much simpler problems,
like the Standard map, on which I worked for 15 years and
where much more tools from solid state physics and quantum mechanics are available.