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4

To start things off I'd say that noting the $L_z$ component is conserved seems to mean pretty much nothing, since you're considering the motion as restricted to the $\mathcal{X}\mathcal{Y}$ plane. If you had assumed the motion along the $\mathcal{Z}$ axis to be possible, then we'd be talking about the spherical double pendulum instead of the planar one ...

0

Myself is working on this area and I will tell you why I found it fascinating. This area contain many interesting new mathematics, for example if you analyze dissipative chaotic system you will encounter fractals in phase-space. Fractals are the beautiful mathematical objects which cannot be treated with standard differential geometry.This type of ...

0

The webcam + monitor loop demonstrates a recursive "Droste effect" http://en.wikipedia.org/wiki/Droste_effect#mediaviewer/File:Droste.jpg which the Wikipedia article describes as a visual example of a "strange loop", a self-referential system of geometry instancing. The resulting still image has self-similarity http://en.wikipedia.org/wiki/Self_similarity ...

5

How far ahead can we predict solar and lunar eclipses? NASA has uncertainty calculations that show how certain we are about when eclipses happen. From a back of the envelope, the eclipses will likely vary by a full day 35 thousand years from now. That said, we have eclipse seasons, so we know eclipses will continue to happen, and at roughly which time of ...

49

On predicting planetary orbits A number of studies have shown that the inner solar system is chaotic, with a Lyapunov time scale of about 5 million years. This 5 million year time scale means that while one can somewhat reasonably create a planetary ephemeris (a time-based catalog of where the planets were / will be) that spans from 10 million years into ...

1

Look I know link only answers are terrible, but I don't want to ruin the surprise. Check out this link. That's on Wikipedia! It's safe to say that if wikipedia knows something, the experts know quite a bit more. We know the dynamics of the Sun-Earth-Moon system really well including the perturbations and the most likely sources of future upsets, so barring ...

-2

Given that we exist at all, we can confidently back-track the positions of the planets (and our moon) for a couple billion years. I see no reason, barring rather massive exo-system-sourced objects showing up unexpectedly, that the positions will go chaotic enough to be unpredictable any time in the next couple billion years. I suppose it might depend a bit ...

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