2 ms·

> There's a lot of reason to think that area should have more structure in it because the proof that there are an infinite number of unstable periodic orbits in

> There's a lot of reason to think that area should have more structure in it because the proof that there are an infinite number of unstable periodic orbits in there starts with knowing there are an infinite number of stable periodic orbits and that there is an unstable orbit on the the separatrix between them. An analogous result is true for the number line: between any two rational numbers, there is an irrational number. But that isn't cause to suspect that, if you look at a random piece of the number line, you should be able to see a lot of rationals. They're there. But you can't see them. The odds of any given number you look at being irrational are 1.

PaulHoule 2y ago
That is very relevant. In a plot like that one those groups of circles that you see are resonances where the winding ratio on the torus (how many degrees you go around per turn) is rational. The area between two resonances (or non-resonant behavior) has a little bit of chaos in those regular regions if you look closely: there is a certain chaotic zone. Those areas of pervasive chaos happen when the chaos area around the rationals covers everything, see https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%8... which involves doing a sum over all the rationals. In the 2 degree of freedom case those tori are a solid wall so even if you have a chaotic zone around a resonance the motion is constrained by the tori. For N>2 though there are more dimensions and the path can go "around" the tori. You could picture our solar system of 8 planets having 24 degrees of freedom (although the problem is terribly non-generic because the three kinds of motion in a 1/r^2 field all have the same period) It sure seems that we are in a regular regime like one of the circles in that plot but we cannot rule out that over the course of billions of years that Neptune won't get ejected. See https://en.wikipedia.org/wiki/Arnold_diffusion https://en.wikipedia.org/wiki/Arnold_diffusion which is poorly understood because nobody has found an attack on it. You have the same problem with numerical work that you do in the plot because sensitive dependence on initial conditions magnifies rounding error. Worse than that normal integrators like Runge-Kutta don't preserve all the geometric invariants of https://en.wikipedia.org/wiki/Symplectic_geometry https://en.wikipedia.org/wiki/Symplectic_geometry so you know you get the wrong results. There is https://en.wikipedia.org/wiki/Symplectic_integrator https://en.wikipedia.org/wiki/Symplectic_integrator but other than preserving that invariance those perform much worse than normal integrators. This is one of the reasons why the field has been stuck since before I got into it. This https://en.wikipedia.org/wiki/Interplanetary_Transport_Network https://en.wikipedia.org/wiki/Interplanetary_Transport_Netwo... came out of people learning how to find chaotic trajectories to use for transfer orbits and is an exciting development though. Practically though you don't want to take a low energy-long time trajectory to Mars because you'll get your health wrecked by radiation.

PaulHoule 2y ago

That is very relevant. In a plot like that one those groups of circles that you see are resonances where the winding ratio on the torus (how many degrees you go around per turn) is rational. The area between two resonances (or non-resonant behavior) has a little bit of chaos in those regular regions if you look closely: there is a certain chaotic zone. Those areas of pervasive chaos happen when the chaos area around the rationals covers everything, see https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%8... which involves doing a sum over all the rationals. In the 2 degree of freedom case those tori are a solid wall so even if you have a chaotic zone around a resonance the motion is constrained by the tori. For N>2 though there are more dimensions and the path can go "around" the tori. You could picture our solar system of 8 planets having 24 degrees of freedom (although the problem is terribly non-generic because the three kinds of motion in a 1/r^2 field all have the same period) It sure seems that we are in a regular regime like one of the circles in that plot but we cannot rule out that over the course of billions of years that Neptune won't get ejected. See https://en.wikipedia.org/wiki/Arnold_diffusion https://en.wikipedia.org/wiki/Arnold_diffusion which is poorly understood because nobody has found an attack on it. You have the same problem with numerical work that you do in the plot because sensitive dependence on initial conditions magnifies rounding error. Worse than that normal integrators like Runge-Kutta don't preserve all the geometric invariants of https://en.wikipedia.org/wiki/Symplectic_geometry https://en.wikipedia.org/wiki/Symplectic_geometry so you know you get the wrong results. There is https://en.wikipedia.org/wiki/Symplectic_integrator https://en.wikipedia.org/wiki/Symplectic_integrator but other than preserving that invariance those perform much worse than normal integrators. This is one of the reasons why the field has been stuck since before I got into it. This https://en.wikipedia.org/wiki/Interplanetary_Transport_Network https://en.wikipedia.org/wiki/Interplanetary_Transport_Netwo... came out of people learning how to find chaotic trajectories to use for transfer orbits and is an exciting development though. Practically though you don't want to take a low energy-long time trajectory to Mars because you'll get your health wrecked by radiation.