Timeline for To see if a system behaves chaotically does one have to vary (in a tiny way) the initial momenta of ALL constituents of the system?
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| Jun 19, 2019 at 7:29 | comment | added | stafusa | @descheleschilder I mentioned it before, but let me emphasize: the usual equations for the double pendulum don't refer to atoms at all. It's a rigid-body or point-mass model so, strictly speaking, asking about changes on one atom is simply an ill-posed question. I've interpreted it so far as meaning an extremely minute change - if you do literally mean an atom, then you need to use a different description for the pendulum, one that includes its atoms. | |
| Jun 19, 2019 at 0:17 | comment | added | Deschele Schilder | I don't mean with "do you think..." that it is a matter of opinion. The small perturbations are put on the whole system, after which the whole system might diverge or not. In trying to show if a double pendulum behaves chaotically one does not put a small change on just one atom but on all atoms of the dp at once, after which the dp will after a certain time behave in a completely different way than if the small perturbations weren't put on the all atoms. Macroscopically this translates itself in small variations of the arms of the dp (the whole dp) and see what happens. | |
| Jun 18, 2019 at 22:04 | comment | added | stafusa | "Do you think [...]" @descheleschilder It's not a matter of opinion. Some systems can be shown to be chaotic, which means that typical, arbitrarily small perturbations put them on new trajectories that are eventually totally different from the unperturbed one. And 'totally different' doesn't mean with more or fewer extreme events - they might just happen at different times, for instance. As for the butterfly wing flap, I believe David Hammen has answered you rather well already, back in 2016. :) | |
| Jun 18, 2019 at 16:19 | comment | added | Deschele Schilder | Likewise, do you really think that the total weather system (in which there are no critical regions) will change if one varies a virtually infinite part of it? Look at films of tornadoes and the enormous destruction they cause. How can this be connected to a butterfly that varies this infinite part in a slightly different way (w.r.t. the total weather system)? If so, how? | |
| Jun 18, 2019 at 16:09 | comment | added | Deschele Schilder | Do you think that that the motion of a double pendulum will change if you make a little variation in the momentum of one atom out of all the atoms that constitute the double pendulum? | |
| Jun 18, 2019 at 14:52 | comment | added | stafusa | Atoms aren't part of the usual model for the double pendulum, but yes, nearby trajectories diverge exponentially fast, so no matter how tiny the initial separation - including one of, say, $10^{-22}$ - one will observe them diverge eventually. I don't know any specific model for a waterfall off the top of my head, but what you describe should admit some chaotic models, perhaps even some displaying spatial-temporal chaos, pattern formation, etc. | |
| Jun 18, 2019 at 13:07 | comment | added | Deschele Schilder | So in the double pendulum, you let a very small part of the momenta of the atoms which make the DP up (with respect to the weather system, a very, very, very small part of all atoms which make up the pendulum) vary in a tiny way (atoms which, as said only form a very tiny part of the DP)? By a waterfall I mean a stream of water that starts somewhere on a high level and streams down to a lower level, meeting stones, obstacles, etc. on its way. I know it's not a real waterfall, but more a stream of water going downwards on a rough surface. | |
| Jun 18, 2019 at 12:12 | comment | added | stafusa | @descheleschilder It's the same for the double pendulum. A chaotic system is sensitive to arbitrary perturbations in the state space, which typically includes perturbations along a single of its axes (corresponding to a given system variable). As for the waterfall, do you mean a chaotic waterwheel? Otherwise I don't know any chaotic waterfall model. | |
| Jun 18, 2019 at 11:49 | comment | added | Deschele Schilder | And what about a double pendulum or a waterfall? | |
| Jun 18, 2019 at 11:08 | history | answered | stafusa | CC BY-SA 4.0 |