joigus

Ok. Before you and I get into a long-winded discussion, why don't we let the OP tell us what concept of chaos they're interested in? I have the suspicion that the subject has evolved and the word has been taken to mean different things by different communities, according to their needs. That's more or less the reason why I said, As to Hamiltonian dynamics. Again, never mind me. I understand your complaint about my not "taking" your argument. I suggested that there is no reason why Hamiltonian dynamics should not be taken as of total generality, even if (and here's the subtle point I may have forgotten to suggest more strongly or suggest at all), from a practical POV, it may not be very useful for open systems. For open systems you could always consider your system, of coordinates (q,p) (many of them, with lower-case letters), plus your "environment", of coordinates (Q,P) (also many, with capital letters). And then you could write your Hamilton equations (formally) as, \[\frac{\partial H}{\partial p_{i}}=\dot{q}_{i}\] \[\frac{\partial H}{\partial q_{i}}=-\dot{p}_{i}\] \[\frac{\partial H}{\partial P_{i}}=\dot{Q}_{i}\] \[\frac{\partial H}{\partial Q_{i}}=-\dot{P}_{i}\] While the total Hamiltonian would really mess things up for your sub-system of interest. \[H\left(q,Q,p,P\right)\neq h\left(q,p\right)+H^{\textrm{env}}\left(Q,P\right)\] with "env" meaning "environment". Any open system can be considered as nested in a larger closed system. The point is very precisely explained in Landau Lifshitz (Course of Theoretical physics) Vol I: Mechanics. Not even fluids, elastic materials, or anything really, escapes this consideration. P(t), Q(t) would make the sub-system non-conservative. Because chaotic behaviour appears in such simple systems as conservative, few-DOF systems, it's only natural to assume that it "infects" every other dynamics that we may consider. Even if it's open (system+environment can always be considered as closed). That was about all my point. But this discussion is quite academic and I wouldn't want the OP to be scared off by it. I'd rather have some feedback from the OP.

Yes, but some are detrimental for the individual, while leaving the reproductive success of the species alone (those are the parasites that thrive); and others aren't. It is entirely possible. I just hope you're wrong, although it seems to be a well-informed guess. 😬

I neither agree nor disagree at this point. But I see no reason why the theory of elasticity or fluid mechanics cannot be put under the umbrella of Hamiltonian mechanics. It's the non-conservative aspect that would make it different from the academic examples of pendula or the like, though. Your definition of chaos seems to be more general. Why would I rush to disagree with you at this point when I'm likely to learn something new?

There are several aspects of your question I don't understand. 1) How can something become lighter by means of electricity? Weight doesn't change by thrust or electromagnetism. Acceleration does. 2) You say "stimulus" as in "stimulus/response". Cybernetics? What is that stimulus? Do you mean push, transfer of momentum? 3) Thrust within. Thrust for a rocket is nothing to do with "within". The exhaust goes away. Maybe someone can understand better...

Sorry, I see it all as Hamiltonian mechanics. 😭

There are things the Bible doesn't say and almost everybody believes it does. There was no apple. It could have been a quince, or maybe a fig, as there were no apples back then in the Middle East. The Bible doesn't say it was an apple, actually. The Bible doesn't say Jonah was eaten by a whale either. The Bible doesn't say there was an angel at the Garden of Eden, but a cherub, which was a mythical animal represented very frequently in the gardens of palaces throughout the Middle East. The Hebrew Bible doesn't say that Mary was a virgin, but a "young woman." ------------------------------------------------ There are things the Bible says and few people know it does. The Bible talks about a pantheon of gods that are subservient to Yahweh. And names God both as Yahweh and El. Is it the same god? I'm not sure. Asherah, the wife of Yahweh, is also mentioned, but the interpretation was presumably changed, as it's mentioned as a synonym for "a stick" in very obscure passages, when she is known to have been a goddess, as archaeology has shown. The stick was one of the symbols of the goddess. Back to Adam and Eve: There's at least one thing the Bible says twice in different (incompatible) ways: Ezechiel 28. Two prophecies, one of them against the king of Tyre. There you can see that the king of Tyre is expelled from the Garden of Eden, on account of his sins. The cherub also appears. Very similar legend; two different narrative uses. Who was expelled from the Garden of Eden, Adam and Eve or the king of Tyre? I'm not so sure. The authors of the Bible seem not to be either. Some scholars believe the Oracles against the king in Ezechiel 28 predate the Adam and Eve story in Genesis. ------------------------------------------------ There are things the Bible says that are taken from somewhere else: The Bible takes the story of Noah from The Epic of Gilgamesh, Utnapishtim , and adapts it to its own narrative needs. ------------------------------------------------ There are blatantly obvious things the Bible is silent about: Omri, big king of Samaria, was a very relevant character of the Assyrian domination period, but the Bible only mentions him in passing, as a baddie. The Bible also plays down the role of many other kings, like Manasseh, although he made Israel into an important olive oil factory and brought a period of peace, contrary to what Hezekiah, his father, did. ------------------------------------------------ And lastly, there are many things the Bible says that cannot be true. Josuah didn't conquer Jericho, as Kathleen Kenyon has proved. Jericho was uninhabited at the time. Plus the Egyptians were in control of Canaan and had the country strongly policed from Beit She'an. I don't believe God gave the law of gravity a suspension for some hours for the benefit of his people to the detriment of the Canaanites either. Plus the Canaanites and the Israelites were the same people: No difference in material culture or belief system, as Israel Finkelstein has shown. Abraham could not have possibly used camels. Camels were domesticated about 1000 years later.

Just change the final height to minus whatever or the initial height to plus whatever and redo the calculation.

The most obvious to me is the misunderstanding of what a scientific theory is. A couple of paragraphs with words about dimensions, parallel universes and such does not make a theory. Where are your predictions? You're not addressing any questions of observational cosmology, so you have no theory, really.

I agree with @Strange that your proposal is littered with misconceptions. Also, with @MigL that a WH is a construct that works very differently to a BB. And as I was reading all your answers I have remembered the particular sense in which the term "parallel universes" appeared in the scientific literature. It had to do with structures similar to the Einstein-Rosen bridge. I think the original motivation was with D-branes, if I remember correctly. So yes, there is a specific sense in which you can talk about parallel universes, but it's not related to extra dimensions, but with curvature. The "funnels" that you see in the picture can extend as to become parallel to each other. Extra dimensions are perpendicular, rather than parallel. No. Dark energy has to do with a scalar field, which is a completely different thing than curvature as it appears in a black/white hole. Speculation in physics with just words is hopeless. You must make calculations, and understand the mathematics. And fit experimental data, of course.

Ok. It seems we disagree about this, even if only mildly. The arguments I've heard or read that have convinced me that some rituals and religious practices may have played a positive part in the remote past are those that contend that some kind of centralized authority, plus a set of rules to decide what to do could have been an efficient way for a group of people in which disagreement can easily emerge, to take a decision and stick to it. But things that stay with us don't have to be good. Parasitic entities have their own evolutionary "agenda." They grow and prosper among us. The only mistake they must avoid making is being so damaging to their host that they manage to extinguish it. Examples of it from biology are the common cold or the measles. Examples from the world of memes are faith-based religions and the Flat Earth Society.

Every universe? How many are there? What's an anti-universe? In what sense is it anti-? What anti-symmetry isn't generally accepted? Anti-symmetry in the sense of matrices or operators? If you mean it in that sense, it is not only accepted, but it is a part of the formalism of, e.g., electromagnetism or General Relativity with torsion. If you mean something like in the sense of Hermann Weyl (symmetry as an operation after which something is unchanged), I'm not aware of any extension to anti-symmetry concept. All of those sign-changing operations are symmetries. Any anti-symmetry is a particular kind of symmetry. I'm not aware of any sense in which an extra dimension can be parallel to the previous dimensions. You can define it as perpendicular, but parallel is the only thing it can't possibly be. If it's parallel, I can see no way in which it could be extra. It would be the same dimension. Also, it's not either "parallel" or "compactified." Rather: It's either infinite or compactified. People talk about parallel universes, I know, but that's a very serious misnomer. Nothing is parallel to our universe in those (hypothetical) dimensions. Misnomers are a problem with popular science (and sometimes with serious science too.) Like, for example, Hubble's constant. It's not a constant. This makes science more confusing than need be. There is, and a very strong one. The metric of space-time and the invariance of the speed of light. There are strong elements of imprecision in your questions. I would like to help, but I don't know how, because I don't understand very well what you mean. Sorry I wasn't able to prove you wrong. I hope this helps, though, to clarify your questions.

Good question. +1 1+1=2 is a definition. The axioms only require the existence of 0 and 1. Were it not for the definitions (symbols, substituters) 2, 3, 4, etc., we would have to write 7 as, 1+1+1+1+1+1+1 If the axioms (associative law for sum) didn't allow for the proof that (1+1)+1=1+(1+1), etc. we would have to distinguish between "these 2 kinds of three": One for the left sum and another for the right. This actually happens in more general algebraic systems, like groups, octonions, etc. I hope that helps.

Sorry. I meant trajectories in the phase space. (q,p) So strict rest would be a point surrounded by asymptotic trajectories (flowing away from it: stable equilibrium; or converging towards it: unstable equilibrium). Rest, OTOH, would be a trajectory in (q,t). All of them would take infinite time, as phase-space trajectories cannot cross (Liouville's theorem.) Nah, this is nonsense. Let me think about it longer. From what I've been able to look up in Euler's theory as applied to engineering, it seems to be not really about strictly statics, but small deviations from an equilibrium position. Am I right? Maybe the subject is suffering some kind of generalisation I'm not aware of. I'll stay tuned.

No. Who said that? 2+2=4 can be obtained immediately from definitions. 2+2 = 4 is the same as (1+1)+(1+1) = (1+1+1)+1 which is obviously true following the axioms. The scope of Gödel's theorem is (presumably) about divisibility, number theory, primes. Things like that. Things that are (or may be) out of reach of finite (algorithmic) proofs from the axioms. x-posted with Studiot. +1 Sorry for overlapping with your answer, @studiot.

I don't think you deserve that title. "Watchful eye" is more like it. And there's certainly nothing wrong with that.

If you allow me a to maintain my analogy a little longer; if the tumor is not malignant, it may just result in giving you an awesome tattoo that distinguishes you from the boring un-tattooed atheists. You are not enslaved and you keep your cool religious gear. What's not to like? I don't see mindfulness or the like as a variation; rather, as a much healthier substitute. But that's just how I view it.

I see, you're right. Thank you, @swansont. +1 To tell you the truth, I was thinking about possible atmospheric drags to make an edit of self-corrections, but I missed the perturbative factors if you want to keep the orbit periodic, which should have been pretty obvious to me.

No thrust at all. Just inertia.

There's one fundamental difference, though. Dynamical trajectories can't cross paths.

I think religion is very much like a skin tumor. It's there for a reason different entirely from what humans need or wish. You get it or not more or less likely depending on your exposure to "the light" as much as on how strong your defense system is, and it can become just a quirk or turn into melanoma. But, as any other self-maintaining, replicating process in Nature, it couldn't care less about what you really want or need. It grows because it can. If you're lucky enough to weed it out, you can concentrate on the much more interesting problem of where it comes from and why it sticks in so many minds (some of them, curiously enough, anything but stupid,) or why it took the form it did in the particular part of the world where you were raised. Why the Bible took the form it did, I think can be understood largely in terms of history and archaeology.

Aaah, yes. I had forgotten that one. +1

We have the same religion. +1.

Although I've been out of touch with programming for a while, I do remember lots of confusion with languages as "cavalier" as PERL, for example. 😮 I don't know how to answer to your interesting "side note/speculation/question"!!! Thank you. Very interesting addition to the initial question. I personally have never found @MigL at fault in rigour. 😲

Not to be a nitpicker, but the main point of chaos theory is about mixing of trajectories, not so much instability, although instability (high sensitivity to initial conditions is a better term) is defined as a necessary condition. So two trajectories might separate from each other for some time in a chaotic system, but then get closer again, and then grow apart again, so as to "almost densely" cover all the phase space. While it is true that it's a matter of definition, look at the revealing fact that the word "mixing" appears 13 times in the Wikipedia article, while the word "unstable" appears only twice and the word "instability", just once in a reference. Certainly instability is a necessary condition for what's called chaos in dynamical systems to appear. Look at what the chaotic dynamics of a system does to a cluster of point in the phase space only after six steps of iteration: Chaotic trajectories look very much like what a child would do when trying to fill in a piece of paper by drawing a line squiggling all around the place.

Actually, @MigL, I think that an excessive attention to precision can be as much crippling as a lack of it. The reason of my post is that I've seen people going around in circles because of their inability to understand that they're using a definition, instead of an equation. That's the opposite end of the spectrum. TBH, I don't think all the distinctions I've made are all that important. But being able to tell a definition from an equation really is a major mistake, that I don't think you, for example, would ever make.