joigus

This is another very good point. On my part, I will brood over it for a while longer, plus go over everybody's comments.

Well, I wouldn't be very good at summing up criteria of philosophical goodness myself, but I cannot deny that there are significant things to be said. Somehow I picture @Eise as the most knowledgeable person I know around here, at least of those I've interacted with. Some of the stickies on the forums rules already cover a number of common fallacies or what an argument in good faith is. Other fallacies could be added, like the argument of authority, or the idola fori (Francis Bacon); that is "many people say or think". If not as explicit prohibitions, at least to make people aware of common sources of error or weak arguments. On the other hand, there are certain philosophical topics that are of interest to science and I think have become universal quality standards for scientific thought: Operationalism (ultimate reference to measurable quantities) Ockam's razor (economy or parsimony of systems of ideas) Falsifiability (K. Popper) --> road to experiments This is a very good point, because if anything, it shows that the distinction is sometimes difficult. I think that dimensions (at least overlapping with @Ghideon, @Mordred) is a concept that generally refers to the ambient space, or space of independent variables, while DOF generally refers to dependent variables that make up the mathematical concept of state, generally a function of the first. So to describe the state of a system you set up a functional dependence Y(t,x), with #DOF = number of independent Y1, Y2,...Yn that you can fix. Any other function of the state would be numerically determined. System, state, variables of state, ambient space I think are the concepts that shape the question. Although there are cases when the distinction can become blurry for several reasons. One of them appears in classical thermodynamics, where the ambient space disappears altogether, and you're left with an implicit relation among the state variables (equation of state): f(P,V,T,n) etc. There you have some kind of cyclicity, in which you can pick any number of these variables to describe the change of the others, and trajectories as abstract (timeless) motions in that surface of state. Then we've got field theory. There the concept of DOF is the set of field variables, so Fa(t,x) would itself be the degree of freedom. A vector field, like the vector potential, mirrors the properties of the ambient space itself, \[A^{\mu}\left(t,x\right)\] but other "internal field variables", like e.g., Yang-Mills fields, have a representation space that is richer. As to GR, you've got the manifold (independent variables) plus a set of fields: g(x), metric; R(x), Riemann; T(x), matter; all of them would add up to the state (dependent variables) as a function of the x's (independent variables). As to classical dynamics, I see the DOFs as the specification of (x1,...,xn,p1,...pn) because, once you fix those, there's nothing else to fix unless you re-define what your system is. But another complication is that there is an existing tradition to call just (x1,...,xn) your DOFs. Then there is @studiot's comments about the equations of constitution. The thing does ring a bell to me, but I don't remember what that is about, so I'd be thankful if he reminded me.

I applaud this idea too. +1 The only thing I find more difficult to establish from a practical POV is the "good" in "good" arguments. You seem to have an idea for when an argument is just too bad quality to be accepted as such... 🤔

Ah. It did ring a bell. +1. I agree. It's a bit outdated maybe, but good stuff.

Glad to be helpful.

I think I can do a little bit more than that. Most, if not all, interesting wave functions in QM have a behaviour that goes to zero as a Gaussian at infinity. If you take a look at most eigenfunctions of "realistic"* Hamiltonians, for example, the harmonic oscillator, hydrogen atom, etc. The all are dominated by exponential damping at infinity. Example: \[\psi\left(x,0\right)=\frac{e^{-x^{2}/2-if\left(x\right)}}{x^{n}}\] Now it's very easy to see that no matter what power of x is integrated against the exponential, the idea works. \[\int_{\mathbb{R}}dx\frac{e^{-x^{2}/2+if\left(x\right)}}{x^{n}}\frac{d}{dx}\left[\frac{e^{-x^{2}/2-if\left(x\right)}}{x^{n}}\right]=\left.\frac{e^{-x^{2}}}{x^{2n}}\right|_{-\infty}^{+\infty}-\int_{\mathbb{R}}dx\frac{d}{dx}\left[\frac{e^{-x^{2}/2+if\left(x\right)}}{x^{n}}\right]\frac{e^{-x^{2}/2-if\left(x\right)}}{x^{n}}\] Watch out for silly mistakes. * Meaning nothing pathological, like Airy functions, or something like that.

Exactly. Under the integral sign, yes. Actually, it's used as a matter of course in all of field theory. Field variables at infinity always go to zero "fast enough", so you can shift the derivative from one factor to the other factor (under the integral sign) by just changing a sign. Sorry. I made a mistake before. The surface term should not be the derivative, but the term that is derived. I've corrected the formula. This is what I wrote: \[\left.\frac{d}{dx}\left(\psi^{*}\psi\right)\right|_{\textrm{infinity}}\] This is what is should be (already corrected in the original post): \[\left.\psi^{*}\psi\right|_{\textrm{infinity}}\]

No, no. Careful. That's not the point. The point is that the integrals, \[\int dx-\left(\frac{\partial\psi^{*}}{\partial x}\psi\right)\] and, \[\int dx\psi^{*}\frac{\partial\psi}{\partial x}\] differ in what is called "a surface term" or "a boundary term". Because in quantum mechanics the boundary is at infinity, they can be identified for all intents and purposes. If you equate one of these integrals to its complex conjugate, what you're saying is that the integral is real. That's not quite so correct. The integrals are equal except terms that vanish at infinity. The point is a bit subtle, but that's the way to read its meaning. Edit: In this case, the surface term is, \[\left.\left(\psi^{*}\psi\right)\right|_{\textrm{infinity}}\]

Consider: \[\int dx\left(-\frac{\partial\psi^{*}}{\partial x}\psi\right)=\int dx\psi^{*}\frac{\partial\psi}{\partial x}\] and what I told you in the other post about fields vanishing fast enough at infinity. You get twice the first integral in 1.30.

It is because you are evaluating the integral at the limits of integration. That's very common in any field theory. The fields are assumed to go to zero fast enough at infinity. In fact, you need that if you want your momentum operator to be Hermitian. If D is any of these differential operators, you need both the i and the vanishing at infinity so that, \[\int d^{3}xF^{*}iD\left(G\right)=\int d^{3}x-\left(DF^{*}\right)iG=\] \[=\int d^{3}x\left(iDF\right)^{*}iG\] I hope that helps. Good question. +1

Thank you for the detailed answers. +1 You mean lambda = 1+ √5 And lambda = 4 is the chaotic one.

That's why I said: I don't know. I wasn't aware that we were talking about the North Atlantic Gyre. Isn't that from another forum, more to do with eddies? Your point does remain indeed. And my point that chaos, the way it's normally taught at universities, is about mixing of trajectories and ergodicity, also remains. That's what my books say, that's what the KAM theorem says, and that's what I thought I knew and I learnt in the classrooms. I must confess I remember only vaguely, but the idea of it was to introduce a measure in the space of parameters of (quasi-periodic) Hamiltonians and prove that the non-chaotic systems had zero measure, while the chaotic ones had the measure of the continuum. I don't recognize that explanation on the Wikipedia article, but I'm quite sure of it. Because the characterization that you seem to be working towards speaks of more general, non-conservative systems, I'm willing to learn more from it. The only thing in which I've disagreed is about chaos being characterized only by instability (Liapunov exponents.) The example I gave, was clear enough. Also, I said, And although from a practical POV it doesn't do to treat some of such systems with Hamiltonians, the fact that everything we know about chaos, as well as the initial motivation by Poincaré's work on the stability of the Solar System, derives from a Hamiltonian, has some bearing on the question,

I think you got it wrong: Gannets and barracuda fish better together Edit: x-posted with Mordred +1. Couldn't have given you better advise.

It's like trying to find your way out of a maze, or a forest. There is no roadmap, and if you've got one it's probably wrong. But understanding topology, reading clues, minor details, can help you a lot. Do you always need a map to find your bearings? In physics the map always comes later. Not very well known fact: Einstein spent one whole year without accepting Minkowski's concept of 4-dimensional space-time (I've heard this in a classroom.) He already had all that was needed, logical fact to logical fact. In the words of Steven Weinberg: "physicists are more like hounds than hawks" Dreams of a Final Theory

And I really recommend that you read the OP: And then: (My emphasis) Now what? Do you realize you haven't understood the OP already? Don't worry, I'm waiting for you to catch up. IOW: Just because we cannot prove everything (following Gödel's theorem), does it follow that we cannot prove that 2+2=4? Should we accept that 2+2=5? Or anything else? That's called "reading between the lines." If you can't read between the lines, the OP looks like a contradiction. Which it isn't. But you must take some time to really read carefully the OP and really want to help. Just trying to appear cleverer than everybody else just because you can quote Bourbaki, or link to it, doesn't really help. Oh, really? I hadn't noticed. I thought we were talking about the history of pudding (sigh).

I already told you about this. I'm certainly not going to repeat just because you don't care enough to read.

Bourbaki, really? The OP has a simple enough question about Gödel's theorem and simple relations between real numbers and your suggestion is a treatise that proposes to redo mathematics from scratch by a group of (brilliant) mathematicians that proposed to change the whole structure of maths under a pseudonym? This is the line that you misunderstood, split into independent lines: 0+0=0 0+1=1 1+1=2 You understood: Incorrect interpretation of OP's question. Honest mistake, so far. But then, You bring up division by zero, which is irrelevant, as it was not implied by the OP. Then you bring up my level of knowledge. Not that I care. I don't. Then you bring up Bourbaki. I happen to know Bourbaki and I coincide with @studiot that it's nothing to do with the OP, nor does it have any bearing on the question, nor is it advisable in order to answer it, among other things, on account of what the OP said, very clearly, All of this when the question had already been answered to the satisfaction of the OP, as I understand. I wonder, what's next? Algebraic topology? Apparently there's no limit to how far off-topic you're willing to go in order to bring more confusion to the discussion or not to recognize that you misunderstood the initial question. Now, I suggest you ask this yourself: Am I really helping here?

Have you seen my profile?: Who says I say I know everything? It's you who seems to think that people are saying things they're not really saying. Read whatever people say and then say whatever you have to say. Enough said.

I'm well educated enough to withhold my opinion of what you (or any other member of this forum) are or are not. I will always concentrate on the arguments and document them properly wherever necessary. I suggest you do the same.

Totally agree. +1. Nobody understood division by zero here, except you, @ahmet. Plus the question, has been satisfactorily answered, I think. Unless the OP has any further question. To me, end of story. Edit: Unless you have any further comments on how Gödel's theorem could imply that we can prove 2+2=5, in which case, I unrest my case. Edit 2: It seems the OP made a mistake in the title. They meant, I think, ¿Can't we prove (from Gödel's theorem) that 2+2=4?

I think it's a good idea. +1. Modern and Theoretical Physics & Astronomy and Cosmology are especially affected in what physics is concerned. Speculations could be another one, because much of what starts on the above ones ends up there. This forums are very interdisciplinary, so maybe there are nuances between our respective current definitions, but I see no reason why I couldn't be worked out.

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...