studiot

At my local supermarket

This is all perfectly true but nothing to do with the question in hand which concerns something which is spinning with angular velocity omega about an axis. Your quote concerns the difference between rotation through some finite angle and some infinitesimal angle whereas the angle rotated thorough by spin is unbounded.

Yes, vector combination is quite basic. So, having sent a sphere into 3 separate rotations would have the same effect as a simple rotation along a single axis ? This is completely counter-intuitive. How a single axis would be selected in relation to the other 3 ones then ? Intuition can be a misleading process. The relevant theorem is the original version of Poinsot's theorem which was originally stated in statics, long before vectors were invented. "Every system of of forces can be reduced to a single force and a single moment (torque) , either or both of which can be zero. " This gives rise the the two laws of statics viz that The sum of forces acting in equilibrium is zero. The sum of moments in equilibrium is zero. This can be widened to refer to vectors in more modern parlance and to dynamics where they become https://en.wikipedia.org/wiki/Poinsot's_ellipsoid The point being that any number of rotations can be compounded to become a single rotation resultant, just as any number of forces can be compounded to form a single resultant or net force.

Agreed it is hard to picture. Do you understand resolution of vectors into components ? And that you can equally achieve the same effect as the one vector by (re)combining those components ? Rotations are more complicated motions than linear vectors but the idea is the same.

+1 Perhaps we can put the whole thing down to a 'Samuel Taylor Coleridge' moment - as in Kubla Khan. Perhaps also your imagination would be better directed towards the arts than towards the sciences ?

There is a good account of this and other infinite products for trig functions in the penultimate chapter of Hobson's Plane Trigonometry, entitled Infinite Products, pages 338 - 373.

If you read the wiki article you should note that there are 3 separate angular velocities, [math]{\omega _1};{\omega _2};{\omega _3}[/math] about three mutually perpendicular axes. You can choose any such set of 3 suitable axes, even not perpendicular ones, but the transformation equations normally work the other way because the 3 principal axes of inertia present the formulae in simplest form (diagonal matrices if you work in matrix format). Of course all that is moot for a perfect sphere because any axis is a principal axis and the others follow by symmetry.

Thanks for replying I was just trying to row back a little on the discussion, which was getting rather esoterical. OK so in lower high school (GCSE) enough Physics is introduced for someone to understand but not fully appreciate the mechanics of flight. Anyone with upper high school ( A level) physics should be able to appreciate fully. For some reason nearly all treatments of flight mechanics launch straight into the issue of what keeps an aircraft up. In my opinion this is the hard way to approach it and also leads to a failure to take note of all the forces acting - something the early pioneers failed to do with as result in so many crashes, despite have enormous (by moderns tandards) wing areas. When a physicist talks about "The Four Forces" she means the four fundamental forces of nature and not what we need here. When an aircraft engineer talks about " The Four Forces" he means the four forces that control the mechanics of flight between them. These four forces are Weight, Lift, Drag and Thrust and all are required. (Before you ask, a glider with zero thrust needs a start velocity and eventually fall out of the sky, hopefully in a controlled manner) In my opinion it is best to start with these, just accept they exist, and postpone how they occur until you a happy with knowing how they interact to control flight. This is the approach taken by flight instructors to din these interactions into pilots, so they can 'instinctively' perform the necessary balancing act between the four. I was planning to offer this in more detail when you had responded, but I am about to go out for the day so I will look again this evening.

As you say the same result can be obtained by different routes. Here is a very simple derivation of the lift force from Bernoulli for @jfoldbar to aim at. It shows the angle of attack, the Bernoulli equation, the bunching of the flowlines I mentioned and explains the circumstances and the limitations of the model. The next thing to then do is to discuss the application of that lift force to the aircraft and the effect upon its trim.

I found the 5 page pdf safe to download and interesting. However the text reads more like a technical advertising document than a technical information document. Long on rhetoric and short on detail. Looks as though it could be a developemnt of mathematical flowgraph techniques, but with a call to manufacturers to implement the necessary hardware. Doesn't the modern GPU go some way towards this goal ?

I think both momentum and energy transfer considerations are needed. But you have to be careful with auxiliary variables such as velocity and resistance to movement as with the situation when calculating the KE of a bullet entering a sandbag or plank and apply the appropriate conservation law. I'm sure we really mean the same thing.

I'm sorry I have just watched that flawed YT video. According to the video if I stopped the flow by blocking off both ends of the pipe the wall pressure along the length of the pipe would vary with the diameter of the pipe. This would be contrary to experience. The problem lies in the explanation as well. It equates velocity of impact with pressure. But this is not the case since the larger diameter portion of the pipe will also have a large surface area to impact on and the impact may result in a large force, but that impact force will be distributed over a larger area and therefore a lower pressure since pressure = force/area. A further problem is that the video describes incompressible flow in pipes. Pipe flow introduces another complication - it is described by the poiseuille equation. There are no pipe walls in the atmousphere. Atmouspheric air flow is not incompressible except in very special circumstances.

I think this question is too wide / open ended to get very far because it has been posed in isolation. So far there have been several references to 'govern' you can't divorce and isolate democracy from government. Also fundamentally involved would be the legal system, and with it the aspects of rights and responsibilities of individuals. I think pretty well everyone would agree that it makes sense to give someone or group the power to decide on 'the rules of the road' for highway purposes, although I have lived in a country (not a democracy) where this was not so. We don't really need a democratic vote to decide which side of the road we drive on every time we go out. But such considerations then lead on to methods of enforcement and the 'powers' granted to persons or groups. And by whom and how these powers are endowed.

Need more details please.

+1 But there is more to the issue than this. The design and implementation of even the simplest computer/software system needs phase of testing on examples with known outcomes. The greater the complexity of the process being computed the greater the risk of unforseen interactions resulting in unwanted outcomes. Thus the greater the complexity the greater the need for more wide ranging yet intensive pre implementation testing. Far too many large 'database' projects have foundered on this rock and AI is no exception. So how much testing would be required for you proposal and how long would that last. How strong would any guarantee of zero to low unwanted outcomes be ? It has been most illuminating (for me) to watch the implementation of computerised robot medicine selection from stock, packaging and labelling, ready for dispensation - and to see all the things that can go wrong with even such a limited objective.

I was eventually coming to that, but you beat me to it. +1 I have a bit of a quandrary here as the OP question was about Bernoulli, not the mechanics of flight, which is all too often misrepresented. I believe @jfoldbar is a student but I am not sure where they are in their studies. Bernoulli assumes what is called laminar flow. - Parallel (not necessarily straight) flowlines. Since the flowlines are parallel they do not get closer together or further apart. Starting with statics we have a theorem that every system of forces can be reduced to a single linear force and a moment. The linear force and the moment can vary independently and in particular each can be zero or any other value. Transferring this to dynamics we have the same theorem that any combination of motions can be reduced to a motion along a line and a rotation. In Laminar flow this rotation is zero. If we place a suitable obstacle in the way of this laminar flow it causes the flowlines to bunch up and/or spread out. This can be equivalent to changing the rotation from zero to some value. I say suitable and can be because some obstacles maintain the flow as laminar. Without going into the maths, introducing this rotation introduces the lift force that keeps the aircraft up. Understanding this then allows discussion to proceed to using Bernoulli locally to describe the bunching and spreading in terms of pressure forces. Fixed wings are always set at some angle to flow to accomplish this. It is also necessary to consider the statics of the aircraft in terms of a rotation and a line force.

Of course there are - that is how fingerprints are produced and how you get coloured fingers if you touch wet paint. But what is transferred, which way it is transferred and how much is transferred depends upon the nature of both your skin(how oily/sweaty it is) and the nature of the surface you touch.

Can you rephrase this into a meaningful question please ? The Butterfly effect is a complicated (please don't use the word complex, that means something else that could be confusing) dynamic.

I hate to rain on the D shaped parade, and note you haven't replied to my earlier post. Nor have I watched the video. You sensibly asked about Bernoulli, and said your aim was to understand how a 'heavier than air' craft can fly. I take it that you have seen some explanations using Bernoulli. Understanding Bernoulli is a good first step, but it is only a first step since lift is much more complicated than Bernoulli and involves other factors, not present in Bernoulli. The D shape is often offered as an explanation but it is not necessary. Take a sheet of writing paper, hold it out in front of your mouth so it dangles down from your fingers. Then gently blow over the top. The sheet of paper will rise and almost straighten out due to the lift force that is generated. This should be you starting point and can be discussed using Bernoulli

You don't need a microscopic scale to understand Bernoulli. In fact such consideration is likely to get in the way as you would then need to start considering statistical mechanics. Plain old standard mechnaics is quite good enough. The law of conservation of mechanical energy states that "The total energy of (an isolated) mechanical system is constant. Note an isolated mechanical system is one that is not busy converting its energy into something else, say electrical energy. That is it is purely mechanical. There are two basic forms of mechanical energy potential energy and kinetic energy. The total mech energy is obviously the sum of the two. So if one goes up, the other goes down. For example as a stone falls under gravity, its potential energy due to gravity reduces but its kinetic energy increases as it speed up. A fluid is a mechanical system. And Bernoulli is nothing more than the above conservation of energy.. A fluid potential energy is represented by its pressure and its kinetic energy is represented in the usual way by the velocity times the rate of mass flow (or the volume flow times the density) So to come back to your question, the total energy is constant so if the speed increases it icnreases the kinetic energy, therefore the potential energy must fall so the pressure falls. A useful term is the stagnation pressure where the fluid has stopped moving and all the energy is potential pressure energy. Mathematically this stagnation pressure becomes the standard expression of variation of pressure with altitude. Does this help ?

I have not looked at this thread before but this post caught my eye in the list. So I looked at the OP and to my suprise the only question I could find in the OP was So as things stand I couldn't understand the OP I don't know who Alan is or was So I am at a loss to proceed. The only thing I can suggest of help is to refer you to the 1960 film "The Bulldog Breed" Which was about a scenario not dissimilar from the one you describe, except it was set at the UK national cold research centre and featured Norman Wisdom, not students. This centre produced over 1000 papers whilst it was active, they must still be a matter of public record so I suggest you start there. https://en.wikipedia.org/wiki/Common_Cold_Unit

Bolded quote And modest with it. I haven't presumed anything. Bbolded quotes I very much doubt this claim, especially since it is offered with no support whatsoever ans just so easy to find numerous counterexamples. Do you understand the geometry of Scottish Dancing or the placement of Tiles, or the arrangement of atoms in molecules ? In other words the geometry of Juxtaposition ? How about Pick's theorem ? Or Mandelbrot's question "How long is the coastline of Britain ?" Bolded quotes Really ? Bolded quotes Perhaps, but not interested in the sincere offerings of others. Good of you to say so So why do you write so disparagingly about them ? This was later shown to be undecidable. That is the both the hypothesis and its negation are consistent with the standard axioms of set theory. (Godel) Is the the mathematical continuity you mean ? First bolding You need to be aware of the very very very great difference between the meaning of Field in Mathematics and in Physics. In Mathematics a Field is a set of entities subject to two binary operations and certain other conditions. In Physics a Fields is a set of elements that are either arguments or values of a function. The union of the range and the domain of a function. Second bolding Of course it is not absurd to ask. In fact most consider asking to be excellent practice. I have tried to offer the beginnings of some answers so it is most disappointing for me to find that you are not receptive to discussion about the topic and only seem to wish to pronounce your views as though no one else has ever had anything worthwhile to say. Others have found them even more upsetting. However I have always treated your posts as those of a thinking adult. Here is some modern (only 55 years old) geometry that is one way to introduce the answers in Mathematics you claim to seek. And yes, it is Scottish Dancing. A final question to think about. You keep mentioning "Natural" - Natural Geometry, Natural Science, Natural Order etc What would then be Unnatural Geometry, Unnatural Science, Unnatural Order ?

Thank you for your posting, I fear it will not be long before the moderators remove your attachment as being against the rules here and against what they have already told you. I see from the attachment that you wish to discuss two and a half thousand year old mathematics. Do you not think we have moved on at least a little bit since then ? You have introduced some more modern terms manifold, group, order but tried to use them in non mathematical ways. The 5 platonic solids you mention form what we now call a 'homotopy group' and it is by this means that we can prove that there are only these 5 regular solids in 3 dimensions. They actually enjoy no particular order (in the mathematical sense). Groups are not, as you suggest, series in mathematics, they have a very special definition. Unfortunately the rest of your article starts to wander off into mystic woo, for instance trying to introduce the so called golden ratio, instead of finding out just how much more modern mathematics in general and geometry in particular has to offer. You may wish some entertaining light reading about geometry. Try perusing The Penguin Dictionary of Curious and Interesting Geometry by David Wells. You may also like The Self Made Tapestry - Pattern formation in nature by Phillip Ball I think you will find many suprises in it especially as it has a similar theme to yours, but with the benefit of modern scientific observations so it represents the best of our knowledge. I suggest avoiding entering a slanging match about Einstein. Although he may well have been the world's greatest Physicist, he was no a Mathematician and had to rely on support form for competent mathematicians. Od course many other scientists have done great things in many other areas of science, both before, at the same time and after. One thing I picked up from your earlier postings was concsrned 'the continuum'; The theoretical nature of the continuum has, as you say, been a subject of investigation since before Greek times and has still not been settled today. But for all current practical purposes the continuum we live in behaves observably like the one you will find in any standard textbook of continuum mechanics. It is only the pure mathematicians that are still arguing over Cantor's 'Continuum Hypothesis' and the practical result will be the same whichever one is eventually proved correct, - if that ever happns.

Instead of the over lenghty preamble, can I suggest you explain/define some of the terms you are hefting about ? I am a retired applied mathematician, who has also (had to) studied lots of other applied subjects to go with the mathematics. So I was always interested in verifiable answers and results. Since I have been retired I have taken the opportunity to look into the foundations of Mathematics to widen my base. And I have never heard of 'Natural Geometry'. So first of all what is Natural Order ? And then what is Natural Geometry ? Never mind your famous solution, that would be for a much later post when I and other members understand what you think is wrong with Mathematics in general and Geometry in particular. In other words what is the problem that needs this 'solution' ?

Well go on then, demonstrate it. Since it is geometry and you are new here, do you need help posting something ?