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Indeed, let us recall how Helmholtz in his ground-breaking paper On Integrals of Hydrodynamic Equations that Correspond to Vortex Motions defines rotational motion of elementary liquid particle. First, every point (x, y, z) in space occupied by incompressible liquid gets associated with the velocity of translational motion of a liquid particle passing through that point at any given time. Then the three components (u, v, w) of that velocity, which are functions not only of (x, y, z) but also of time t, are assumed to be continuous and differentiable almost everywhere with respect to space coordinates. It is important to note that in this setup material point is not capable of rotation, but elementary liquid particle is! How is this possible? The rotational motion of each elementary particle is derived from the translational motion of the neighboring points! Here is how it is done:
∂v/∂z - ∂w/∂y = 2ξ,
∂w/∂x - ∂u/∂z = 2η,
∂u/∂y - ∂v/∂x = 2ζ.
That's the definition of mathematical operation curl (a.k.a. rot) which plays a pivotal role in Maxwell's electrodynamics
Helmholtz doesn't dwell too much on this derivation of rotational motion from translational one, but it amounts to a major hypothesis: (1) each liquid particle is a small rigid body, (2) those tiny rigid bodies move in such a way that neighboring ones slide (mostly continuously, but possibly discontinuously along some surfaces) against each other, and (3) there is no resistance to sliding in the case considered by Helmholtz where the liquid is assumed to be frictionless.
This hypothesis seems quite reasonable for the dynamics of incompressible liquids. But it does not seem appropriate at all for the dynamics of electromagnetic medium: particles of electromagnetic field behave in such a way that the neighboring particles neither slide against each other (as in liquid model), nor stick to each other (rigid or solid model), but rather rock and roll, as it were, over each other. This is probably the key to understanding why all mechanical models of electromagnetic aether, which are based one way or the other on liquids or elastic solids, have failed! As Henri Poincare put it: "All those [models] that have been proposed have a savor of artificiality which is repugnant to the reason".
The other side of the prevailing paradigm coin is that twisting action is not perceived as basic and irreducible action, but rather as a derivative action, namely, a pair of pushing or pulling actions.
Another characteristic of this paradigm is that it speaks in the language of forces in the Newtonian sense of that word.
Hertz has attempted a major shift in this paradigm with his forceless mechanics, which, in my opinion, has not received yet the attention it deserves.
While I concur with your analysis and complaint , I have to point out that mathematical descriptions are always reducible to translational lines in orthogonal orientations.
As a young aspiring mathematician I did not understand, nor was I taught that this was and is a convention only. Certainly I and many scientists were of the opinion that curves can be approximated differentiable by a sequence of orthogonal steps, and further that this was the reality of curved motion!! Thus I was shocked to find that Newton, Wren and many others were not restricted in this way at all. In fact the concept of curvature enjoyed equal validity and existence with that of the straight line. Newtons Lemma 1 , and 7 rely on the validity of curvature.
It took me a while to accept that torque or moments of force were an inadequate and misleading model of rotation, and one which Newton did not ascribe to. Newton makes it plain that curvature is valid and irreducible to a straight line even in the infinitesimal form of Lemma 1 . Rather his argument is that the ratio tends to unity, not that a curved line loses its curvature even in infinitesimal quantities!
Recently I came to understand that Newtons centripetal and centrifugal force with the tangential motion that defines circular motion actually are a contra pair of forces with an intermediary tangential velocity: two cancelling accelerations and an instantaneous velocity that acts orthogonally to the forces, and tangentially to any curvature ! The sequence of motions that take any object on a curved path cannot be reduced to anything less! Newton thus stated that curved motion could equally be continuos or impulsive, it was not possible to decide which it was beyond the empirical situation.
Many have simply ignored Newtons actual advices or devices, and in particular they have ignored the different natures of line segments used by all geometers from Pythagoras onwards.
Thank you for the insights you derive from the Newton's original writings. I gather that my second hand opinions about Newton do not reflect, or reflect poorly at best, the real Newton. I have to confess that I have not studied the original works of Newton; perhaps I should have.
I have a high regard for his genius , but as I have pointed out, he is not infallible. In particular he is not very communicative of his actual methods, thus leaving all but a few ardent students who served as acolytes with very little insight into how he achieved his results. But certainly I have gained much from reading in his Principia in his own words.
Hamilton was hailed in his day as the New Newton, amongst other things, but a lot of his rumination that lead to Quaternions is based on DeMoivre and Cotes excellent work. Of course these were the two most famous students of Newton, who received personal instruction from him.
In regard to the modern lineal algebra we have to turn to Grassmann for insight. If you google jehovajah Grassmann you will find what research I am doing on him.
∇x[(u∇)u] ≡ (u∇)(∇xu) + (∇u)(∇xu) - [(∇xu)∇]u,
where u is the velocity field vector (or any field vector for that matter), and x stands for the cross-product operation.
That might save time for those of you who decide to digest Helmholtz's paper, which is definitely worth it.
Note also that excluding pressure from the equations of motion is per se a realization of Hertz's forceless ideal! Perhaps it is no accident that Hertz was Helmholtz's student.
http://en.m.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookII-IX
Thanks for the pointer to Helmholtz whose work I had on my list to read through.
This and many other problems Heaviside, Hertz and the Maxwellians had to contend with. They heavily redacted his work based on his expressed hypotheses not on his mathematical devices.
However his idle wheel mechanism would become mechanically important and highlights a fractal element that Newton missed and Maxwell incoherently exploited.
Fractal idle wheels at all scales would generate vibrational excitation in the material and resistance would generate heat. His initial stipulations would thus remove 2 associated phenomena from electricity!
It is to be noted also that magnetic behaviours are left to the second part of his book and treated as electric phenomena a la Ampere. I do not perceive any modelling of Amperes circuits except in the behaviours of these " idle" wheels.
In the preface by Helmholtz to Hertz's The Principles of Mechanics we read:
http://www.ivorcatt.com/em_pref.htm
And to be sure when I first found his work I was intuitively excited by it. It was the mere possibility that an alternative explanation existed to basic physical phenomena that gave me cautious optimism. My purely mathematical and applied mathematics research based on foundational issues in mathematics had led me to suspect a much simpler connection than what I was led to believe conventionally. In fact I had fallen into a warm bath I called Shunya, after the Sanskrit word for " fullness of everything". At its heart was the dynamic sphere as a fundamental vortex dynamic. It was the most special vortex we could conceive!
My interaction with Ivor revealed how ignorant I was about " electricity".i thought I was ahead of the game through listening to Eric Dollard. But my most fundamental concepts were all wrong! I had to start from a different paradigm, that of fluid mechanics.
This was a major paradigm shift. Nothing was solid. Everything was fluid. Thus no particles were admissible as fundamental. It was then I began to notice how careful Newton had been! Corpuscles are not billiard balls! They are biological entities . By choosing these as his fundamental particulate primitive he implied nothing solid about form. Rather all was trans mutable nd transformable, as all Alchemists believed on oath!
Thus his Alchemy underpinned his primitive concepts, but he could not give voice to themin the then hostile climate of Britain, where Alchemy was socially akin to Devil Worship!
On the continent this was not the case, as Boyle at once perceived on coming to Cambridge from the continent! Thanks to Boyle the law in the British empire against Alchemy was eventually repealed, but Newton publicly had little to do with that campaign.
Newton however contributed to the corpuscular theory of matter. Up until the quantum era, this was the only theory of matter all scientists subscribed to.JJ Tomson only added a detail to that theory based on his experiments with electrified Plasmas.
The Rutherford Bohr model of the atom was a significant step backwards! And yet it has proved to be the perfect smokescreen behind which physicists have redacted their knowledge and beliefs without public debate!
Allowing the most vicious and vitriolic response to simple unorthodox enquiries on so called physics forums has been a tactic to keep the masses in line while the "gurus"pontificate all sorts of non sensical things, titillating the general enquirer ith technological " magic"!
Nevertheless the facts , historical, seem to be that by the time of Helmholtzthe intellectual conception of how natural phenomena worked and were related was at its social peak!
But then commercial values and patent law was introduced to allow greed it's full effect! The social intellectual communities of artists poets and philosophers was shattered by hard nosed business and profit oriented capitalists. Suddenly the " truth" became a commodity and it could be owned by vast corporations! We the consumers of these corporations did not need to know the truth! That social educational contract was gone. Trademark and commercial interest rights prevailed over the simple right to know!
The simplest formal models we can conceive will always be based on the sphere and it's parts. That is the most fundamental form we can conceive. But in its dynamic multiplicity such a fluid form can account for everything by fractal principles.
This is my understanding and my goal. The fundamental primitive fluid element is not a material point but a fractal region of spheroidal (or better trochoidal) dynamic space. All is in all and of all.mthat is to say that a fundamental attribution I have to give to space is self relative motion.
Once that is accepted, it is my contention that that self relativity is rotational at is most primitive behaviour.
Allowing just these 2 fundamental primitives is sufficient to explain all else, I believe.
What is beyond me is any power to declare my model as the truth,or divine or nothing else we use to provide spurious authority to the cogitations of a human mind.
This is a fractal loop theory , by which I mean the distribution of loops have a structure that is scale free and a distribution that is everywhere at every scale but not necessarily uniform or contiguous.
Now Ampère's circuits are closed loops, essentially, based on Volta's concept of a circuit containing at least 3 different elementary pieces one of which is moist. The later development of Resistance Capacitor/ inductor circuits thus feed back directly into Ampère's theoretical scale free primitives!
However, it is clear or should be that the model of the so called magnetic lines of force around a wire supposedly carrying a " current" , or as otherwise designated Rotating magnetic field lines; that this model as a closed or circular loop is inaccurate.
The better 3d designation of these lines of compass behaviour is as a helical spiral or vortex, and not one, as in the one fluid theory, but 2 contra spiralling behaviours one designated north and the other south.
I may dispense with this older terminology and state that two contra vortices of plasma pass around a wire in such circuits where the wire is merel the "form" around which these vortices develop out of the general dynamic environment, which in my philosophy is essentially a fluid dynamic entity.
When such a vorticular structure develops we call it induction. It is only due to certain forms bing particularly good at inducting that we term these conductors and the process of vortex structure formation in these material " conduction" .
What our teachers have done is to so term thins that we are surprised when poor conductors or insulators actually appear to conduct! In the case of a capacitor, or a voltaic cell. Or a Leyden jar conduction in otherwise insulating materials is made to appear miraculous and mysterious.
The basic, and fundamental " induction" that occurs in all material was termed Electra by our ancient forbears but more particularly by Gilbert as a designation. Of a type of magnetic behaviour . Magnetic behaviours itself was a term yo designate the behaviour of the crystalline rock lattice Magnes, which turned out to contain ubstantial amounts of ferrous ores.
The ferromagnetic behaviours then are undeniably and fundamentally the sound magnetic base of all our theoretical musings about electromagnetism, and eventually all our modern " particle" physics and chemistry, and thus material sciences and biology.
It is thus fundamentally important to understand Ampère's theoretical model and the improved models developed by Joseph Newman, professor Eric Laithwaite, a metallurgist Vladimir Ginzburg in his book Prime elements and finally the Helyx Toryx erosion of the same,which are all fundamentally fractal helical theories which are scale free.
http://www.rexresearch.com/newman/newman.htm
http://www.angelfire.com/biz/Newman/section2.html
http://helicola.com/?p=about
http://www.iseti.us/WhitePapers/MARS2005/MarsSociety-2005-(DoestheLaithwaiteGyroscopicWeightLosshavePropulsionPotential-2005-09-11).pdf
I might add that redirecting forces is precisely the role of rotation, while pressure, which I say is the fundamental concept from which force may be derived, provides a motive potential in all directions INCLUDING rotational ones. It is by differential application that pressure gradients and thus acceleration that may lead to a force measure in a particular symmetrical body , or at a Barycentric centre.
http://en.m.wikipedia.org/wiki/Galvanometer
It turns out Ampère suggested the name! As a consequence of his detailed use of the compass needle in all parts of a circuit including over the battery teminals.
I have some hypotheses as of what takes place in Laithwaite's experiment.
Hypothesis 1. Laithwaite has discovered a nearly reactionless mechanism for converting the energy of object's rotational motion into gravitational potential energy of that object. The key words here are nearly reactionless. Converting the energy of rotational motion to and from potential energy of gravitation is no big deal provided there is no shortage of gross matter to be used as reaction base. Indeed, that is exactly what Maxwell's Pendulum does – it converts back and forth the energy of rotational motion to and from potential energy. But it is entirely different story when there is no gross matter in abundance, like earth, to thrust from.
If this hypothesis turns out to be true, then Laithwaite's mechanism could be used for converting the energy of rotational motion into the energy of translational motion in open space, i.e. it could become the basis for constructing a propulsion system where a relatively small amount of jet reaction is used as a trigger mechanism for converting the energy of fast rotational motion into the energy of translational motion of the rocket, while the rotational motion itself is sustained by mass-less, or nearly mass-less means, like electricity or nuclear energy.
I can't stress enough the importance of the underlined part of the previous sentence: the conversion of the energy of fast rotational motion into the energy of translational motion is not possible in open space without using a relatively small amount of jet reaction as a trigger mechanism. That is exactly what the large army of so-called inertioid inventors are missing: all those wonderful "reactionless" devices always work in terrestrial conditions, no matter how smooth the surface of contact is, but they always fail when there is no surface of contact. That's why no inertioid will ever work in open space without that tiny amount of jet reaction to be used as a trigger mechanism. In the terrestrial conditions the role of the trigger mechanism is played by the forces of friction, the tiny amounts of which are always present no matter how hard one tries to get rid of them by using lubricants, rollers, or any other means.
The above statement requires an important clarification. I do not believe for a second that Newton's third law is violated in Laithwaite's experiment, or could be violated for that matter. That is to say, if we have a completely isolated system then there is absolutely no way to convert the energy of rotational motion of the parts of the system into the energy of translational motion in a way that upsets the state of rest, or uniform and rectilinear motion, of the barycenter of the system. But the fact is that there are no completely isolated systems anywhere in the universe. While speaking of isolated systems, we usually think in terms of gross, i.e. ponderable, matter. Excluding aether from the picture is what makes apparent violation of Newton's third law possible.
As far as I know, it had never occurred to Laithwaite, or anyone else who repeated his big wheel experiment, to measure the spin rate of the wheel at the high point and compare it to that at the bottom. If the two rates are equal, that would refute my hypothesis, but if the wheel's spin rate is slower at the top and, in addition, the difference could account for the difference in gravitational potential between the lowest and the highest points, that would boost the credence of this hypothesis.
Hypothesis 2. Laithwaite has discovered perpetuum mobile. Let's assume that the spin rate of the wheel turns out to be the same at low and high points. Then since we have managed to lift the spinning wheel almost without any muscular effort, we basically have a perpetuum mobile for we have gained in potential energy without any apparent expenditure of work or energy: just let the wheel drop from high point and collect its gravitational energy potential when it hits the ground, then lift it again effortlessly, and keep repeating the cycle.
This statement also requires a major clarification. I do not believe for a second in the feasibility of violation of the energy conservation law. If this hypothesis turns out to be true, that simply means that Laithwaite has managed to tap into the practically bottomless energy reservoir of imponderable matter.
Hypothesis 3. The effect has something to do with earth's magnetic field. I would suggest repeating the experiment with a heavy wheel made of nonmagnetic and nonconducting material, and see if that makes any difference. This would help to form an opinion of this hypothesis' credibility.
In my judgement, Hypothesis 1 is by far the most credible.