A reference frame is a mentally constructed tool used to communicate, record and store referential information about anything at all.
The common reference frame that goes unnoticed is a fractal pattern we learn to call our own language. Such an example serves to illustrate not only the complexity possible in a reference frame but also the prototype of all reference frames.
I want to restrict my discussion to those reference frames hich are astrological in origin, and metrical in procedure, requiring Metrons and presenting information in a quantified way.
I can go way back to find the artistic origins of the astrological reference frames and the tools and devices used to quantify and compare quantities of orientation, position motion etc. Babylonians Egyptians Dravidians Harappians Chinese Greek and Arabic empires have brought to us a wealth of such reference frames so by no means does Newton shine brighter than any other star in yhis galaxy of predecessors. In fact Wallis and Descartes should be mentioned.
It is my device to start with Newton, because he devised a philosophy of Quantity in his work the Method of fluents. However, again it is as if he had never read Plato, Socrates, Euclid, Archimedes and many other Greek and Arabic luminaries including Hooke and others! This is why Newton always said he stood on the shoulders of giants.
Newton's work then provides a useful and enlightening introduction to the principles of the Astrological reference frame.
We must also acknowledge later great introductions in particular Joseph La Grange Celestial Mechanics.
Newtons take on the reference frame is still a delightful foundation.
The mechanics of the method of fluents, though detailed is not complete. It relies on a prior assumption of a subject called misleadingly Geometry instead of Gematria or combinatorics. It is misleadingly linked to the Stoikeioon of Euclid, which indeed covers the philosophical reasonings and skills required to form an adult and adept view of the social cultural matrix in which one lives, and to properly respect the " gods" as things yet to be understood but never to be ignored!
I find therefore that LaGrange particularly draws from Newtons collection of ideas a more complete method of fluents, and that Kant initiated a debate that underpinned the Humboldt education reforms in the Prussian Empire , because of the enlightenment within Newton's body of works including the Astrological Principles.
Thus Newton had a direct influence on De Moivre and Cotes and an indirect one on Grassmann nd Hamilton through Kant and La Grange.
Euler perhaps is the most astonishing of the heirs of Newton, but in no way is he any less of a productive genius than Newton. In many ways he fills out and bursts through the Newtonian concepts!
Ok, so the fundamental aspect of Newton's reference frame is the "vector " designation and combinatorics.
Based on the Pythagorean school, the sphere and the circle are fundamental Metrons within which all forms can have a multiple reference frame, and similarity and proportion figuratively explained.
It is the spherical reference frame that Thales reportedly brought back to Greek culture and society from Egypt. This is the source of the so calle Pythagorean theorem . It is the ground of the Eudoxian system of proportions and his theory of proportionality. It is the ground of trigonometry and the centuries long endeavour to calculate the sine ratios. It is the source of Indin astrological theory of measurement of star and plant motions and thus the very foundation of the records caled Tymes! It is therefore heartening to find that it was at the very heart and core of Newtons principles of Astrology as a fundamental organising principle.
Now Newton was not alone in revering and relying on the spherical and circular form, but it was Newton that fundamentally understood its mechanical importance to unifying disparate ideas. And yet, as much as he grasped and taught De Moivre it was Roger Cotes who had the insight that later Euler fully exploited.
Newton would have marvelled at Eulrt's developments of his work, just as he hoped Cotes would have revealed to him.
Now Newton proposed one framework, usually described as his third law of motion, none of his laws of motion are to be isolated from the sphere and the circle nor from each other. Newton describes an equilibrium in space relative to the observer in which each action has an attendant dual and opposite action all motion, strength , liveliness is relative to this sphere of influence and consciousness. In this set up Newton observed the fundamental importance of the right triangle itching the sphere. He also in his work on the motion of heavenly bodies, explains the importance of. The parallelogram.
These fundamental tools were essential for his propositional and phenomenological explanation of motion. Metaphysically he is no different to his peers, supposing god ordained properties or attributes such as celerity and vis, that is strength of motion, viva liveliness and excitability.
However, his one empirical difference, based on experimental work inspired by earlier philosophers like Gallileo , and Hooke, Huygens, wren etc was to notice a celerity source, which he called motive. This source provided what he called added celerity, translated as acceleration. So now he had a cause behind a cause, and that was revolutionary!
His method of fluents may have inspired this insight or been inspired by this insight, but certainly this step of relating the increase in celerity demonstrated by the increase in velocity of a falling body to a cause at once brings the thinking in his time up to date with the way we think. This is because we adopted Newtonin thinking on this one matter to do with velocity..
Of course the wag in us immediately generalises to a cause for this cause called motive, and then a cause beyond that etc. this is perfectly acceptable and is to be encouraged because it demonstrates the fractal scale relationships all around us. However. Religious dogma discourages this type of thinking! The " correct" thinking is to ascribe all cause to the first Cause, hat is god. To a very great extent, this was Berkleys beef ith Newtonian Fluxions, they denied god the right to be a first cause, pushing hom back into the distant end of a fractal scale chain of causes.
Newtons frame of reference was the sphere. Straight lines tangents radii of the sphere and right triangles and parallelograms within the sphere were fundamental. When Wallis fixed Descrtes flexible coordinate system, based on the acknowledged Greek methods Bombelli re discovered for the west, because the west unlike the east was in academic abd religious darkness, this formalised the Newtonian reference frame into the form we use today. For some reason we call it Cartesian.
The centripetal and centrifugal descriptors are fundamental to Newtonian reference frames. Some today attempt to decry the centrifugal descriptor, but these pedagogues have perhaps never read the principles of Astrology by Newton. In any case , as a force or action system they absolutely are both required to explain motion..
The Newtonian combinatorial algebra for vectors when laid out in full covers every conceivable motion. Like the Greeks curvilinear vectors were included, even though no child today is taught about curvilinear angles or arcs etc.
Admittedly they are in the Stoikeioon, an introductory philosophy course, but in mechanics, especially fluid mechanics curvilinear vectors are everywhere in use. The horn or the curvilinear vector is usually presented as a tangent to a curve, and the tangent is emphasised. It is this insistence on straight lines that obscures the adaptive power of Newtonian reference frame, and hich lead Grassmann uptown state that his next task was to develop his analytical method for the circle and thus the general curve. He did not complete that work, but suffice it to say that Huygens , Newton and others already used the curvilinear vector in the circle reference frame called the radius of curvature.
Hamilton and Clifford have moved to a more general algebra essentially furthering the groundwork of the Grassmanns, although Hamilton certainly can claim an independent derivation of his Quaternions.
Clifford's algebra is reputedly the most general, but only a few seem to be enamoured of it, let alone the Measurier Numbers
Back to centripetal and centrifugal. When describing circular motion one particular resolution of the motion is chosen, the tangent and the radial. This is an inadequate resolution for circular motion. You see this simply by applying the parallelogram combinatorial rule: the combined forces do not follow a circular path! Yo need the centrifugal force to bring the resultant back to anywhere near the circle!
Why are we taught this incorrectly? Newtons explanation of motion were not always understood even by his supporters. Circular motion is not a static force system it is a dynamic equilibrium system requiring 3 forces, one of the forces , the centrifugal one reflects the tension keeping an object in orbit! The other tension tries to pull an object out of orbit, and finally the tangential force moves the object along the orbit, but that is resisted by what Newton called an inertial force trying to keep it at a fixed velocity in that orbit. In the meantime the object is attempting to rotate about a centre of rotation due to vorticity and this is being resisted by a contra rotation, all these are impulse actions which either occur sequentially in a ctcle(as in a computer simulation) or instantaneously,
Instantanaeity is being severely challenged because we now have cameras fast enough to capture the wave or shear progression of stresses abd strains as they move through a medium. Thus sequential cycles seem to control thes behaviours.
What this means is that fluid mechanics is a more accurate descriptive paradigm than ever, and the impossible motion of fluids are now conceivable.
Back to centripetal and centrifugal. It is common to direct centripetal along a radius toward the centre and resolve it to the two axesc when this is done it becomes clear that one axis shows a centripetal action, while the other a centrifugal action. We cannot remove these fundamental centrifugal actions from Newtonian mechanics,
But also we se that circular or rotational motion is entirely described cy these two contra actions on axes whatever the angle between the axes. The tangent therefor is a resultant of the centrifugal and centripetal action dynamic.
A long while go, on Fractalforums.com I defined a term called Spaciometry. That general term I use today intuitively. I also defined spaciometric rotation as apprehending the boundary of a form.
This notion of rotation was meant to be arbitrary and general and include partial rotations. Any motion that allows me to apprehend the whole boundary of a form is a rotation of or around that form. Thus a partial rotation does not give me complete apprehension, but it gives me a sense of getting there. . This definition does not make sense of "infinite forms " because such forms are oxymorons. However a form that has infinite distinctions subjectively asserts that the form is bounded and can be infinitely distinguished. It is the bound which characterises the form, and one distinction I can make is the largest extent on the boundary. To do this I must be able to rotate the form spaciometric ally. This rotation is therefore fundamentally a part of the process of establishing a Metron.
Essentially then I cn only descrim
Be the measurement of forms that I can actually rotate.
The group or ring theoretic models of rotation therefore must be finite groups, because I must be able to apprehend all the aspects of a form to know I have rotated it.
Secondly the rotation action must be a closed result if modelled by any operation in the group or ring.
Thirdly there must be a sequence of operations in the ring that take me to every element in the ring or group for it to be described as a complete rotation.
Considering these requirements it becomes clear that boundless infinities are not able to model rotation. Bounded infinities also cannot model systems the rotate multiple times . We are therefore restricted to bounded finite descriptions of rotation.
Rotation, as Newton observed really pins our imaginings to specifics!