X-No-archive: yes The Least Action Consistent Stable Universe and the Mathematics Modified June 6, 2009, October 21, 2009, October 23, 2009 John Lawrence Reed, Jr. Section 4
In the Beginning Isaac Newton defined centripetal force in terms of his second and third law, to act at a distance, by setting his first law object on a circular path of motion, at a uniform orbital speed. Newton allowed the moving inertial object to impact the internal side of the circle circumference at equidistant points to inscribe a regular polygon. He dropped a radius to the center of the polygon from each vertex (B) of the polygon to describe any number of equal area triangles. “...but when the body is arrived at B, suppose that a centripetal force acts at once with a great impulse...” (Principia) To argue for his supposition, Newton took the triangle base length, toward the infinitesimal limit approaching zero. The base length, and the infinitesimal arc of the velocity driven and time consuming trajectory of the moving inertial object, can then be represented as arbitrarily close in length as desired. The velocity-acceleration vector (v/t), or (dv/dt) at the vertex (B), is by definition consistent with the continuous and efficient curvature of the circle, and is ultimately directed along the radius toward the center of the circle and represented as centripetal acceleration (v^2/r). This time- space mathematical property of the perfect circle and perfect motion, serves as the least action consistent, mathematical carrier for “inert” mass, by assigning inert mass as the cause of the defined centripetal acceleration, and designating (mv^2/r) as centripetal force. This is done mathematically by multiplying two sides of a least action consistent equation by unity, in the form of [m/m]. Note again that Newton used a perfect circle and perfect motion, to derive his supposition for a mass driven centripetal force from a defined instantaneous acceleration (velocity), where the only change in velocity is direction. Here the equal areas in equal times falls out of the perfect orbit, as a least action (efficient) mathematical artifact of the efficient (least action) area enclosing circle itself. Consider: A circle is an efficient enclosure of area. That is, the circle circumference is the shortest line length to enclose the greatest area. Equal arc lengths from the same circle will radially enclose equal areas, just as equal time intervals from the same orbit will radially enclose equal areas. Recall from Ptolemy (see johnreed take 1a) that when we take the efficiency ratio of the circle as the quotient [circumference/area] or [2pir/pir^2] and reduce it, we have [2/r]. When we take the quotient of a circle’s [arc segment length to its radially enclosed area] we also reduce that to [2/r]. This is an efficient area enclosing, symmetrical property of the circle itself. As I have previously noted, this is, on the face, trivial and rather mundane, as it follows from the perfect spatial symmetry of the circle. With the real world orbits this symmetric efficiency is retained in terms of time and space. We have the efficiency ratio here as the quotient [the period/the area enclosed by the orbit]. The reduced quotient here when we take [r] as the average distance of the planets from the sun (as is done in the introductory physics texts), is [2/ rv]. This is a real world orbit time-boundary to enclosed space, analog of the circle’s length-boundary to enclosed area, efficiency quotient [2/r]. I’ll leave it to the reader to show that Kepler’s law of areas proves that the analog of the symmetry of the ‘circle’ efficiency, in the real orbits, is maintained. All you need to show is that the efficient symmetry quotient for any Kepler swept out area is [2/rv]. [Arc segment interval length to its radially enclosed area]. We can see that the efficient symmetrical property of the circle analog, is reflected in the real elliptical orbits as Kepler’s law of areas, where acceleration includes change in both the magnitude and direction of the object’s motion, and that the magnitude changes even as the direction changes, such that the efficient symmetrical, area enclosing property of the orbit is maintained. Newton extended his mass generated least action consistent property, to include the trajectory of two bodies in elliptical orbit. “Every body, that by a radius drawn to the center of another body... and describes areas about that center proportional to the times, is urged by a force...” (Principia) Newton further generalized the efficient equal areas in equal times property of the supposedly inert mass driven object’s perfect circular path, together with his centripetal force, to any curved path directed radially around a point. "Every body that moves in any curve line... described by a radius drawn to a point... and describes about that point areas proportional to the times is urged by a centripetal force... to that point” (Principia). Newton tied his “least action” mathematical model for a supposed “mass” driven centripetal force to gravity (our tactile sense of attraction to the Earth that we feel as resistance and quantify as weight [mg]). “For if a body by means of its gravity revolves in a circle concentric to the earth, this gravity is the centripetal force of that body.” (Principia) Note that Newton accepted as an a priori given, the resistance he worked against and called gravity, as a fundamental (soon to be , mass driven universal gravitational) force. Where the resistance he works against is either his own inert mass or the inert mass of another planet surface inertial object, that he senses (feels). It is of special significance that Newton formally generalized Kepler’s law of areas to the entire universe as the carrier for his mass driven centripetal force. “...because the equable description of areas indicates that a center is respected by that force... by which it is drawn back... and retained in its orbit; why may we not be allowed... to use the equable description of areas as an indication of a center about which all motion is performed in free space?” (Principia) Note that Newton asks, “why may we not be allowed… to use the equable description of areas as an indication of a center about which all motion is performed in free space?” There is no reason whatsoever that we cannot, “use the equable description of areas as an indication of a center about which all motion is performed in free space?”, provided we are referring to stable system orbital motion. Newton is describing “least action consistent motion”. However, there is a reason we cannot use our feel of force as the cause of that motion. There is a reason why we cannot use Kepler’s law of areas as a carrier for the force that we feel. Newton used Kepler’s law of areas, as the mathematical carrier for his assumed mass driven centripetal force. Therefore I conclude that Kepler’s laws are required for Newton’s mass driven centripetal force. Where conversely, Kepler’s laws do not require Newton’s mass driven centripetal force. Rhetorical Questions: Since this is in fact the case, how is it we say that Kepler’s laws "require" Newton’s mass driven centripetal force? That is, how is it we say that prior to Newton, Kepler’s laws were entirely empirical and that these empirical laws can be “derived” from Newton’s universal law of gravitation? How is it that we can hijack the time controlled least action consistent celestial object motion, and define it in units of [mg], as a mere “consequence” of the least action consistent resistance we feel as a planet surface inert object? The answer to this question will show the importance of acquiring a clear and unambiguous conceptual understanding of the phenomena, and of the applied mathematics. For the phenomena see my June 2009 post: “The Principle of Equivalence Explained”. Newton’s universal law of gravitation is defined consistent with least action motion. I will explain the least action consistent mathematics that supports this conclusion as follows: Recall that Kepler’s laws are consistent with least action, stable system orbital motion, and that least action orbital motion is a _requirement_ for a stable orbital system (see my post, “The Mathematics and the Universe, Part 5”). Kepler’s laws do not address the dynamics of the stable system. Rather, we have quantified the local planet surface object, conserved, least action consistent, dynamic mass magnitude, in terms of its resistance and its motion, based on our measure (and feel) of resistance, and we have proportionally applied this magnitude to the celestial least action universe, celestial object resistance and motion, as its controlling cause. We have done this even though inert mass does not enter into the planet attractor mathematics during freefall, during orbit and during escape velocity experiments. Because what we feel (inert mass) appears to control us, and because inert mass (what we feel) is conserved in the local planet or moon surface, least action consistent motion frame, we have proportionally generalized its local least action consistent magnitude to the celestial least action consistent universe motion. Since inert mass does not enter into the planet attractor mathematics, but does enter into our mathematics (what we feel), the entire Newtonian Classical, least action consistent, gravitational system model, is proportionally based on the magnitude of force “we must apply”, in units of [mg]. This is like saying, since two triangles are similar they are congruent. It is _the classical_ case of confusing correlation with causation. Outside a perfect circle and uniform motion, the "equable description of areas indicates" more than just a center "respected by that force". Kepler's laws show that a time function accompanies the force. Where is the time function in Newton's perfect circle and uniform motion derivation for centripetal force? Where is the time function in Newton's universal law of gravitation 1)? The only force that Newton entertains is the force that we as planet surface inert objects “feel” and quantify as accelerated mass [ma] and/or [mg]. On what basis do we conclude that the state of surface planet matter is the state of planet, and/or star core matter? Where experiment only shows quantitatively that our resisting mass represents the least action consistent, conserved cumulative resistance, of a planet or moon surface, inert object’s atoms (See: The Least Action Consistent Stable Universe and the Mathematics, Section 5, The Principle of Equivalence Explained, johnreed). Consider the mathematics: 1) F=GMm/r^2 Newton defined a universal gravitational force between two objects as a function of the product of their mass, where the function is attenuated by the inverse of the square of the distance between the mass centers. The centers of mass of the moving objects describe the least action consistent trajectory of those objects. I note that [1/ r^2] is an efficient quantitative, general (least action) property of the surface area of a sphere. I also note that the mass density of the objects is an “invisible” shared variable, semi-dependent on the least action consistent, inverse of the square of the distance [1/r^2] between the objects, and that the scale of proportion is set by the (subjective) local measure and quantification of inert mass [m], and the local constant of proportionality [G], originally measured as a function of torque (Cavendish) between suspended planet surface inert mass [m] objects. The celestial quantity [Mm] is ultimately scaled proportionally to the local planet surface mass [m] magnitude by the so called constant of proportionality [G] measured as a function of torque between two suspended planet surface inert mass objects. The quantity [GMm] is then further attenuated by the quantity [1/r^2] which is consistent with the efficient surface area of a sphere. Consider the mathematics: 2) F=4pi^2mr/T^2 The right side of 2) reflects the efficient least action consistent properties of perfect circle and perfect motion orbits, where planet surface object inert mass [m] has been included by using the mathematical technique of multiplying both sides of an equation by unity (in this case [m/m]: this operation is not shown here but accessible in many introductory physics texts). Then the introductory text will set 1) equal to 2) as: 3) GMm/r^2=4pi^2mr/T^2 Where on rearranging and simplifying we have: 4) T^2/r^3=4pi^2/GM The introductory physics text will now argue that 4) shows that Kepler’s third law [K = T^2/r^3] is “merely a result” of Newton’s gravitational law 1), and “... although this derivation uses perfect motion and perfect orbits, it applies equally well to real orbits in real motion provided we use the average distance from the sun to the planet for [r].” (Paraphrased) The introductory physics text states that the derivation here uses perfect circles in perfect motion (where we have the efficiency quotient as either [circumference/area] or [the period/area]). And then it states that the derivation applies to real orbits as well, provided we use the average distance from the sun to the planet for [r]. So that the efficiency quotient in the real orbit case is: [2pir/ pir^2] or [T/pir^2]. Clearly, nothing has changed mathematically. They each reduce to the efficiency quotients [2/r] or [2/rv]. Where the time function is obscured as it remains joined to, and, as an artifact of, the perfect circle in uniform motion. In 2) we have the perfect orbit and perfect motion where we allow our local quantity for resistance inert mass [m], a free ride, by multiplying both sides of a least action consistent equation by unity [m/m]. Then we use 3) and 4) to eliminate inert mass [m] from the derivation, while including the local inert mass [m], empirical “scaling” measurement [G], and applying it to the celestial least action measurements that accompany the least action orbits, to proportionally define the magnitude of mass for the celestial object [M]. In other words, we arbitrarily assign as a proportionally controlling property of the celestial body least action orbits, the constant quantity [G] measured locally as a function of a quantity that we feel, inert mass [m], which quantity is independent of the planet attractor mathematics. We call [G] a constant of proportionality and wonder why we are missing what we call dark matter and dark energy (see the paper by Andre Michaud at: http://www.wbabin.net/science/michaud1.pdf). Since inert mass is an influential quantity (what we feel) in our mathematics, but is an independent quantity in the planet attractor mathematics, how is it we measure [G] as a function of planet surface object inert mass, and assign it as a constant of proportionality for an inert mass, that applies to all celestial bodies? The planets do qualify as celestial bodies. Since inert mass is independent of the Earth attractor action, it is a reasonable generalization to conclude that inert mass is independent of the action of all celestial attractive planets and moons. Again, it’s like saying since two triangles are similar they are congruent, and again, it’s _the classic_ case of confusing correlation with causation. We are inertial mass objects. When we are in contact with the planet surface we feel the Earth attractor pulling action on our bodies as resistance. We can measure this resistance on the balance scale by comparing it to a planet surface, inert mass object with a known standardized resistance. The balance scale compares the resistance of planet and moon surface bodies. The balance scale does not show that the attractor action is on mass or weight, where surface planet object mass and weight are quantities that apply to the effort we feel and expend to lift the resisting surface planet body. The balance scale action is anonymous with respect to the Earth attractor. This action compares two planet surface inert mass objects, where each inert mass is equal and opposite to the effort we expend in lifting that inert mass object, and which we quantify as weight [mg], and call force. Again, we conclude that the Earth attractor acts on our inert mass [m] as a function of a force (our quantified weight [mg]), which we measure and feel, where inert mass [m] is conserved, independent of the Earth attractor focus. Since what we feel [m], is conserved, independent of the Earth attractor focus, again I ask, how is it we can assign what we feel [m] as the controlling cause of the order we observe in the least action consistent celestial universe? Because the resistance of a planet surface body, its mass, is equal to our expended effort to lift the body, we conclude that our effort is equal and opposite to a universal gravitational force we respond to, rather than merely a force we exert? We say that the Earth attractor applies just the necessary amount of “our notion of force” to equal our weight and our effort. We call the force we think we respond to, “gravity”, and define it mathematically as equal and opposite [mg] and as [GMm/r^2]. How much clarity is required here? Both formulations are consistent with least action motion. Both are equal and opposite to the effort we apply to a resistance. Consider the mathematics: In (1) where [M] represents the mass of the planet and [r] represents the distance to the center of the planet from the planet’s surface, the resistance we as inert mass objects work against at the Earth’s surface is formulated as: 5) F=mg We must exert effort (force) to lift, to overcome the resistance [m] of the planet surface inertial object. That is all we are doing. We must exert an effort to do this. This effort begins and ends in our bodies. No inanimate object exerts effort. We quantify the resistance to our effort in terms of the object that we lift, and in terms of its location, as it’s weight [mg]. We say that the Earth attractor focus is on the object’s weight [mg], and the magnitude of that Earth attractor focus is on, and is equal and opposite to, the weight [mg] of the planet surface object that we lift. Here we are the living constant of proportionality. The so called universal force of gravity is tailor made for our effort. Where the effort we apply is equal and opposite, only to the resistance of the object that we lift. The focus of the Earth attractor on the object, can be objectively explained as a super electromagnetic action on the atom, using functional mathematics that is consistent with the present subjective inert mass (resistance) gravitational paradigm. We already know that the quantity inert mass is independent of the Earth attractor action. Whereas the Earth attractor action on the atom unambiguously explains why all objects fall at the same rate, without recourse to the third law, which law again, defines force [mg] solely in terms that are consistent with the resistance (our effort) that we quantify in units of [mg], and work against as inert mass objects. We call our effort force. That is reasonable. We feel force. The Earth attractor pulls on atoms and we pull back. However, we have assigned our quantified equal and opposite inert mass defined “pull back” to the entire universe, and we call it a universal gravitational force (or, even more obtuse, a consequence of a curved space-time). So that we set (5) equal to (1) as: 6) mg=GmM/r^2 Although we have defined two least action consistent formulations for a supposed mass generated force, when we set them equivalent in 6), planet surface mass [m] is not a functional part of the least action consistent formulation. We interpret this as though it is a mysterious consequence of the fact that all objects fall at the same rate. Where inert mass [m] is simply not acted upon by the Earth attractor. There is no mystery here. The reason little [m] divides out of the equation is because we have defined what we feel as resistance, as a universal force that is a controlling function of inert mass [m] in both cases. When they are set equivalent planet surface object inert mass [m] vanishes. Where each definition separately, applies to us as inert mass objects and the conserved least action consistent interaction with planet surface inert mass objects and planet surface objects least action consistent, equal and opposite interaction with celestial objects, but fails to address the time controlled aspect of the least action consistent celestial universe. Consider further: We can define the least action behavior of the celestial universe in terms of the force we feel and quantify as [mg] (which again, operates anonymously within celestial least action parameters), and set both formulations equivalent in 6). So mass [m] divides out of the calculation as a matter of definition, that is consistent with the fact that all objects fall at the same rate, which, in this view is another obscuring correspondence, rather than an amazing co-incidence, that completely excludes the significance of the universally general, celestial orbit time function. The fact that inert mass objects fall at the same rate independent of their mass means that with respect to celestial bodies, inert mass is insignificant, and emergent with respect to the interaction between planet and moon surface bodies. Inert mass is conserved within super electromagnetic celestial time controlled least action parameters. Which electromagnetic time function, when it does eventually appear as controlling, appears as another dimension in general relativity as a "space-time" curvature? A consequence of Newton’s original invisible inclusion of time [T] as a mere artifact of the circle itself. We divide the quantity mass [m] out of the equation, but we retain the proportionality of the attendant planet surface object, least action properties that define the independent, inert mass [m] magnitude object motion, as a function of the local least action planet surface object resistance and least action consistent motion measurement, and again, apply it proportionally as a hard wired property of the universally general, celestial least action consistent kinematics. We are defining the celestial universal order, after our own local, least action consistent, planet surface inert mass image, in terms of a force we exert and consequently feel. Newton defined gravitational force in terms that are proportional to the local least action consistent empirical measurements accompanying planet and moon surface mass [m]. This includes the equal and opposite comparative behavior of impacting inert mass objects, the comparative measure of inert mass on the balance scale, the uniform time controlled planet specific anonymous property [r] and [t] dependent, accelerative action (g), and the gravitational constant [G] measured as a function of planet surface object, “independent” inert mass [m]. The magnitude of [g] varies from location to location so that the attraction between celestial bodies is defined in dynamic terms that are proportional to the resistance we feel, using the similarity of measurements accompanying least action consistent motion, where the quantification [m] of that resistance is independent of the celestial attractor action. Little [m] divides out because we have defined force as a function of [m] in both cases. However, the so called universal gravitational force is solely defined in terms that are proportional to the resistance we feel, which does not divide out, and does not include, but is consistent with the least action time function. Again I ask, is it any wonder that we have a dilemma regarding our cosmological calculations for celestial object, mass that requires the proposed dark matter and dark energy? Consider the mathematics: 7) g=GM/r^2 When we divide little [m] out of [6], we are left with [7]. Note again that [G], [g], and [1/r^2] are empirical measurements that accompany local planet surface least action consistent processes. Recall that the law of areas is a consequence of a time controlled least action orbit. So, when we divide [m] out, the result in [7] leaves [M] proportionally hardwired to our empirical measurements that accompany the least action physical processes involving [m], and extend to [M] via [1/r^2], a property consistent with a least action process. Again, in other words we have defined a universal celestial force in terms of the resistive properties of planet surface inert objects (which we qualify as and which we work against) that function “anonymously” with respect to (independent of) celestial bodies, solely within least action parameters. The least action parameters are today extended within a 4D mathematical framework called general relativity. These parameters are now known as “geodesics”. A term that simultaneously clarifies, even as it further obfuscates the underlying least action principle. johnreed Friday, October 23, 2009 Author’s after note: This is not to say that celestial bodies have no mass. It is to say that the true mass of some planet and all star cores cannot be based solely on the anonymous least action consistent behavior of planet surface object mass. The celestial core inert mass while causing the eccentricity of the super electromagnetically Sun, time controlled orbits, and possibly causing the planet precessions, and the "least action" inclination of Uranus, have no further visible consequence, at the present time. And it is in part, to propose as an alternative possibility, that in the case of active stars and some planets, their present inert mass is a consequence of the structural build of their super electromagnetic cores, which super electromagnetic cores would be the primary source of the celestial controlling attractor action. (See johnreed Take 23 - Dark Matter and johnreed Take 24 - What is Super Electro-Magnetic 'Gravitation'). These super atomic star cores are created in an action-reaction process that I call constructive electromagnetic fusion in response to extreme pressure, and are the probable cause of the red shift we note in distant galactic spectra. Analogous here to the similar but displaced vibrations emitted by different sized tuning forks., etc. A closing note to Section 4: The reader’s indignation runs high with many of my posts. I can understand that. I challenge the very foundations of physics, as we know it. I am only the messenger and if the message is valid, personal attacks on me serve no purpose. If the message is invalid, then again, the message should be attacked, not the messenger. To avoid the issue by saying the post is too speculative because it opens a rational, factual based door for thought, is without merit. The proper questions to consider are: Are my base arguments valid or invalid? If they are valid, are they significant or insignificant? I note, but do not limit these arguments, to the following: Argument 1: Isaac Newton defined a mass generated centripetal force in terms of his first law object moving along a perfectly circular trajectory at a uniform (perfect) speed. I show that this is true directly from The Principia. Argument 2: The law of areas falls out of a perfectly circular trajectory and uniform speed as a property or artifact of the efficient area-enclosing circle itself. I see this as self-evident but I have explained it on the first page. Argument 3: Kepler’s law of areas is an efficient symmetry analog of the circle. I show this by first showing that the efficiency quotient of the circle reduces to the efficiency quotient of any radially enclosed area of the circle. Then I show that Kepler’s law of areas proves the analog case for the real orbits. Argument 4: Isaac Newton connected this efficient property of the circle to its analog in the real orbits and used Kepler’s law of areas to carry his idea of centripetal force, mathematically, to the entire universe. I show that this is true directly from The Principia. Argument 5: I show that our mathematical derivation of Kepler’s laws from Newton’s universal law of gravitation ultimately rests solely on least action motion. The fact that all objects fall at the same rate also shows that mass operates solely within least action motion, “anonymously” with respect to celestial bodies. Argument 6: I show that setting the quantity we work against and call weight [F=mg], equal to Newton’s universal law of gravitation [F=GMm/ r^2] co-opts the least action properties attendant to an anonymous object in motion, and proportionally renders these properties solely in terms of the resistance we work against and call force. johnreed --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Epistemology" group. 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