Or, by what formal means can we comprehend and communicate the nature of Nature?
From Mathematics: Nature and physics are “independent of mathematics”
From Physics: “We need the criteria of something that actually works, not just math rigor.”
“I appear to be caught between two successful disciplines, mathematics and science - rigor and usable results.”
[For full versions of the quotes in justice to the individuals review “What Is a Linear Representation of an Essentially Quadratic Phenomenon?” in Q&A on ResearchGate. Applied Mathematics is not specifically mentioned in the posts there.]
There exist phenomena in nature, for example kinetic energy of a mass in motion, which may be represented by the product of the mass times its present velocity times the linear average of its velocity between zero and its present value (K.E. = m*v*v/2 to clarify.) Now, ignoring the abstractions of mathematics and the accurate-enough theories of physics, what are the axes of any space in which kinetic energy may be plotted, noting that in that space energy is represented by a volume, area times length, and the area may be linearized with v as the square root of the area? Note that two axes of v and v/2 are needed to plot the area. What’s up? What are area and length in nature that are plotted in such a space? Is it significant in the natural process that v*v is an area in the plot? Is energy an essentially quadratic phenomenon of nature, or even essentially cubic? Are these matters so answered already they may be dismissed, or may some of us explore them for the insight that may bloom (flowers “blow” in some poems)?
These are questions of Natural Philosophy in the mind of a person that are not answered by the rigid explanations of the current state of ignorance that feels so knowing in each era. Aristotle’s view was great in his time, like ours is to us now. (The accepted solution is for an academic to fix the person who errs with an inoculation of facts “it is known that,” a familiar experience. In Charles Ives musical work, “The Unanswered Question,” the trumpet remained curious after the explanations.)
Looking to the 4thmillennium in the Gregorian calendar, will this inquiry have been resolved as natural processes are then comprehended? Given scales and data a number may be assigned to a measured kinetic energy that is accompanied by units to associate it with nature. The issue seems moot, but in fact, . . .
The Schrodinger Eqn. is a linear representation used in calculation that must be squared to be real, that is, the real result is a squared value, calling it quadratic will serve. The square root of two is formal, but the Schrodinger Equation produces a real value in the end after multiplication by its complex conjugate, “squaring.” The physicist is correct that in Physics it doesn’t matter “what the imaginary solution was” during calculation since only the result matters and other representations exist. Is that okay with you? I wonder, since another formal linear representation of an essentially quadratic phenomenon, the square root of two, cannot be written down as a completed decimal value, and during processing the imaginary Schrodinger expression requires representation as an area via a + bi, or (a, b), or . . ., which expressions are not real.
[Remember – ResearchGate encourages discovery, as if current knowledge could not only be incremented, but also that a new arena for thought could be discovered. “It is known that” must not be the whole answer, and exploration is in the charter.]
So, by what formal means can we comprehend and communicate the nature of Nature?
Happy Trails, Len
Howdy Robert A. Phillips,
We share a sense of Nature and of knowledge. Your preprint of "The Cause of the Gravitational Effect" reminds me of Albert Einstein's first cut in 1911 at the bending of the path of light in a gravitational field gradient. Your observation on KE in a fluid is appropriate to my fluid dynamics focus, but I would word it "Kinetic energy involves more than just the body. In any case our agreement is quite pleasant.
"Using a natural fluid model to interpret the nature of cosmological interactions resolves many of the "big questions." " You might enjoy my condensation cosmology concept. The Universe did not "inflate," but as primal energy condensed to matter, light slowed (same speed always is measured) and the Universe has appeared to expand.
The issue of whether natural processes "are quadratic" represented by v2 in K.E. = mv2/2 remains of interest to me in forming an object oriented model of flow in a mountain stream rapids. What is happening "in" a turbulent flow structure that must be specified in its object class data and functions, whether it is processed analytically or by a computer program. Imaginary and irrational number expressions may freely be used during calculation as long as they are made rational in the end. Do we need to notice that; does nature notice that; does it matter? Daniel Boone was never "lost" in the Kentucky wilderness, but on occasion he was "mighty bewildered" for days. Maybe I'm just bewildered.
Happy Trails, Len
Howdy Robert A. Phillips,
Okay, so, were we to start with a sense of momentum of an object, the changing of which we experience as inertial resistance, and note it by mv per convention, then to multiply it by v/2 would represent a change of momentum by 1/2 of its velocity, independent of the manner of the velocity change. Now its inertial resistance to that acceleration has been overcome by an effort equal to the change of momentum over the interval of the effort, strong effort - short interval and vice versa, but again the change is independent of the manner of the effort. ("Effort" here is called impulse.)
The new momentum will be experienced as if it were intrinsic to the object, so what is stored and how is it stored? I like to bring in the Fitzgerald-Lorentz contraction here, and justify its usage by noting that the object is composed of waveforms, many of which are orbital, and their survival requires internal connection and resonance. Therefore, they must individually shorten at the higher speed to "keep their head from abandoning their tail," that is for an orbital wave motion to overlie itself precisely. (The calculation of a light speed orbital oscillation accelerated to the speed of light yields a zero-length orbit at light speed.)
So, why multiply mv by v/2 to represent the new state of the object? Well, if the new state of the object were considered primary, and it were given by mv2/2, which is known as kinetic energy of the object, then we would have that momentum is a linearized representation of a fundamentally quadratic phenomenon, m*v*(v - 0)/2. With differentiation this is a simple step that yields mv, but why does differentiation mimic linearization - is it because the rate of change important to the effort is not contained in the energy? I'll have to review that. Does this mean that v2 is fundamentally quadratic?
This has all been fresh, here and now, except the F-L contraction ideas. It is exploring, and rescue will be welcome,
Happy Trails, Len
Howdy Robert A. Phillips,
Further thought has clarified and corrected parts of my preceding post as will be explained after an observation.
Perhaps I am just up against "wrongs making a right" in efforts to explain and calculate, but I feel we are looking at symptoms of a new understanding we fail to recognize - valid intuition versus wishful thinking? Either may be the route to insight, perhaps in another's mind.
So, seeking expression of the nature of Nature amenable to calculation and prediction, we seek kinetic energy of an object of mass m as our objective and start in a reference frame with our object at rest but free to move. Work as force times distance rings true enough, while force measured as change of momentum with time also rings true as a fact of natural processes. Work on the object will increase its momentum and be stored as kinetic energy of the object. Motion of the object will begin with zero velocity and increase to some value, v, over the distance that the force (assumed constant) is applied. The distance will be given by the product of the average velocity, (v + 0)/2, and the interval involved, say delta-t. The change of momentum will be given by m(v - 0), again over the interval delta-t. Thus, Force = m*v/delta-t, and Work = Force*v/2*delta-t = (m*v/delta-t)*v/2*delta-t = m*v*v/2 = mv2/2 in this case of acceleration from rest. The factor of 1/2 is clearly a result of the using average velocity over the distance of the applied force.
Now, all this is well known in more normal presentation but this approach does separate the contribution of v as the source of the distance in work from the presence of v in momentum, indicating to me that kinetic energy is not fundamentally quadratic in v2. It also indicates that it is inappropriate to think of momentum as a linear representation of kinetic energy. However, as observed above, "the trumpet remains curious."
Happy Trails, Len
Howdy Robert A Phillips,
It seems my inquiry was too succinct. It did include an elfin (lfhin) quality.
I was mystified by the non-intuitive details of a formal treatment after your observations on the "Gravitational Effect" bending light paths. I did not memorize MTW's "Gravitation" when I read it after spending myriad mystified moments in tensor analysis and numerous treatments of Relativity by Albert Einstein and others, but your treatment looks valid within that formalism.
Now, "a mass in motion has the same energy as a stationary one." requires specification when we are attempting to communicate the nature of Nature. Mass energy, kinetic energy, potential energy, "free" energy, etc., must be sorted out to communicate in such a phrase. "The energy of acceleration merely moves the time-space vector ..." describes a form of energy (work) storage to me with different wording. Oh, and "Note that an equal amount of energy must also be added to the system to decelerate the body." needs explanation if "added" were valid instead of "subtracted" in a simple mistyping.
I meant to request explanation with "Hunh?" and do so herein. For instance, the potential energy of my great-granddaughter at the top of her swinging arc is proportional to what clock rate?
Happy Trails, Len
Yes, we can understand and talk about how nature works using math and science. Math helps us describe things precisely, and science helps us discover the rules of nature through observations and experiments. By using math and science, we can appreciate and share the amazing details of the natural world.
Howdy Dharmadas Mardanya,
"understand," "talk about" and "describe things precisely" are very different concepts. To "talk about" or "describe precisely" the event of lunar cycles and motion certainly may be done well, although the latter requires additional data in most cases; to "understand" the lunar cycles and motion is very different. Just in the West, the discussion one may imagine among Aristotle, Ptolemy, Copernicus, Kepler, Newton, and Einstein when expressing an "understanding" on the topic has many different forms. It is unwise, even today, to claim that we "understand" lunar motions.
I used "comprehend" in my original question in an attempt to encourage thinking beyond talk and description. Even within an era dominated by a single paradigm, say Copernican, there is sufficent variety to indicate that understanding is not present. And details: he declined to use anything but natural motion of circles - did he "understand" or "comprehend" the nature of Nature in this case? Ptolemy's theory using mathematics and science resolved the location of the moon in our sky quite accurately, but its size varied by a factor of two in the calculations, so the "rules of nature" to which you refer are not all that accurate.
My question is about how we might comprehend and communicate the nature of Nature with mathematics independent of nature and physics and physics pursuing useful results. This paradigm's textbooks won't work to explain (understand) the next paradigm's approach unless we think in a larger space.
Happy Trails, Len
Howdy Robert A Phillips,
Not to worry about timing, to be involved in these discussions is part of a plan for my life after-wife and necessary adjustments are coming into place to make this timely.
Clarifications: "space" as the translational component of orbital motion, "time" as the oscillatory component - motion of orbit or motion in orbit? - plane orbit or spherical? Does orbital motion occur through space and over time, or produce them? I suspect the former, but "space" ("time") as . . . component leaves it open.
: "I found that a velocity vector (speed of matter) and time rate vector (speed of time) must have a resultant vector of length = c, due to the energy of matter being mc2." For me the mc2 energy is affected by apparent mass that changes with velocity (See my Wave-In-A-Well model on inertia) so your "due to" conclusion is not clear.
: " The work applied rotates the velocity/timerate arrow without changing total energy." As I asked last time, why isn't this rotation a form of stored energy.
: " It seems that the work changes the direction of the arrow, always decreasing oscillatory energy as translational energy increases (and vice)." Work done from the enironment accelerates, work done on the environment decelerates; the direction of work is the same, but acceleration increases the energy (rotation) of the body and the work it does on the environment decreases the energy (rotation of the body). It is all positive work, on the body and then on the environment.
Decades - my explorations leading to my Wave-In-A-Well model started in 1973, so I am familiar with the experience. There is a lot packed in each. My objective was/is to comprehend inertia; what is your guiding objective in this treatment of energy?
It is time to get back in the saddle and enjoy new Happy Trails. Thank you for the posting, Len
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