johnreed 1st addendum to "johnreed Catch 22" modified July 5, 2007

johnreed 1st addendum to "johnreed Catch 22" - June 6, 2007

Kepler's laws are thought to be the consequence of Newton's universal
law of gravitation. I will show in this post that this is incorrect. I
will show that Kepler's laws follow from the efficient, least action
motion, common to stable systems in our universe. I will show that
Newton's universal law of gravitation operates within, and in fact co-
opts, this least action motion.

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 an
imaginary circular path of motion, at a constant orbital speed. Note a
perfect circle and perfect motion. 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 infinitessimal 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 assigned carrier for "inertial" mass, as the cause of
the defined centripetal acceleration and is designated as centripetal
force (mv^2/r). Note again that Newton used a perfect circle and
perfect motion to derive his supposition for a mass driven centripetal
force from instantaneous acceleration where the only change in
velocity is direction.

Here the equal areas in equal times falls out of the perfect orbit as
a mathematical artifact of the efficient area enclosing circle itself
(See Take II). This efficient property of the circle is reflected in
the real elliptical orbits as Kepler's law of areas, where velocity
includes both magnitude and direction, such that the efficient area
enclosing property of the orbit is maintained [1].

Newton generalized the efficient equal areas in equal times property
of the supposedly mass driven 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 extends the mass generated 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 ties his "least action" mathematical model for a supposed mass
driven centripetal force to gravity. "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
accepts the resistance he feels and calls gravity, as a fundamental
given.

It is of special significance that Newton 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).

A circular orbit implies a centripetal force. However it does not
necessarily imply a mass generated centripetal force, nor does it
necessarily imply a centripetal force of the type we feel. The fact
that we can quantify the resistance we feel in terms of inertial mass
and call it gravitational force does not require that the earth
attractor act on the quantity of resistance we feel.

Kepler's laws reflect efficient, least action motion common to stable
systems in our universe. Newton generalizes to the entire universe,
and co-opts, Kepler's law of areas, as the carrier for his mass driven
centripetal force. Since Kepler's laws are required for Newton's mass
driven centripetal force, 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? The brief answer to this question shows how important
our definitions and conceptual understanding of the words we use with
the applied mathematics, is. Consider:

1) F=GMm/r^2

We can see from (1) that Newton defined the gravitational force
between two objects as a function of the product of their mass where
the function is solely attenuated by the inverse of the square of the
distance between the masses. Note that [1/r^2] is an efficient least
action property. Note also that mass density here is a variable,
solely dependent on [r]. Consider:

2) F=4pi^2mr/T^2

The right side of (2) reflects the efficient properties of perfect
circle and perfect motion orbits, where mass has been assigned to
apply by using the mathematical technique of multiplying both sides of
an equation by one. 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

Author's Note: In (2) we have the perfect orbit and perfect motion
where we allow our sensory quantity [m] (for resistance we feel), a
free ride. Then we use (3) and (4) to eliminate [m] from the
derivation while including [m's] empirical measurements and the
measurements that accompany the least action orbits, to define [M]. In
other words, we assign the resistance we feel and quantify as mass
[m], as a controlling property of the least action orbits. Then we set
the formulations equivalent where [m] divides out of the equation. We
say this is to be expected since all objects fall at the same rate.
This is functional in terms of time and space only folks. Not
necessarily functional in terms of the dynamics of planets, moons and
stars, which must include density as an attendant consequence or cause
of the controlling attraction, rather than as a mere function of [r].

The introductory physics text will now offer that (4) shows that
Kepler's third law is merely a result of Newtons gravitational law.
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 last paragraph is rather interesting. It 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. They each
reduce to [2/r] or [2/rv].

Newton's centripetal force is defined within the parameters of a
perfect circle and perfect motion. A circle is efficient. Newton
connects this efficient property of the perfect circle in perfect
motion to its analog in time-space elliptical orbits. My analysis of
centripetal force as put forward by Isaac Newton revealed that the law
of areas falls out of Newton's perfect circle and perfect motion as an
efficient property, or artifact of the circle itself. Newton used this
property of the real orbits to generalize his supposition for a mass
generated centripetal force, to the entire universe.

Kepler's laws have since been regarded as mere empirical facts, that
are a consequence of Newton's laws. True, it is not the law of areas
that is fundamental here. Rather, it is the principle the law of areas
obeys. That principle clearly does not depend on mass. That principle
results in time controlled efficiency. We see it now as the universal
carrier for Newton's notion of a mass driven gravitational force. When
Newton asked "...why may we not..." generalize the law of areas to the
entire universe, as a carrier for his defined force, it almost seems
as though the subconscious half of his brain suspects something is
wrong. Doing so will carry his idea of a mass generated centripetal
force with it. Making it clear to me that the least action, time
controlled property of stable systems are used as the carrier for
Newton's idea for a mass generated force.

The introductory physics text approximates the orbits as circular and
notes that a circular orbit implies a centripetal force. It is
important to note again that while such an orbit implies a centripetal
force, it does not necessarily imply a mass generated centripetal
force, nor does it necessarily imply any force of the type we feel.

Consider:
In (1) where [M] represents the mass of the earth and [r] represents
the distance to the center of the earth from the earth's surface, the
resistance we work against at the earth's surface is formulated as:

5) F=mg

We must exert effort to lift, to overcome the resistance of the earth
surface inertial object. We call this effort force. (The earth
attractor pulls on atoms and we pull back. We have assigned our "pull
back" to the entire universe and we call it gravitational force.) So
that we set (1) equal to (5) as:

6) mg=GmM/r^2

Although we have defined two different formulations for a mass
generated force, when we set them equivalent in (6), mass [m] appears
to not be a part of the formulation. We see this again as a
consequence of the fact that all objects fall at the same rate.
Therefore the mass of the inertial object divides out of the equation.
The fact of the matter is, that although mass is not acted upon by the
earth attractor (see johnreed Catch 22), Newton has defined
gravitational force in terms of the local empirical least action
measurements accompanying mass [m]. This includes the gravitational
constant [G]. The magnitude of [g] varies from location to location so
that the attraction between celestial bodies is defined solely in
terms of the least action measurements accompanying a resistance we
feel. Then we simplify (6) to arrive at:

7) g=GM/r^2

To close for now, then, again consider [6]. Where when we divide
little [m] out, we are left with [7]. Note that [G], [g], and [1/r^2]
are empirical measurements that accompany least action processes. Note
too that the law of areas is a consequence of a least action orbit.
So, when we divide [m] out, the result in [7] leaves [M] hardwired to
our empirical measurements that accompany the least action physical
processes involving [m] [endnote 2], and extend to [M] via [1/r^2],
also a property attendant to a least action process. In other words we
have defined a universal gravitational force in terms of the resistive
properties of inertial objects (which we qualify as and which we work
against) that function solely within least action parameters.

Endnotes
1) A circle is an efficient enclosure of area. That is, the circle
circumference is the shortest line length to enclose the greatest
area. Nothing is wasted here. 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. 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 (see Take II). This is, on
the face, trivial and rather mundane, as it follows from the perfect
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, 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] (see Take II). 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. Just as in Ptolemies model it is the consistent efficiency
of the orbits that enable the model to be as useful as it is. The same
efficiency carries Newton's mass driven centripetal force to the
entire universe, as well as Einstein's geodesic.

2) In the post "johnreed Catch 22" I have shown that inertial mass is
"emergent" in the classical gravitational frame.
johnreed

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