The following are a few excerpts from Newton's Principia, along with my comments:
SECTION 11.
Of the Invention of Centripetal Forces
PROPOSITION 1.
THEOREM 1.
The areas. which revolving bodies describe by radii drawn to an immovable centre of force do lie in the same immovable planes, and are proportional to the times in which they are described...
johnreed wrote>
Newton is bringing his feel of gravity into his argument by referencing "revolving bodies" and an "immovable center of force", in combination with a fixed magnitude for [r] and a fixed plane orientation. The combination speaks to a permanent physical system that functions in terms of a force. The only empirical knowledge of such a thing as force is through our sensory detection of touch, or as I like to refer to it: FEEL.
Critic
Newton is describing _any_ kind of revolving motion. We know this because he says "revolving bodies". "Areas proportional to the times" could be any proportion, esp. if the object is changing speed.
johnreed wrote>
Newton is describing the path of any object in circular motion with a fixed value for [r] and a constant orientation wrt the plane of rotation. In any rotation so described, the area swept out will always be a function of [r] and [v]. However, it need not reflect the efficiency ratios of a circle, or a time controlled ellipse. Even so, it is clear that Newton is leading to planets and moons in orbit. We also know that he is speaking about the attraction he feels toward the Earth, since this is the only manifestation of "force" that he could have experienced that applies to this description.
Critic
And forces that he has experienced are irrelevant;
johnreed wrote>
You don't think that Newton's feel of attraction toward the Earth is a bit subjective? And that it might influence his conclusions, not to mention all the rest of humanity? In any event, as long as my observation, that the only force Newton could be thinking of, is the force of attraction he felt toward the Earth, is agreed upon, I will show that it is relevant. I take it you agree that this was his focus? His assumption.
Critic
only the physics matters.
johnreed wrote>
We may have some disagreement here as to what the physics is. Is it: a mathematical description of a physical event? Is it: comprehension of the physical event conceptually with respect to the rest of the universe that immediately applies? Or might it be both?
I am aware that the only thing that matters in the mathematics of it, is its predictive power. If that were the only important aspect, we could have remained with Ptolemy. His mathematics was far better than Big C's.
Whatever. As the next passage indicates, to create his idea of centripetal force, Newton utilizes a circle within which he inscribes an equal sided polygon. He proceeds to have an object with inertial mass (implied by his First Law) travel along each polygon side until it strikes the circle circumference, whereafter, it is deflected along the adjacent polygon side, until it again strikes the circle, etc., and so on, around the polygon.
…. But when the body is arrived at B, suppose that a centripetal force acts at once with a great impulse …
johnreed wrote>
From each vertex of the polygon, Newton drops a radius to the center of the circle. This creates any number of congruent triangles within the inscribed polygon. Newton then takes each polygon side to the limit towards zero. Newton is using a circle here rather than an ellipse, but the equal areas in equal times functions as a result of the inertial object's constant velocity.
Newton has implicitly assumed that the attraction he feels toward the Earth attractor is the same force that applies, in his mathematical description above. Now, provided Newton can show a local, proportional, equivalence, wrt different magnitudes of the force, as it acts on local objects, that depends on local consistencies, that also exist in the frame he wishes to generalize to, he can set the universally generalized aspect of the force, proportional in magnitude to the locally derived magnitudes of the object. We can safely conclude that he believes that he can prove this assumption.
PROPOSITION 11.
THEOREM 11.
Every body that moves in any curve line described in a plane, and by a radius, drawn to a point either immovable, or moving forward with an uniform rectilinear motion, and describes about that point areas proportional to the times, is urged by a centripetal force directed to that point.
johnreed wrote>
Here Newton postulates that any stable orbit, by implication, is the result of a centripetal force. Again, Newton's only physical knowledge of a force is through FEEL. He proceeds to show this by invoking the first law of motion, which deals with inertial mass and momentum.
…Case 1. For every body that moves in a curve line, is (by Law 1) turned aside from its rectilinear course by the action of some force that impels it.
johnreed wrote>
Newton is defining his idea of centripetal force in this entire passage. He comes to this after laying down his 3 laws of motion. Law 1 tells us that a body at rest will remain at rest, and a body in motion will remain in rectilinear motion, until a force acts upon the body to change that motion. Law 1 directly refers to force as it applies to inertial mass. generalizes Law 1 to support his notion of centripetal force, Now Newton justifies centripetal force by invoking Law 1.
Law 1 speaks to objects in motion and force, in a space that has no influence on the objects. This can be proved at the local terrestrial frame, accounting for friction, using billiard balls, springs, and the like, because the Earth attractor does not influence the quantity inertial mass. The Earth attractor and free space are identical wrt inertial mass in this regard.
The only experimental basis for Law 1, are these local inertial mass experiments. Law 1 is therefore not to be generalized to stable cyclic motion, without noting this limitation.
To define centripetal force in terms of a Law that is verified by a force that applies to inertial mass, is to assume that the inertial mass and centripetal mass function according to that Law.
So far the empirical basis for these arguments are Newton's FEEL, and the mathematical basis for the arguments utilize the efficiency ratio properties of a circle, which Newton knows conform to the efficiency ratios of the orbits, ie: Kepler's laws.
Critic
It's the metalaw at work. Force is force. You can push a toy truck, you can push a neutron star. Apples fall, moons fall. Now we know what centripetal forces do, which aside from what generates it, is all we need to know about a force.>>
johnreed wrote>
You speak of push and push and fall and fall. The moon obeys the time distance function of the Earth Attractor. The apple obeys this same function. Free fall, orbit and escape velocity, of inertial mass all depend on the same principle. This principle shows that the Earth attractor, like free space, does not distinguish between inertial mass magnitudes.
Since inertial mass does not change, the only thing that changes are multiples of distance and time. Newton's definition for force depends on multiples of distance and time, either in terms of inertial mass, or centripetal mass.
PROPOSITION 111.
THEOREM 111.
johnreed wrote>
Newton now generalizes more expansively. But it is important to note that he has laid the foundation for centripetal force, using a revolving inertial mass, deflected, by a "supposed" centripetal force that is asserted to exist, by virtue of Law 1. Where Law 1 is a general enough axiom to apply, once it is stated, its only experimental basis in fact, is as it applies to inertial mass. The mystery now becomes: "Why does it work so well as inertial and centripetal?" not: "Why is inertial mass equivalent to gravitational mass?"
The answer to the first question will be shown to be a result of the natural motion in the universe and the ease by which the mathematics can represent it. The universal attraction axiom will be shown to be the result of our FEEL of attraction toward the Earth, in combination with the above.
The answer to the second question is: Inertial mass and gravitational mass were each defined in distance time terms that apply to inertial mass. Distance and time terms, also apply to gravitational mass. Distance and time are properties of motion. Motion in the universe can be represented as conic sections. Any conic can result but they all represent the least action principles that apply to the distance and time functions, wrt motion.
Every body, that by a radius drawn to the center of another body, however moved, and describes areas about that centre proportional to the times, is urged by a force compounded out of the centripetal force tending to that other body, and of all the accelerative force by which that other body is propelled.
Critic
Yes, that's good. You can also back-derive this, as many physics textbooks do, and show how a centripetal force, or just gravity in general, produces those equal area-equal time ellipses.
johnreed wrote
This supports my case precisely. So far Newton has not required mass to directly define centripetal force here (altho' his second law has been stated earlier). Newton has mathematically described what would be centripetal acceleration if not invoking the notion of a centripetal force on the basis of Law 1.
Critic
Yeah, there's centripetal acceleration, which is produced by a centripetal force. If you'd like to know how the object accelerates, then you need to use F = dp/dt and involve inertial mass in the calculuations.
johnreed wrote>
Any object will do. All Newton has described mathematically, is constant motion around a circle. He causes the motion by a centripetal force acting linearly on a deflecting object (this deflecting object is the object in Law 1),along the radius r.
Critic
I don't know if he's got it down to constant speed yet, but circular motion seems to be the intent of the passage you quoted. And that motion by definition is produced by an acceleration, which is by definition produced by a net force, which we call "centripetal" here for no other reason than it's cleaner than saying "center-directed force" over and over.
johnreed wrote>
Sounds like your just running your jibs here. Constant speed is implied by the use of equivalent triangles and areas proportional to the times. Anything else would require a fluctuating centripetal force, which by Law 1, would complexify the problem unnecessarily. The rest is, as you say, merely a matter of definition. He is going to ultimately define centripetal force in terms that can only provably be shown to apply to inertial mass.
In the last passage, Newton describes a more specific case of orbit, and extends the idea of centripetal force to all bodies that rotate stabilly around one another (by implication), based on the previous conclusions. He also calls on Law 3 here.
Cor 1. … if one body L, by a radius drawn to the other body T, describes areas proportional to the times; and from the whole force, by which the first body L is urged, we subduct that… accelerative force by which the other body is urged; the whole remaining force by which the first body is urged will tend to the other body T, AS ITS CENTER.
Critic
Stable orbits fly out from the mathematics, and not from the real world.
johnreed wrote>
But the properties of the real world orbits closely approach the properties that fly out from the mathematics. This is the real world property that enables the mathematics to function so well. See my post "math and the universe" for a popular presentation of this idea.
Critic
We don't know why math describes the universe so well. It'd be nice if we did.
johnreed wrote>
I guess, but apparently not nice enough since you have not read my post that explains it clearly.
Critic
However, if the Earth weighed as much as the Sun (binary star system), then Newton's equations _still_ work perfectly when applied to this real (just not _here_) world. Newton's equations also work in ternary and higher systems, they just can't be solved _exactly_.
johnreed wrote>
That's a pretty loose way of describing "works". But here is the essence of my argument right in your last statement. You mention the weight of the Earth and Sun. Weight is a quantity that applies to small local terrestrial objects (on the Earth, moon or other large orbiting bodies). Weight does not apply necessarily to the fundamental properties that gocern planet sun inetraction. So saying it works there just means that the distance time properties of these orbits are also conic sections.
Critic
The mathematics can start with equal areas in equal times or with just the gravitational force.>>
johnreed wrote>
Again. My point precisely. The force is generalized to the entire universe by virtue of the efficiency principle that is optimized in the ratios of the circle. Since this is a consistent property of natural spatial motion, it applies to local terrestrial action and the action of planets and stars. Since inertial mass does not enter into the distance time properties of the action locally, it can be freely applied to all other action trajectories in terms of its local measured magnitudes.
Critic
But not necessarily, and your interpretation seems to lead to trouble. You might be better off if you don't think about "optimization" and "efficiency" and just do the equations.
johnreed wrote>
Yes, as Ptolemy might have said to Copernicus. Obviously, if any merit exists in this parallel, doing the equations based on the blind interpretation of the laws, fails to provide vision.
Newton continues developing his mathematical argument with some sophisticated circular reasoning, the kinematics of which are guaranteed by the conic sections in the sky.
Cor. 2… And, if these areas are proportional to the times nearly, the remaining force will tend to the other body T nearly.
Cor. 3... And vice versa, if the remaining force tends nearly to the other body T, those areas will be nearly proportional to the times.
Critic
Newton doesn't use circular reasoning in the Principia.
johnreed wrote>
Please. Newton feels a centripetal force. He defines it in terms of near efficient conics (necessary for stability in the field wrt cyclic motion). The force depends on the conics and the inconsequential (to the real force) inertial mass, so completely that the conics can now be derived from the force Newton feels, in terms of the local mass magnitude quantities. The only thing we have for certain is the necessary motion for stability in the field, and the proportional behavior of inertial mass in the Earth attractor field.
Critic
No, I wouldn't say that. Centripetal forces, he said, are generally any force that keeps an object moving in a circle at constant velocity.
johnreed wrote>
He started with areas proportional to the times, which does not require constant velocity. He brought constant velocity in mathematically as a property of a circle with a radius [r] that represents an infinite number of triangles, the base of each being equivalent to the infinitessimal arc of the circle. Since the motion of the inertial object (First Law) must have constant velocity or further complexify his mathematics (by the First Law), equal areas and times apply equivalently to each part of the orbit, as they apply to the whole orbit.
Critic
Gravity can do this and can act as a centripetal force. Also, because of the way gravity falls off with distance, gravity can keep other kinds of orbits too.
johnreed wrote>
Gravity can do this true. But the issue is, is this what the universe does?
Surely you are not referring to triangular or square orbits. So you must be referring to ellipses. More broadly conics. But no distinction can be made here between the ellipse and the circle because the definitive properties Newton uses here, ie: constant velocity and equable areas in equal times, reflect the efficiency ratios of the circle. Where in elliptical orbits, variable velocity and equable areas duplicate that efficiency.
Critic
Einstein explained this seemingly miraculous thing, that gravity falls off as r^2, because we live in a universe of 3 spatial dimensions.
johnreed wrote>
Who can argue with that? Efficiency in 3 spatial dimensions can be represented in various ways, all of which are consistent with r^2. Gravity falls off according to the same function that affects a spheres area, or the dispersion of EMR. Namely r^2. The sphere is an efficient enclosure of volume as the circle is an efficient enclosure of area as the dispersion of EMR is an efficient ... etc.
It all comes down to an aspect of the anthropomorphic idea, but based on the simplest of considerations. A systems potential for stability in the field dramatically decreases in proportion to any untoward or inefficient action that accompanies it. This tends to eliminate any system that does not operate in a near stable manner.
SCHOLIUM
Because the equable description of areas indicates that a centre is respected by that force with which the body is most affected, and by which it is drawn back from its rectilinear motion, and retained in its orbit: why may we not be allowed, in the following discourse, to use the equable description of areas as an indication of a centre, about which all circular motion is performed in free space?
johnreed wrote>
Newton's request above is an excellent use of words. One almost has the idea that Newton is completely aware of the geometrical basis of his idea for centripetal force and is laying the foundation to enable the jump that connects the force he feels to the natural orbits of the planets.
Especially note that Newton recognizes that the equable areas and times do not speak to constant velocity alone. They only speak to a fixed center of revolution. Also note that this is all he asks of the reader here.
The entire mathematical basis for the generalization of centripetal force (and therefore gravity) rests on the equable description of areas in equal times in circular motion, and on his FEEL. By circular motion, Newton also means the planet orbits, which turn out to be, circles in time.
Author's note: The segment of a circle circumference (arc) and its radially enclosed, subtended area, is equal to the ratio between the circumference and area of the circle. This is an equable description of areas in equal arcs and defines the efficiency ratio of the circle. .
Critic
Stable orbits fly out from the mathematics, and not from the real world.
johnreed wrote
One can hardly avoid noting the empirical fact that the properties of the real world orbits closely approach the properties that fly out from the mathematics. This is the real world property that enables the mathematics to function so well. Again, see my post "math and the universe" for a popular presentation of this idea.
Critic
We don't know why math describes the universe so well. It'd be nice if we did. The mathematics can start with equal areas in equal times or with just the gravitational force.
Newton defines a force like this: Force is a change in momentum over a period of time. He states this in calculus as F = dp/dt and enshrines it as Newton's Second Law of Motion. That's _it_. That's the only definition of a force that he ever used, or that anyone ever needs to use in Newtonian mechanics. Conic-looking orbits, stability, cyclic motion, these have _nothing_ to do with the definition of force. Force is a more general concept than gravitation.
johnreed wrote>
The conics may never be mentioned by Newton. They may never be considered in the wrote application of his laws, but if you can continue to say that they have no bearing on the success of his laws in the real universe, give it a couple years and the ideas will start to gell. Your left and right brains will start the gathering of evidence, even as you carry on in your normal endeavors. One day you will derive the conclusion yourself. That's the way it sometimes works.
Critic
You can play with Newton's Three Laws of Motions in a universe entirely FREE from gravitation, in which orbits (other than electromagnetic) and gravitational mass simply DO NOT EXIST.
johnreed wrote>
You have a tendency to put forward my argument as your defense of mainstream opinions. I completely agree here. In fact that's the reason Newton's Laws apply to our universe.
The force depends on the conics so completely that the conics can now be derived from the force Newton feels.
Critic
You see circular reasoning because you think that force is defined by referring to orbits. There is no circular reasoning because that, as I said, isn't how Newton defines what force is.
johnreed wrote>
Wait a minute. Newton states 3 laws that apply specifically to inertial mass. In the passages I have included he puts forward his definition of centripetal force, which also uses his idea for inertial mass. Right before your very eyes, in the instant post, are passages from Newton's Principia where he specifically sets out to define centripetal force.
Furthermore, Newton has not mentioned an orbit. He has used mathematical orders of form to define centripetal force. Force is a quantity that Newton feels. It is a FEEL that he can quantify against inertial mass magnitudes. First in terms of the manner of weight, since the Earth attractor divides out leaving only the comparative unchanging masses, to balance. Second in the manner these masses interact horizontal to the Earth attractor component, while the Earth attractor again divides out, thus approximating free space, wrt inertial mass, and enabling the natural motion in the universe to occur..
The only thing we have for certain is the ease by which the mathematics can represent symmetry. Which happens to be the ideal and which is necessary to approach that motion in the field in order to obtain a degree of perpetuity in the field.
PROPOSITION IV.
THEOREM IV.
The centripetal forces of bodies, which by equable motions describe different circles, tend to the centers of the same circles; and are one to the other as the squares of the arcs described in equal time applied to the radii of the circles.
johnreed wrote>
Newton defines his centripetal force as mv^2/r. He defines the force in terms of the mathematics of the circle and the equable motions of bodies. Equable motions wrt different circles refers to the proportion of the orbit wrt to entire orbit. So that if one half of a small orbit is represented as mv^2/r, it can be set equivalent to the mv^2/r that represents the magnitudes of the quantities in the large orbit..
The question here is whether Newton means that the orbits have a common center, or whether its just a redefinition for the center of force, and applies to all circles anywhere. I don't think Newton meant the latter. I think he knew that T^2/r^3 did not apply outside a given system as the same constant that applied within the system.
In any event, Newton has so far defined centripetal force in terms of his FEEL, inertial mass, and the regular mathematical properties of the circle.
Critic
This is wrong, because Newton never did this. Otherwise, why would Einstein need to axiomize this nearly 300 years later, if Newton had already _proven_ it? You never axiomize something that's been proven, that's a waste of precious axioms.
johnreed wrote>
No Newton did not intentionally do this, nor did he or anyone else ever think he did this. This is why everything appeared mysterious. Gravity is so much a part of our subjective experience as a force we feel, it was easily used to give us a notion of a flat world (self evident from our feel), a notion of heaven above and hell below, a notion that we are the center of the universe, and today, the notion that what we feel acts on every object in the universe.
The manner Newton developed it from the mathematics and the belief that Euclidean geometry represented reality, tied our FEEL of gravity to the mathematics. It served as proof that gravity was the same force here as everywhere.
So the conics were assumed to be generated by gravity. Later (La Grange and Hamilton) showed that the inverse square orbit was a function of mass (ditto) and the equivalence by definition between inertial and gravitational mass was never noted.
Its that damned FEEL we have. Worse than 30 eye witnesses describing an accident. Its everywhere. But QM can't handle it anywhere. GR can't handle it where QM handles everything else. And we still think its everywhere. Our biggest problem and the answer is right there. Its just not everywhere...
So far Newton has defined the centripetal force in terms of the efficiency properties of the circle.
Critic
Well, if there's an orbit (any kind) at work, then there's a centripetal force at work. If the orbit is terrible and crazy and loops all over the place and is generally nasty, then you know what that means? The centripetal force isn't constant. But we don't expect it to be.
johnreed wrote>
Look lets not dream up absurd universes. You're starting to sound like a crank. Physics is the science that trys to understand the universe. If we had wild and crazy forces controlling stable systems, instead of forces that result in efficient action, those wild and crazy guys would either be the express work of God, or they would no longer exist.
Even if the ranging turns and curves of an orbit could not be predicted, a cyclic regularity must accompany that orbit for stability in the field.
You can't just dream up any gedanken defense. Its got to tie in with some kind of reason. The mainstream physicists have come to live in their own interpretations of the math, so much, that any thing seems reasonable as a defense. But nothing seems reasonable that challenges your ideas.
Critic
You seem to be trying to limit Newton's results only to orbits we usually see, when a lot more are possible.
johnreed wrote>
You mean many conceivable universes can be represented mathematically. And because of that, they are possible. Being possible is not the criteria here. Half the cranks have part of their ideas as possible. Anything is possible. But let us return to the master.
SCHOLIUM.
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.
johnreed wrote>
And the package is almost complete. From the First Law and the inertial mass of the revolving body, using the template provided by the natural efficient motion in the universe, Newton defines a centripetal force, that he sets gravity equivalent to.
Critic
... his theory of gravity and forces rests not on orbits, but on his axioms, which we call capital Laws (three of Motion, one of Universal Gravitation). He then builds _from_ his axioms, ... All that is necessary in any branch of physics is the particular set of axioms used, plus the standard logic and math.
johnreed wrote>
His definition of centripetal force, indeed, does rest on his axioms. These axioms have a physical experimentally verified counterpart that are the basis for his laws, in fact. The basis in fact speaks to inertial mass.
By definition, rather than any coincidence (as is the often stated case) , the linear definition for force in terms of inertial mass (1st law verification), is generalized as equivalent to the linear measure of a centripetal force (as the radius of time controlled 2D conics), and set equivalent to our FEEL of gravity.. By making centripetal force a function of the time controlled conic, Newton insures that his description of gravitational mass is identical to inertial mass.
Critic
This is wrong, because Newton never did this. He didn't accidentally do it, either. Starting from Newton's Three Laws of Motion and the Law of Universal Gravitation (the only axioms, if I remember correctly, you need in Newtonian mechanics), you nor no one else can rigorously show that inertial mass is equivalent to gravitational mass, because that doesn't come out of the axioms. Only through a long contrived argument with plenty of vague generalizations are there sufficient opportunites for error that inertial mass can be "shown" equivalent to gravitational mass, just as 1 = 2 cannot be done without going through a long series of steps, one of which contains the hidden flaw.
johnreed wrote>
Most of the long and contrived aspect of this argument is coming from you. I am just pointing out Newton's manner of inventing centripetal force. If what I say about Newton's words is false... then it should be apparent. I am only pointing out what He said, and explaining what it represents.
Critic
I welcome you to try to start with the axioms (the equations) and show how inertial mass is equivalent to gravitational. Remember, inertial=gravitational is _still_ an axiom in GR, so even GR physicists can't show that inertial mass is equivalent to gravitation.
johnreed wrote>
There is no way to use the mathematics to prove that its underlying axioms are true. However, an underlying equivalence can exist in a system of mathematics, without being recognized as axiomatic. It can be regarded as an unprovable consequence of that mathematics. And it can be raised, or lowered, to axiomatic status.
In this case, it is a matter of definition.
Critic
The generalization of the specific instances of gravity we see is Newton's Law of Universal Gravitation, F[grav] = G m[1,grav] m[2,grav] / R^2. This is _provably_ more "general" than conic orbits, you see.
johnreed wrote>
Let's make it a simple case. We discover an empirical fact. Let's call this fact Kepler's Law of Areas. We interpret this Law in a somewhat mystical manner, ie: equal areas in equal times. I have shown that this Law reduces to the efficiency ratios of a circle.
Newton, OTOH, defined the force he felt from the Earth attractor (you can say no he didn't, he meant any force and I say, OK take away the Earth attractor force. What is left for Newton to recognize as a force) in terms of the mathematical properties of a circle. He then quantified the inconsequential inertial mass magnitude locally and inserted it in the conic sections in the sky. Instead of efficient motion, we have universal gravity based on our feel.
Critic
Why? Because not only does Universal Gravitation explain the existence and properties of conic orbits, Universal Gravitation is applicable in free fall, rocketry, ternary systems where conics fear to tread, and generally... universally. Anything relating to conic orbits is derived from Universal Gravitation and NOT the other way around.
johnreed wrote>
You recognize of course that what you are saying is that the efficient motion in the universe is do to a quantity we feel, and not to a property required for stable motion or perpetuity in the field. But let us continue.
PROPOSITION VI.
THEOREM V.
In a space void of resistance, if a body revolves in any orbit about an imovable center, and in the least time describes any arc just then nascent; and the versed sine of that arc is supposed to be drawn bisecting the chord, and produced passing through the center of force: the centripetal force in the middle of the arc will be as the versed sine directly and the square of the time inversely.
johnreed wrote>
We have something new here. Newton includes the phrase "in the least time". Why do you suppose he did that? He is aware that the equable description of areas in equal times, as it applies to the part and the whole, is an efficiency property of the orbits.
By making centripetal force a function of the time controlled conic Newton describes gravitational mass identical to inertial mass by measuring it's resultant force on the linear plane between object centers. These centers are points along the time controlled conic. Newton defined gravitational force consistent with the way we feel it, as inertial mass..
my regards,
johnreed