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Unsourced assertions

Dick: I don't know where you get the idea this stuff you deleted is unsourced. Just prior to the footnote the numbers are provided with their sources. Then in the footnote itself a pdf file is cited from the geodesic society that gives the accepted number for 1/f. So the two estimates are sourced and irrefutable. Is the problem that calculating the 23% difference is unsourced arithmetic? Would you like a sub-section with sources explaining how to calculate a percent error??? Or could it be that you just did a knee-jerk reaction here without realizing the numbers all are sourced?Brews ohare (talk) 22:36, 25 June 2009 (UTC)

The footnote read "The error looks worse if the common measure of flattening f is used (see Clairaut's theorem), the ratio of the difference between semi-major and semi-minor axes divided by the semi-major axis. The flattening by Newton is f = 1/230, while that from the measurements is f = 1/298.4623304, an error of about 23%. See also Table 1.1 of the report by the International Association of Geodesy."

The assertion that "the error looks worse" is unsourced. The number 1/298.4623304 is unsourced, even though a statement of 10 significant digits practically screams for a source. The linked article has a different number, and fewer digits. The "error of about 23%" is your interpretation of the relationship between the numbers, I presume, but is there a source for that way of comparing flattening? No source provided. I don't understand how you don't understand what unsourced means. Read WP:V and WP:RS. Dicklyon (talk) 05:12, 26 June 2009 (UTC)

Well Dick here is how it goes. The modern estimates of semimajor and semiminor axes are sourced as the values 6356.77 km & 6378.14 km leading to an f of (6378.14 - 6356.77)/6378.14 = 0.003350 or 1/298.4623. The cited Table 1.1 has f = 1/298.25642±10, not enough different to matter here. Newton's numbers are sourced as a ratio of diameters of 229 to 230 or an f =(230-229)/230 = 1/230 = 0.004348. Taking the modern estimate as the more accurate the size of percent error is (|0.003350- 0.004348|)/0.004348 × 100 ≈ 23 %. How would you like this information to be presented? For a definition of flattening f see Clairaut's theorem. Brews ohare (talk) 05:38, 26 June 2009 (UTC)

Inertia

Dick, what you have done is to replace a more specific meaning (momentum) with a more general term (inertia) that has got at least two meanings. Even if you look at the wikipedia article about inertia, you will see that some editor has realized that the word inertia has got a common meaning of momentum which differs from its more accurate meaning of 'resistance to change'.

As regards 'centrifugal force', when we use the word inertia in this context, we are talking about momentum as opposed to the more accurate meaning of inertia which is 'resistance to change'.

I think that your reversion of my clarifying edit was motivated more by your own inertia (in the more accurate sense of the word) rather than by any desire to help clarify the wording.

Your point about the momentum being orthogonal to the centrifugal force is a true fact in its own right but it in no way diminishes the accuracy of stating that centrifugal force is an outward force that is associated with momentum.

It can of course be argued that the whole essence of momentum is its resistance to change, hence making the two meanings tend to blend into one. But there is still a subtle distinction between the two meanings because by the same token, we can equate inertial mass with inertia, yet nobody is trying to say that momentum is the same thing as inertial mass. The word inertia truly is a classical archaic broad spectrum terminology. I was merely trying to write the introduction using modern scientific terminologies.

On reflection, how about simply stating that centrifugal force is the outward force that arises in connection with rotation, and totally dropping all mention of either inertia or momentum? The simpler the introducing line, the better. David Tombe (talk) 11:34, 5 July 2009 (UTC)

Planetary Orbital Theory

FyzixFighter, you have removed a fully sourced section on the false grounds that Leibniz's views are already covered in the history section. That has got nothing to do with the fact that the planetary orbital equation in question is still used in modern textbooks. This is nota matter of history. I am therefore going to restore the section because it is accurate, fully sourced, and totally relevant to the topic. What legitimate reason can you possibly have for objecting to its inclusion? Even if you think that the centrifugal force in question is a manifestation of the fictitious force, that is still not a reason to remove the information in question. David Tombe (talk) 16:17, 9 July 2009 (UTC)

If it's going to be in there, it needs to point out that it's talk about a pseudo-force induced by the fact that the r coordinate is measured along a vector that is co-rotating with the planet, and that it's just a special case of that rotating system approach. Probably should be integrated into the relevant section. Dicklyon (talk) 17:47, 9 July 2009 (UTC)

I agree with you in part Dick. The planetary orbit equation is a subset/special case of the fictitious force concept. It really is only of interest in this general article in terms of the history of centrifugal force. Multiple sources have been provided that explicitly state that Leibniz's centrifugal force is tied a rotating frame. For example, from the Swetz reference about Leibniz's and Newton's early concept of centrifugal force:

"Considered as an endeavor of the circulating body, or a force acting on the body itself, does not exist. But if we consider a reference frame fixed in a the body and rotating with it, the body will appear to have an endeavor to recede from the centre. This of course is a fictitious force reflecting the acceleration for the reference frame."

Therefore, I'm going to fold some of the disputed section into the history section where most of the material in the disputed section appears. --FyzixFighter (talk) 19:43, 9 July 2009 (UTC)

I agree, but I wouldn't object if it also had a brief mention in the section on fictitious force in rotating frames. Dicklyon (talk) 19:56, 9 July 2009 (UTC)

The fact that planetary orbits can be dealt with without involving rotating frames of reference means that rotating frames are not an essential part of the analysis. It may well be that some textbooks have tried to integrate the Kepler problem into the rotating frame of reference analysis, but that is no reason for consigning the topic of planetary orbits to the history section.

You have both stated your own opinion that the centrifugal force in planetary orbits is a special case of the fictitious force in a rotating frame of reference. Since you have got sources which agree with your opinion, you are entitled to add that opinion to the section which I put in yesterday. However, I will also add on top of that my opinion that planetary orbital theory does not require a rotating frame of reference in the analysis, and I will also cite sources to back that idea up. At any rate, there was absolutely no justification whatsoever for deleting that new section on planetary orbits lock, stock, and barrel, without any discussion on the talk page. David Tombe (talk) 11:51, 10 July 2009 (UTC)

Hi David: As you know, the Lagrangian approach does not explicitly involve any reference frame: it just picks out the Jacobi coordinates and cranks away. So to that extent the rotating frame is not "an essential part of the analysis". However, the analysis can be done many ways, and while the Lagrangian method may have the advantage of being an approach with wide application, the other methods based upon explicit use of rotating frames arguably have more (or at least different) intuitive content. Evidently, intuition is a fallible guide, but it is a great source of innovation. Brews ohare (talk) 14:58, 10 July 2009 (UTC)

Brews, I have no major quarrel with the Lagrangian approach in this respect. However I do think that Lagrangian can be seriously lacking when it comes to gyroscopic analysis. Lagrangian is an 'energy accountancy' system, but it is totally silent as regards crucial causative forces such as the axial Coriolis force which prevents a spinning gyroscope from toppling under gravity.

As regards rotating frames, it does indeed seem that some attempts have been made to do the Kepler problem within the context of rotating frames. But I can't see how the angular velocity that is associated with a rotating frame can be in any way adequate to deal with all the permutations of pairs of mutual angular velocity as would occur in two adjacent two body Kepler orbits. This scenario is what Maxwell used to generate the very real centrifugal force of repulsion between his vortices to account for magnetic repulsion.

Furthermore, trying to strap a rotating frame of reference around a two body planetary orbit would be a most cumbersome endeavour as it would involve a variable angular velocity. Why bother? As far as I am concerned, we only need to introduce rotating frames of reference if the actual physical scenario being analyzed contains one naturally. For example, in meteorology, we have the Earth and the entrained atmosphere. That is ideal for the introduction of a rotating frame analysis. Radial water pipes on rotating turntables would be another such example. David Tombe (talk) 15:36, 10 July 2009 (UTC)

The main reason for my removal of the planetary orbit section is that it is a special case of either one of the two more general sections. And actually Brews has summed up one of the reasons why I didn't fold the planetary orbit stuff into the rotating frame fictitious force section. This is supposed to be a general article which should cover the material generally and direct to the more specific articles. That means we describe the two Newtonian mechanics uses of the phrase, the Lagrangian mechanics use of the phrase, and the history of the term and its importance in the whole absolute rotation debate. I don't see the centrifugal force of planetary motion as a separate and distinct topic from these general topics. The planetary motion figures prominently in the history section, so that's why I merged the info into there. But I don't see why it should get special mention outside of that section. Again, it's a special case of the general fictitious/inertial/pseudo- centrifugal force - associated with a non-stationary frame (or in other words, refers to terms moved from the acceleration to the force side of F=ma) in Newtonian mechanics or to extra terms that appear in the generalized force of Lagrangian mechanics.

So because this is the general centrifugal force article, I'm going to again remove the special case subsection - remerging the information into the history section. I'm also going to try my hand at expanding the intro and the fictitious force section to include the three contexts that Lagu brought up. --FyzixFighter (talk) 18:36, 10 July 2009 (UTC)

I think your argument about the role of this page is valid. The example appears in some form on the rotating CF page, but I'm not sure it is all covered there.

The "three contexts" introduces under the guise of polar coordinates what is really a Lagrangian attack on the problem, and discussion of this aspect should go there. See the following comments under #Three contexts. Brews ohare (talk) 21:57, 10 July 2009 (UTC)

Hey Brews, I hope you don't mind me merging the planetary orbit example into the main subsection. I think a separate sub-subsection of an example starts getting too far into a specialized article, but to come to a compromise we can all agree on I left in a good portion of it but moved it to were I felt it would flow well with the more general text right after the central potential discussion. I also trimmed some of the historical Leibniz stuff since (1) it's covered in the history section and (2) I think Leibniz didn't derive it using Lagrangian mechanics, but with a formalism that more closely paralleled Newton. In your opinion is this an equitable solution? If not, why and what would you suggest as a compromise? --FyzixFighter (talk) 22:32, 10 July 2009 (UTC)

There is only one universal centrifugal force. The planetary orbit is a very important manifestation of that force and hence it needs to have a section of its own. It is in fact the most general manifestation of the centrifugal force. The planetary orbit example can then be extended to the rotating spheres example by attaching a string between the two objects when they are in a state of mutually outward motion. The string will be pulled taut. This is the so called 'reactive centrifugal force' kicking in , which in turn induces an inward centripetal force due to the tension in the string.

It is wrong to claim that the centrifugal force in planetary orbits is a special case of the 'rotating frame' centrifugal force. We don't need a rotating frame in order to analyze the planetary orbital problem and the centrifugal force in the Leibniz equation is a 'polar coordinates' centrifugal force measured relative to the inertial frame. The centrifugal force in a planetary orbit is the centrifugal force that is built into the inertial path. That centrifugal force is just as much the so-called 'reactive centrifugal force' as it is any other kind of centrifugal force. Indeed, it was in the context of planetary orbits that Newton concocted the concept of the 'reactive centrifugal force'.

Hence, we put in a short and simple section on planetary orbits, much as Brews has just done, and we give that section the appropriate title. There is no need to juggle it all around and merge sections together in order to try and dilute the planetary orbit concept and the Leibniz equation. David Tombe (talk) 00:45, 11 July 2009 (UTC)

Brews's alternative section

Brews, your alternative section covers the main points. I will not make any amendments to it until I have studied the details. At the moment, I am wondering whether or not the variable r is the distance to the centre of mass as you say, or if it is the actual distance between the two objects. I have a feeling that it is the latter, but I need more time to think about it. David Tombe (talk) 12:09, 10 July 2009 (UTC)

On thinking more about it, it's probably OK because I can see that you have used the language of reduced mass. David Tombe (talk) 14:22, 10 July 2009 (UTC)

David: Are the initial energy and angular momentum sufficient to determine the solution? The angular momentum is subsumed under the parameter ${\displaystyle \ell }$, but the equation is second order and so appears to need another initial condition. Brews ohare (talk) 15:07, 10 July 2009 (UTC)

Brews, The general solution is a conic section. A conic contains two parameters that need to be determined. One is the eccentricity and the other is the semi-latus rectum. As you say, we are dealing with a second order differential equation, and hence we will have two arbitrary constants that need to be determined. I do believe from memory that the two arbitary constants in question are indeed the eccentricity and the semi-latus rectum. The eccentricity is determined by the initial kinetic energy for the particular radial distance at the kick-off point. The semi-latus rectum is determined by the angle of projection.

It's thirty years since I did these problems in applied maths and I'm rusty. If you have any more questions, I'll look it up for you. But I can now see that we are dealing with three factors that determine the full solution. Not only do we have the initial kinetic energy and angle of projection, but also the initial radial distance. However, I think that when all combined, this will reduce to initial energy and initial angular momentum. I remember a formula for the eccentricity which involved speed and radial distance. David Tombe (talk) 15:23, 10 July 2009 (UTC)

Three contexts

The third context is equivalent to the Lagrangian discussion and duplicates points made in that section. I have added the references.

This topic is separate from the "rotating reference frame" topic because it (i) doesn't invoke a rotating frame (ii) has applicability where no rotating frame is involved and (iii) leads to endless confusion when believed to have some connection to Newton's fictitious forces. For example, the "third context" centrifugal force is not "fictitious" because it doesn't vanish when the frame doesn't rotate. Brews ohare (talk) 21:47, 10 July 2009 (UTC)

Thanks Brews for the edit. I really prefer how you've connected the "third context" to the Lagrangian formalism than how I had included the idea. My only minor qualm is more about policy than content. The Bini article which introduces the "three contexts" does all three from a Newtonian mechanics viewpoint. In Bini's third context, F=ma is written out in polar coordinates and then, to make the equation look like the Cartesian coordinate counterparts (ie ${\displaystyle m{\ddot {\eta }}=\sum F_{\eta }}$), we move Goldstein 3-12's "centripetal acceleration term" over to the force side and call it a force. Personally I much prefer your redirection to the Lagrangian formalism since, IMO, arbitrarily moving a term from one side of the equation to the other does not suddenly transform it from a term in the radial acceleration to a force term. When this is done, it completely throws the Newtonian definition of "force" out the window. But the Lagrangian formalism gives moves all the arbitrariness to the choice of coordinates (where it should be) and in a consistent and logical fashion cranks out the "generalized forces". However, again I worry if casting the third context as an extension of the Lagrangian formalism strays to far from the (IMO somewhat lacking and disingenuous) description in the Bini source. --FyzixFighter (talk) 22:26, 10 July 2009 (UTC)

FyzixFighter, you are making alot of unnecessary complications. There is only one centrifugal force. I'm happy enough to have a section on the Lagrangian treatment of centrifugal force. But we do not mix such a section with the Leibiz treatment of the Kepler problem. The Leibniz equation is not a Lagrangian equation. It is a force equation. It is a second order differential equation in the radial distance. Lagrangian mechanics is about conservation of energy.

And as regards moving things from one side of an equation to the other, that never converts a centrifugal force into a centripetal force. You are getting the inertial path equation mixed up with the two body Kepler problem. The former does not involve a gravitational field and it is a hypothetical situation which never exactly happens in nature. The latter has a gravitational field involved. The term that refers to centrifugal force in the latter refers to centripetal force in the former. There is no moving terms to the other side of the equation going on. They are two different equations for two different physical secnarios. David Tombe (talk) 00:53, 11 July 2009 (UTC)

David, Leibniz's equation can't be a force equation, at least not in the standard, Newtonian mechanics definition of force. There are only two ways to get Leibniz's equation from first principles: using Lagrangian mechanics or Newtonian mechanics. When the two body Kepler problem is done using Newtonian mechanics, the only force that needs to be included in F_net is gravity - no centrifugal force is included in the sum of forces. As Bini and Strommel note, the ${\displaystyle -mr{\dot {\theta }}^{2}}$ term in the radial acceleration (which Goldstein calls the centripetal acceleration term) is moved to the force side of the equation - that is how the centrifugal force term arises. It's not a real force, it's just a term from the radial acceleration that we've moved. See the end of Strommel's discussion on pg. 36-38. Bini also classifies it this as a fictitious force that is implied by the frame of reference. --FyzixFighter (talk) 03:43, 11 July 2009 (UTC)

FyzixFighter, you are just playing around with words. The Leibniz equation and planetary orbits are the most general way of explaining centrifugal force. You don't want it in the article because it doesn't involve the use of a rotating frame of reference. It's as simple as that. All your arguments above are totally specious. David Tombe (talk) 09:00, 11 July 2009 (UTC)

At least three references explicitly support the "fictitious" interpretation of Leibniz's centrifugal force (two of which you initially provided):

• From Swetz, "Learn from the Masters!", pg 269

Considered as an endeavor of the circulating body, or a force acting on the body itself, does not exist. But if we consider a reference frame fixed in a the body and rotating with it, the body will appear to have an endeavor to recede from the centre. This of course is a fictitious force reflecting the acceleration for the reference frame.

• From Linton, "From Eudoxus to Einstein", pg 413

Newton had realized crucially that it was much simpler to consider things from a frame of reference in which the point of attraction was fixed rather than from the point of view of the body in motion. In this way, centrifugal forces - which were not forces at all in Newton's new dynamics - were replaced by forces that acted continually toward a fixed point.

• From Aiton, "The celestial mechanics of Leibniz in the light of Newtonian criticism"

Leibniz viewed the motion of the planet from the standpoint of a frame of reference moving with the planet. planet. The planet experienced a centrifugal force in the same way that one experiences a centrifugal force when turning a corner in a vehicle. From the standpoint of an observer outside the vehicle the centrifugal force appears as an illusion arising from the failure of the traveller to take account of his acceleration towards the centre. Although both standpoints are valid, Newton, in the Principia, always used a fixed frame of reference.

and

Leibniz's study of the motion along the radius vector was essentially a study of motion relative to a rotating frame of reference.

So if we get to Leibniz's equation from Newtonian mechanics (ie the traditional definition of force), his centrifugal force is a fictitious force that vanishes from the list of forces acting on an object in Newton's 2nd law when the dynamics is described from the inertial frame. --FyzixFighter (talk) 13:47, 11 July 2009 (UTC)

FyzixFighter, centrifugal force only appears to vanish in the inertial frame of reference when the inertial path is described in Cartesian coordinates. It shows up when we use polar coordinates. And yes, the centrifugal force in planetary orbits is the same centrifugal force that arises in the Lagrangian formulation, and in the rotaing frames formulation in the special case of co-rotation. But that is not a basis for hiding a section on planetary orbits inside the Lagrangian section as you have been attempting to do. This article is about 'centrifugal force'. The sections of the article are for the purpose of illustrating centrifugal force. Planetary orbits present the best directly observed illustration of centrifugal force as an outward inverse cube law force. The equation which was used in that section by both myself and Brews was Leibniz's equation. Leibniz's equation is not Lagrangian mechanics even if it is being used to describe something that can equally be described using Lagrangian mechanics. So you have got absolutely no grounds whatsoever to hide this section inside the Lagrangian section.

Likewise with the centrifugal force in the inertial path. Polar coordinates do not involve a rotating frame of reference. The 'inertial path' centrifugal force is indeed the same centrifugal force that arises when an object co-rotates with a rotating frame of reference. But that is not grounds for deleting all mention of the treatment of centrifugal force in connection with polar coordinates in the absence of rotating frames.

As for your extension to the introduction, you cannot be serious. It doesn't exactly read very well. What is the point of your extension? David Tombe (talk) 19:02, 11 July 2009 (UTC)

Look again at both the Bini and Stommel reference which talk about the polar coordinate centrifugal force. Both call it a fictitious force. For example from Bini after discussing the different CF contexts including the polar coordinate context:

"In other words, the "fictitious" centrifugal force is a convenience that only has meaning with respect to some implied reference frame"

And from Stommel:

"Sometimes equation (2.14a) is written with one of the acceleration terms on the righthand side

${\displaystyle {\ddot {r}}=r{\dot {\theta }}^{2}+F_{r}}$.

The term ${\displaystyle r{\dot {\theta }}^{2}}$ then looks like a force, and it actually has a name: "the centrifugal force" . It is always positive and directed away from the origin. But it is really not a force at all, and so if we want to make use of it in a formal sense, then we could call it a virtual, fake, adventitious force."

This is because, as Stommel states, ${\displaystyle {\ddot {r}}}$ is not the true radial acceleration, the true radial acceleration is ${\displaystyle {\ddot {r}}-r{\dot {\theta }}^{2}}$. While Leibniz's radial equation is mathematically valid, since ${\displaystyle {\ddot {r}}}$ is not the radial acceleration, then we cannot use Newton's 2nd law to interpret the other side as a true force or sum of forces. It's that simple. In polar coordinates in the inertial frame, the ${\displaystyle mr{\dot {\theta }}^{2}}$ term is part of the acceleration, not a force acting on the circulating body. In all the fictitious CF cases, both the rotating frame contexts and the polar coordinates, the "centrifugal force" and other fictitious forces arise when we take Newton's 2nd law and start moving terms that appear in the acceleration, ${\displaystyle {\ddot {\vec {r}}}}$ and move them to the force side of the equation and interpret them as "forces". In the rotating frame case, the terms arise from the time dependency of the basis vectors for the rotating frame; in the polar coordinates, it's the centripetal acceleration term as Goldstein calls it. --FyzixFighter (talk) 21:24, 11 July 2009 (UTC)

RfC: Content dispute on Leibniz equation and inclusion of planetary orbit equation

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Ongoing content dispute over the different treatments of the centrifugal force and whether Leibniz's centrifugal force represents a distinct and separate concept. FyzixFighter (talk) 03:49, 11 July 2009 (UTC)

FyzixFighter, the centrifugal force in the Leibniz equation does not represent a distinct and separate concept as compared to any other approach to centrifugal force. There is only one centrifugal force. You are misrepresenting what this dispute is about. The dispute is about the fact that you have consistently attempted to remove all mention of the centrifugal force in connection with situations that don't require a rotating frame of reference in the analysis.

This latest round began when you removed my new section on planetary orbital theory. At first you tried to argue that it belonged in the history section. Brews then restored his own amended version of what I had written. You removed that too. After your attempts to remove it failed, you then blended it with the Lagrangian section when in fact the Leibniz equation has got nothing whatsoever to do with Lagrangian mechanics. David Tombe (talk) 08:57, 11 July 2009 (UTC)

The comparative table

One way or the other, something needed to be done about the comparative table because it didn't cater for situations in which centrifugal force is treated outside of the context of rotating frames of reference. An alternative approach would be to reduce it once again to two columns but amend the bit where it says 'rotating frame' in relation to the fictitious force, to read 'inertial frame'. In Lagrangian, polar coordinates, and co-rotation, the centrifugal force is measured relative to the inertial frame. It is only in non-co-rotation that any motion can be considered to be relative to the rotating frame of reference. David Tombe (talk) 19:23, 11 July 2009 (UTC)

FyzixFighter is trying to provoke an edit war

FyzixFighter, you wrote this when you did you reversion,

"I still think planet orbits are a special case and should not be mentioned - - -".

Planetary orbits are indeed one specific illustrative case of centrifugal force. That is not a reason for arguing that they should not be mentioned in the article.

You began this edit war over a year ago on the grounds that my attempts to insert the planetary orbital approach were original research. Now that we all know that you were wrong in that regard, you are scraping the barrel. You are now down to the pathetic argument that planetary orbits shouldn't be mentioned because they are a special case.

Every example of centrifugal force is a special case. David Tombe (talk) 21:10, 11 July 2009 (UTC)

You do realize with that last revert you just surpassed the three reverts in a 24-hour period:

• 1st revert 23:24, 10 July 2009
• 2nd revert 09:03, 11 July 2009
• 3rd revert 18:39, 11 July 2009
• 4th revert 21:02, 11 July 2009

--FyzixFighter (talk) 21:32, 11 July 2009 (UTC)

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