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Below is the proposal with some annotation:

Centrifugal force can often arise in the analysis of orbital motion and, more generally, of motion in a central-force field. The symmetry of a central force lends itself to a description in polar coordinates. Thus, the dynamics of a mass, m, in a central-force field, expressed using Newton's second law of motion, becomes:

${\displaystyle F(r){\hat {r}}=m(({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {r}}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\theta }})}$

where F(r) is the central force.

Because of the conservation of angular momentum and the absence of a net force in the azimuthal direction, the angular momentum, L, remains constant. This allows the radial component of this equation to be expressed solely with respect of the radial coordinate, r, and the angular momentum, yielding the radial equation:

${\displaystyle m{\ddot {r}}-{\frac {L^{2}}{mr^{3}}}=F(r)}$.

The ${\displaystyle -L^{2}/mr^{3}}$ term in the radial acceleration is often called the centripetal acceleration.(Taylor,pp.29 & 359)

• This sentence is a bad idea- the centripetal force in circular motion is the entire inwardly directed radial force, and so is both terms. In other words. the central force F(r) is the centripetal force. I don't know what Taylor means here - maybe context would make sense out of it.

The equations of motion for r that result from this are the same that would arise from a particle in a fictitious one-dimensional scenario under the influence of a force:(Goldstein, Ch 3)

${\displaystyle F'(r)=F(r)+{\frac {L^{2}}{mr^{3}}}}$

where the additional term added to the central force is called the centrifugal force.

• I don't find this very helpful: what is the connection to the "centrifugal force" of the article?

For the one-dimensional scenario, the radial equation then becomes:

${\displaystyle m{\ddot {r}}=F'(r)={\frac {L^{2}}{mr^{3}}}+F(r)}$.

Expressing the radial equation in this way physically corresponds to describing the dynamics within a non-inertial frame that co-rotates with the particle.(Tatum)(Whiting) Thus, the centrifugal force is unnecessary when describing the motion in the inertial frame; the influence ascribed to this fictitious force in the rotating frame is expressed by the centripetal acceleration term within the radial acceleration in the inertial frame.(Taylor,pp.358-359)(Goldstein(2002),pp.176) When the angular velocity of the co-rotating frame is not constant, such as for elliptical and unbound orbits in orbital mechanics, other fictitious forces - the Coriolis force and the Euler force - will arise but can be ignored since they will cancel each other.(Whiting)

• Again, a bit obscure. The point to be made is that in the co-rotating frame the object is stationary, and so straightforward application of Newton's laws requires zero net force. However, in the co-rotating frame the central force is clearly present (as it is for every observer in every frame) so zero net force cannot be obtained by the co-rotating observer unless the centrifugal force is postulated to enter Newton's laws to balance the central force in the co-rotating frame. Brews ohare ( talk) 13:39, 16 April 2009 (UTC)

True. The added outward pseudo-force is the one Goldstein calls centrifugal force. Since terminology is not uniform, we should specifically attribute that description to him. Then we can converge you guys, I think. Dicklyon ( talk) 14:22, 16 April 2009 (UTC)

Brews, I'm glad that we are agreeing on this most important point of all. The inverse cube law term is most certainly not the centripetal force. The only time that centripetal force can take on that expression is in the special case of circular motion when it is numerically equal to the centrifugal force. In non-circular motion the two are not in general equal, and it is the outward centrifugal force which is induced by transverse motion and written in the form mrω^2 (or in the inverse cube law format when Kepler's areal constant is invoked). The centripetal force is never induced by transverse motion and we all know that in a planetary orbit, the centripetal force is inverse square law.

If we can correct that issue, then FyzixFighter's proposal begins to look reasonable. One wonders why anybody who was copying from Goldstein should suddenly want to put in a 'Taylor' reference which was almost certainly totally out of context. Why is there this determination to mask the involvement of the centrifugal force?

And yes, you asked the perfect question. What is the relevance to the article of telling us that Goldstein said that the radial equation is the same equation as would hold for the fictititious one dimensional problem. We do not need to add that information into the article?

And agreed again. FyzixFighter's last paragraph is unnecessary. He is merely telling us that the problem can be dealt with using a rotating frame of reference. But it doesn't have to be dealt with using a rotating frame of reference and it never was when I did the course, and Goldstein doesn't use rotating frames of reference to analyze the planetary orbital equation. So why bother with this paragraph? When we remove all the bad parts of FyzixFighter's proposal, we are basically back to my proposal with a little bit extra about polar coordinates which is informative but unnecessary. Polar coordinates are normally taken for granted in celestial mechanics. In fact have you ever seen gravity formulated in anything other than polar coordinates? David Tombe ( talk) 14:43, 16 April 2009 (UTC)

@Brews: Thanks for the comments. I knew this was going to need some refinement and would benefit from multiple eyes. I'll try to address your three points below:

• Yeah I think I see where the confusion is. I've kind of muddled Taylor a bit. I think it might be best to move the discussion of the angular momentum until a bit later and stick with the r's and theta's. How's this for a replacement to the three sentences preceding your comment:

Looking at just the radial component of this vector equation gives us

${\displaystyle m({\ddot {r}}-r{\dot {\theta }}^{2})=F(r)}$.

The ${\displaystyle -r{\dot {\theta }}^{2}}$ term that appears in the radial acceleration is sometimes called the centripetal term.(Taylor, footnote: The term centripetal acceleration is often used to refer to the total acceleration when the central force is inward. Taylor also uses the phrases centripetal acceleration and centripetal term to describe this term that appears when describing general motion in polar coordinates)

I'm still a little torn about removing/including the centripetal acceleration statement, but I see how it is causing confusion. The reason I'd like to keep it is because I'm trying to follow Taylor's explanation that in the inertial frame, this behavior is accounted for by a centripetal term in the radial acceleration, and by a centrifugal force term in the rotating frame.

• Would removing the use of the angular momentum, and writing the term as ${\displaystyle mr{\dot {\theta }}^{2}}$ help. I think moving the Goldstein reference to the end of the sentence would also make it clearer where this nomenclature comes from. I think also moving the rewritten radial equation up before this might help:

Moving the centripetal term to the other side of the equation allows the radial equation to be rewritten in the form:

${\displaystyle m{\ddot {r}}=F(r)+mr{\dot {\theta }}^{2}}$.

From this it is easy to see that the equation of motion for r is the same as that for a mass in a fictitious one-dimensional scenario under the influence of the central force and an additional radially outward force of magnitude ${\displaystyle mr{\dot {\theta }}^{2}}$.(Goldstein) Expressing the radial equation in this way physically corresponds to describing the dynamics within a non-inertial frame that co-rotates with the particle.(Tatum)(Whiting) In this frame ${\displaystyle \Omega ={\dot {\theta }}}$, so that the magnitude of the 1D problem's additional force can be expressed as ${\displaystyle mr\Omega ^{2}}$, the centrifugal force for the rotating frame. However, the centrifugal force is unnecessary when describing the motion in the inertial frame; the influence ascribed to this fictitious force in the rotating frame is expressed by the centripetal acceleration term within the radial acceleration in the inertial frame.(Taylor,pp.358-359)(Goldstein(2002),pp.176) When the angular velocity of the co-rotating frame is not constant, such as for elliptical and unbound orbits in orbital mechanics, other fictitious forces - the Coriolis force and the Euler force - will arise but can be ignored since they will cancel each other.(Whiting)

• On your third point I'm going to have to disagree with you. Only for circular orbits does the particle remain stationary in the corotating frame. The only requirement for the corotating frame is that ${\displaystyle {\dot {\theta }}'=0}$ but r can do whatever, so the object is not by definition stationary. This paragraph overall is necessary to answer the previous question of how Goldstein's use of the 1D fictitious problem and use of "centrifugal force" ties into the general "centrifugal force" idea of rotating frames which this article is addressing. Tatum, Whiting, and Taylor (in ch 9) clearly casts Goldstein's formalism of centrifugal force in describing planetary motion as a special case of mechanics in rotating frames, which I'm hoping to do with this paragraph. The last sentence could be sacrificed, but I think it provides a nice closure/reason for why we can ignore the other fictitious forces - a point commonly skipped in central force problems. -- FyzixFighter ( talk) 16:02, 16 April 2009 (UTC)

Yes, I was thinking of circular motion. You're thinking of things like this image? Is that necessary here? Brews ohare ( talk) 18:25, 16 April 2009 (UTC)

I believe it is necessary for the sake of generality. If we limit our discussion to just circular motion, IMO that leaves the question open/unanswered to the reader on whether or not a similar approach is valid for non-circular motion in central-force fields. At least in the case of planetary orbits, the centrifugal force concept and the associated 1D equivalent problem are used to analyze/describe elliptical and unbound orbits, and not just circular orbits. Therefore, I think that it is important that we also be general in describing how centrifugal force appears in central-force problems (specifically the 1D problem) and how this approach fits as a special case within the larger rotating frame paradigm of the centrifugal force. Whiting does a good, quick summary of this connection in the 1983 Letter in Physics Education, and addresses why the other fictitious forces can be ignored. -- FyzixFighter ( talk) 23:19, 16 April 2009 (UTC)

FyzixFighter, I presented a very simple equation,

${\displaystyle {\ddot {r}}=-G(M+m)/r^{2}+l^{2}/r^{3}}$

out of a textbook. We all know that the inverse cube law term is the centrifugal force. It acts radially outwards and it is induced by transverse motion. The gravity term is the inward acting centripetal force. And you are now trying to tell us all that the inverse cube law term is the centripetal force? David Tombe ( talk) 19:34, 16 April 2009 (UTC)

I am simply reporting what reliable sources such as Tatum, Whiting, Taylor, and several others state, namely that your simple equation is arrived at when we (quoting Whiting) "reduce the orbital motion into a one-dimensional problem by setting up the equation of motion in a reference frame rotating about an axis perpendicular to the orbital plane and passing through one of the foci, such that the azimuthal angle of the object is constant; that is, the rotating frame moves round in step with the direction of the position vector r." In the inertial frame, the only term on the force side of the equation is the gravitational force; the inverse cube term appears as a negative term within the total radial acceleration in the inertial frame. Some, like Taylor, call this a centripetal acceleration term or simply a centripetal term. -- FyzixFighter ( talk) 23:19, 16 April 2009 (UTC)

Brews, now that we have got the k in, do we really need that section about reduced mass? Remember that we are really only interested in the radial position variable, r, for this article. David Tombe ( talk) 20:08, 16 April 2009 (UTC)

FyzixFighter, I saw many different ways of analyzing that equation back in 1980. The Goldstein method was only one of them. Never did I see an analysis that involved rotating frames of reference. Of course it can be done that way if you like. But why would you bother? Can you not clearly see what is going on without using a rotating frame of reference? We have an equation for radial acceleration. There are some here who have been trying to tell us that it is only a radial acceleration in relation to the inward gravity term, but not in relation to the outward centrifugal term. That doesn't make any sense.

This equation embodies centrifugal force in its most general form. If two bodies are moving in mutual transverse motion, there will be both an inward gravity force (inverse square law) and an induced outward centrifugal force (inverse cube law) which is induced by the transverse motion. The two different power laws give rise to a stability node which makes planetary orbits stable. Hence if the objects get closer to each other, the centrifugal force will dominate and cause a recoil. Interestingly, if by chance, gravity had been an inverse cube law force, then the planets would spiral inwards and collapse into each other. I was once shown that analysis.

Now let's imagine that gravity is negligible. We will then have a hyperbolic orbit. Let's then attach a string between the two planets. The outward centrifugal force will pull the string taut. It is this effect which is currently being catered for over in the other article called 'Reactive Centrifugal Force'. But it is not a reaction. The reaction to this effect is an induced inward centripetal force due to the tension in the string, which then causes the orbit to become circular (providing the string doesn't snap!). Centrifugal force is all one single topic.

And can you not also see that Coriolis force goes hand in hand with the conservation of angular momentum and Kepler's second law, and that it occurs in all vortices and non-circular orbits. It may cancel mathematically with what you call the Euler force, but it can still be individually observed. We can see the inward radial motion being continually deflected into the transverse direction. Hence all the confusion in meteorology and also over on the Coriolis force page. They have confused the cyclonic mechansim which determines the initial angular momentum direction with the Coriolis force itself. The cyclonic mechanism is more a case of inertia causing apparent deflections relative to a rotating frame of reference. David Tombe ( talk) 11:55, 17 April 2009 (UTC)

Thank you for your comment. However, it is primarily your personal original interpretation, which is very much a minority view. Wikipedia is an encyclopedia, and it's purpose is to report the current state of human knowledge. Wikipedia is not a tool for persuading or converting people to another viewpoint. As such, please provide reliable sources that support your interpretation. I will gladly discuss reliable sources with you. I refuse to waste time with drawn out discussions where one of us is trying to teach the other one correct physics. From this point on I am basically going to refuse to engage in discourse with you on this topic unless such a discussion is based on reliable sources. If you feel that this position is unreasonable, then there are avenues other than this talk page to resolve such issues. -- FyzixFighter ( talk) 19:33, 17 April 2009 (UTC)

FyzixFighter, considering that you are only learning the subject now, I don't see how you could possibly know whether my edits are original research or not. You have only just realized that the inverse cube law in the equation is not a centripetal force. When did you last do a physics edit that wasn't for the sole purpose of trying to undermine my edits? You clearly haven't got a clue about this subject. That was a most horrendous error on your part to suggest that the centrifugal term was a centripetal term. And you made that error so boldly. David Tombe ( talk) 13:54, 18 April 2009 (UTC)

David, there's no need to attack a well-meaning editor who is making progress straightening out the long festering problem between you and Brews. He's doing a good job, taking input from all, basing his proposals on sources, etc. I sympathize with his frustration in dealing with you, since your unique viewpoint based on the phrase "the familiar centrifugal force" in Goldstein is at odds with all the other sources, and you won't listen to reason about that. I think we need to move forward. I will defend the inclusion and attribution of the equation from Goldstein, but within a context that agrees with sources in general, which the centrifugal force is what is termed a "fictitious force". Let's move forward, get it done, and as FF says, ignore pleas not backed up by sources. Dicklyon ( talk) 17:47, 18 April 2009 (UTC)

No Dick, there is no evidence that he is well meaning. You can check his record. He hasn't made a single physics edit since I opened my account last April, that hasn't been to undermine my edits. And you can see for yourself that he doesn't know what he is talking about. He tried to tell us that the inverse cube law term is centripetal force. That is so wrong. You know it's wrong. Brews knows that it is wrong. How could anybody who knows about the subject possibly make an error like that?

The main argument between myself and Brews is over the issue of whether the centrifugal force, in the context of planetary orbits, is real or fictitious and there is no textbook that says that it is fictitious in this context. We cannot have the discussion derailed by FyzixFighter. If you and Brews hadn't challenged him on that point, it would have prevailed because there are others who would have readily supported him. And then the article would have been worse than ever.

As it stands, the section which Brews wrote on planetary orbits is technically correct. I have been through it carefully, and together we tidied up the symbolism. But it's a bit long winded and Brews has not highlighted the star piece. Instead, he has put his own opinion below it to say that it is not an impressed force. There is nothing wrong with simply stating that it is the centrifugal force term and saying nothing more. David Tombe ( talk) 18:28, 18 April 2009 (UTC)

The various uses of terms like centrifugal "term", centrifugal "force" and centrifugal "acceleration" are discussed in terminology. One point to be kept in mind through the confused use of the same terms for completely different things (a feature of the published literature) is that Newtonian forces (real forces) exist outside of any coordinate system. Moreover, they have transformation laws when coordinate are changed (they are vectors). In the present article, vector forces and accelerations are the topic. However, the other terminologies (like those mentioned in your discussion of Taylor) refer to "accelerations" and "forces" that are not vectors, do not transform like vectors, and have different magnitudes and directions in different coordinates. For example, they depend upon whether Cartesian or polar coordinates are used, and may vanish in Cartesian coordinates and be non-zero in polar coordinates. Brews ohare ( talk) 13:18, 17 April 2009 (UTC)

I recall an article where the author suggested that the phrases "centripetal acceleration" and "centripetal force" should be abolished from physics education. Although the alternative proposed was to use the wordier phrases "acceleration/force in the centripetal direction". Taylor's use of "centripetal acceleration" makes me cringe too. I'll try and remove the use of "centripetal" in my next revision to avoid the ambiguity of the terminology. -- FyzixFighter ( talk) 19:49, 17 April 2009 (UTC)

That's good FyzixFighter. What ever made you think to call the inverse cube law term 'centripetal force' in the first place? You have finally conceded that the radial equation,

${\displaystyle {\ddot {r}}=-G(M+m)/r^{2}+l^{2}/r^{3}}$

is not my original research. But now you are trying to suggest that my interpretation of it is original research. Well there only are three terms in the equation. One term is gravity which is a radial acceleration. That means that all the terms have to be a radial acceleration, otherwise we would have a problem with dimensional consistency. The rω^2 term (the inverse cube law term) is an outward radial acceleration. And your entire presence on this page is for the sole purpose of trying to deny that the rω^2 term is a centrifugal acceleration. It will be interesting to see what name you would like to call it. How about 'Harry'? David Tombe ( talk) 13:42, 18 April 2009 (UTC)

David: one term is gravity, which is a radial acceleration. That means that the sum of the other two terms ${\displaystyle {\ddot {r}}-l^{2}/r^{3}}$ also is a radial acceleration. Acceleration is a vector: the individual terms in this sum are not individual vectors. The sum with its radial orientation, like the gravity term, is a physical vector that is the same in every coordinate system. The individual terms change with a change to (for example) Cartesian coordinates or elliptic coordinates. In the general case:

${\displaystyle {\boldsymbol {F}}=m{\boldsymbol {a}}=m\sum _{k=1}^{d}\left({\dot {v}}_{k}\ +\sum _{j=1}^{d}\sum _{i=1}^{d}v_{j}{\Gamma ^{k}}_{ij}{\dot {q}}_{i}\right){\boldsymbol {e_{k}}}\ ,}$

and

${\displaystyle \sum _{k=1}^{d}{\dot {v}}_{k}{\boldsymbol {e_{k}}}\ }$

by itself is not an acceleration vector, but a "generalized acceleration". In polar coordinates, for this example, this expression is ${\displaystyle {\ddot {r}}{\boldsymbol {\hat {r}}}\ }$.

Brews ohare ( talk) 15:07, 18 April 2009 (UTC)

Brews, this really is playing on words now. The whole equation is the radial component of the acceleration vector. So every term in it is radial acceleration. But let's get to the main point. What would you like to call the inverse cube law term? We are agreed that it is not the centripetal force. So what is it? David Tombe ( talk) 18:18, 18 April 2009 (UTC)

David: The transformation properties of a vector under coordinate transformations is a defining property, not semantics. The individual terms do not transform like a vector. For a particular example see here. Thus, you can call things what you like, but it introduces confusion when it appears to suggest that vectors are described. Brews ohare ( talk) 18:59, 18 April 2009 (UTC)

David, it doesn't really matter what Brews wants to call it or what you want to call it. We need to just report what others call it. There's no problem reporting Goldstein's calling to centrifugal force, and no problem reporting that most call this a "fictitious force" due to the fictitious 1D system in which it arises. Let's all just stick to reporting and we won't have a problem. Dicklyon ( talk) 19:15, 18 April 2009 (UTC)

@David - I don't believe I ever said that it was the "centripetal force". I did use "centripetal acceleration" in a similar manner that Taylor uses it in his "Mechanics" textbook (although he keeps it in terms of r and theta and I tried to express it in r and L where I think I lost a factor of m on the bottom, which may account for the force/acceleration confusion - sorry, it's what I get for editing late at night). He's not entirely consistent, sometimes he calls it the "centripetal acceleration", "centripetal acceleration term", or the "centripetal term". The last is would probably be the easiest to incorporate as it does carry "acceleration", which carries with it specific meaning in physics as Brews pointed out. I agree with Brews that using "centripetal acceleration" to refer just to that term is confusing. As I've been unable to find a similar nomenclature in additional texts, it would appear as if it might be unique to Taylor. I do understand why Taylor would use, but for us to use it would require superfluous explanation of how such usage compares to more traditional usages of the phrase. I find it just as easy to refer to it directly (as I've done in the latest revision) as a term within the radial acceleration in the inertial frame. I guess we could call the term a centripetal contribution to the radial acceleration in the inertial frame, but that seems a bit wordy and borders on inventing our own neologisms rather than relying on how it's treated in reliable sources. Either way, as Taylor indicates in chapter 9 (pg 359), this accounts for the behavior in the inertial frame that is attributed to the centrifugal force in the co-rotating frame.

At everyone in general - does the latest shortened revision have any of the terminology hang ups? Any other terminology-related feedback? -- FyzixFighter ( talk) 20:38, 18 April 2009 (UTC)

Dick, Let's get this business about the one dimensional fictitious problem sorted out once and for all. You keep bringing it up, and you are doing so opportunistically. Goldstein deals with the planetary orbital equation (3-11 and 3-12) before he mentions anything about equivalent fictitious one dimensional problems. The planetary orbital radial equation is what we are dealing with here. We are not dealing with the one dimensional fictitious problem.

All Goldstein said was that the planetary orbital equation is the same as the equation that arises in the equivalent one dimensional fictitious problem. And you have been twisting this so as to interpret Goldstein as having said that centrifugal force is fictitious. Goldstein has said nothing of the sort. Fictious forces don't enter into orbital mechanics.

I'm referring to what other say more than Goldstein. It's a force that appears in the 1D equation for r, but not in an equation of coordinates in an inertial frame, and that's what people call a fictitious force. Dicklyon ( talk) 23:28, 18 April 2009 (UTC)

Goldstein mentions the equivalent fictitious one dimensional problem as an aid for doing a qualitative treatment of the problem. He draws a graph, V against r, which shows the stability node between the outward inverse cube law term and the inward inverse square law term.

We are dealing with the real radial planetary orbital equation. We are not dealing with the fictitious one dimensional equivalent problem. David Tombe ( talk) 23:05, 18 April 2009 (UTC)

Yes the equation is real; but r is measured along a rotating direction, which is why the equation looks the way it does. Dicklyon ( talk) 23:28, 18 April 2009 (UTC)

FyzixFighter, I could list about ten methods that are used to wriggle out of facing up to the reality of centrifugal force. One of them is to make a quibble over whether we are talking about force or acceleration. That's the one that you've just used. It doesn't matter whether we say force or acceleration for the purposes of this argument. We only have to use F = ma to convert between the two.

Now that you have agreed that it is not centripetal acceleration, what is it called then? Harry? David Tombe ( talk) 23:14, 18 April 2009 (UTC)

Dick and Brews, I'm going to copy equation (3-11) straight out of Goldstein, exactly as it appears in Goldstein, so as nobody can make any accusations of musical chairs. Nobody can make the accusation that sources are not being used, or that I am working from first principles or engaging in original research. Here is the equation,

${\displaystyle m{\ddot {r}}-mr{\dot {\theta }}^{2}=f(r)}$

Goldstein calls the term on the right hand side of the equation a 'central force'. That means that all the terms in the equation must have the dimensions of force. This is the real planetary orbital equation for the radial component of force. This is not the fictitious one dimensional equivalent equation which Goldstein talks about a few pages later. This is the real thing.

The term that we are interested in here is the ${\displaystyle mr{\dot {\theta }}^{2}}$ term. This term is a radially outward term with the dimensions of force, and it becomes induced by angular speed. What name would you like to call this term? And what name does Goldstein use for it? David Tombe ( talk) 11:59, 19 April 2009 (UTC)

David, yes, the terms all have dimensions of force; the outward one is called centrifugal force. However, this net "force" is NOT the radial component of force on the planet; the only force on the planet is gravity, which is why it moves in a curved instead of straight line. This equation is an equation of motion for r, the single coordinate in the co-rotating 1D system, which is not an inertial system. It is exactly the kind of system and force that physicists generally refer to as "fictitious", since in an inertial system there's no force that corresponds to this centrifugal force. There's nothing to decide here, as it's all in reliable sources. Nobody but you believes that this f represents "the radial component of force", so we can safely ignore that distraction. Dicklyon ( talk) 15:49, 19 April 2009 (UTC)

Dick, I'm glad that you have finally acknowledged that the controversial term is called centrifugal force. So why can we not use Goldstein's exact words and introduce it as 'the familiar centrifugal force'?

I agree with you that it is exactly the same force that occurs in the topic of rotating frames of reference in the special case when we are dealing with co-rotating objects. And I agree with you that most modern textbooks will refer to it as being a fictitious force in that context. But most modern textbooks don't use rotating frames of reference when dealing with the planetary orbital equation, and we most certainly don't need to impose a frame of reference around that problem. We can all see the problem quite clearly within the context of polar coordinates in the inertial frame, which is how Goldstein does it.

In relation to planetary orbits, there is absolutely no requirement to describe the familiar centrifugal force as being fictitious. And at any rate, can you see anything remotely fictitious about it? It will pull a string taut or it will push against a car door. And to add to the confusion, you have put these secondary effects into a special article with the misnomer 'reactive centrifugal force'. The confusion is appalling.

I have been trying to tidy this article up into one single, easy to read, coherent article, but I have been persistently obstructed by a large group who clearly know very little about the topic. This group barely understood much beyond simple circular motion before this debate began. They tried to prevent the planetary orbital equation from being acknowledged, and so it's not at all surprising that when we finally get it acknowledged that there will be attempts to deny the presence of centrifugal force in the equation (such as FyzixFigher's attempt to call it centripetal acceleration). And when it is finally accepted that centrifugal force is in the equation, and that this centrifugal force can push and pull on other objects, it is to be fully expected that they will still try to argue that it is only a fictitious force. David Tombe ( talk) 22:02, 19 April 2009 (UTC)

Did I ever deny that it's called centrifugal force? What would be the point of that? It's there in black and white. I see no problem quote Goldstein's exact words; but if he doesn't clarify that it's what most physicists call a "fictitious force", then we need to also quote someone else on that. The r coordinate in polar coordinates is the same as a regular cartesian coordinate in a rotating frame, which is why it's a fictitious force; that and the fact that in an inertial frame, the only force acting on the planet is gravity. I don't think the large group you're fighting is ignorant of the issues here; they just have a hard time working around your unique point of view. Dicklyon ( talk) 01:54, 20 April 2009 (UTC)

Dick, It was never referred to as a fictitious force in that particular context in any source which I ever read. The term 'fictitious force' seems to be a relatively new thing. Rotating frames of reference as a topic seems to have evolved out fixing frames of reference into rotating rigid bodies. And yes, the modern topic of rotating frames of reference uses the term 'fictitious' boldly. I think that too many editors here only know about that particular topic and as it is presented in modern textbooks, and don't know much if anything about planetary orbital theory.

As for planetary orbital theory, the modern trend seems to be to totally play down the centrifugal term and to never mention it by name. It is often hidden from view inside a general radial vector box. FyzixFighter tried that one out on 1st February 2009 on the Coriolis force page. Another trick is to use the integral form of the equation so as to mask the centrifugal force term behind the transverse component of kinetic energy.

Anyway, the important thing is that you do appear to understand the underlying physics. But don't forget that the gravity force, which is not fictitious, is also rotating in a planetary orbit and why therefore should the centrifugal force be fictitious just because it is rotating? My main argument is that it is the actual rotation which induces the centrifugal force to begin with.

I'm glad that you didn't jump on the bandwagon with FyzixFighter's attempt to call it centripetal acceleration. There are others who would readily have done so. If you and Brews hadn't faced him down on that point, the situation would now be totally confused and hopeless. David Tombe ( talk) 11:21, 20 April 2009 (UTC)

David, the "particular context" you refer to is the co-rotating frame. Certainly it's common in circular motion where the pseudo forces are balanced, keeping r constant. I'd be surprised if Goldstein is the only one who has used the equation you like so much, but even if he is, and doesn't mention "fictitious force", it's stil the same thing that other physicists call a fictitious force, since it disappears in an inertial frame. Here is another mention of that context and rotating frame together. As to your comment about gravity, it does appear in the inertial frame; using it unchanged in the rotating frame is what requires an additional fictitious force. Dicklyon ( talk) 14:12, 20 April 2009 (UTC)

Dick, it doesn't disappear in the inertial frame. Both gravity and centrifugal force are rotating, and we analyze them both using polar coordinates in the inertial frame. That's what Goldstein has just done and he doesn't involve rotating frames of reference or mention the concept of fictitious forces.

Ask yourself what could possibly be fictitious about centrifugal force in a planetary orbit. In what respect does it differ fundamentally from the gravity force such that you want to see one as fictitious and the other as real? The centrifugal force can pull a string taut. What is fictitious about that? David Tombe ( talk) 16:50, 20 April 2009 (UTC)

The outward force on the string is real; it's the reaction to the inward force on the weight that the string is pulling into a circular path; it's not a force on the object moving in a circle, it's the force that object exerts on the string. In the planetary case, the correponding reaction force is the force that the planet exerts on the Sun; that's real, too. But the only force on the planet is gravity, and its orbit is exactly what the acceleration due to gravity alone would predict. But you know that already, right? Dicklyon ( talk) 18:01, 20 April 2009 (UTC)

Dick, You are totally confused. In the gravity orbit, the gravity and the centrifugal force are totally independent of each other. One is never a reaction to the other. In the case of a circular motion in which the centripetal force is caused by the tension in the string, that tension is a reaction to the outward centrifugal force. You can see quite clearly that the string only becomes taut in the first place because of the outward centrifugal force. You cannot say that the centrifugal force is a reaction. The outward centrifugal force comes first and the inward centripetal force which is due to the induced tension in the string then follows.

Reading your entry again, I can see that you accept that the object pulling on the string is real, but you are trying to say that the outward force that acts on the object itself is only fictitious. That's like saying that weight is real but gravity is only fictitious. How on Earth can you make a conclusion to the extent that the outward force which causes the pull on the string is only fictitious whereas the pull on the string itself is real? David Tombe ( talk) 18:33, 20 April 2009 (UTC)

I'd like to discuss the reversion of the text below. (An attempt to entertain such discussion is wandering off track in the above section.)

A co-rotating frame is one rotating with the object so that the angular rate of the frame, ${\displaystyle \Omega }$, equals the ${\displaystyle {\dot {\theta }}}$ of the object in the inertial frame. In such a frame, the observed ${\displaystyle {\dot {\theta }}}$ is zero, so the term ${\displaystyle r{\dot {\theta }}^{2}}$ in the acceleration is zero. ... Because the object is not seen as rotating in the co-rotating frame, the ${\displaystyle {\dot {\theta }}}$ parameter in this term is not observable directly, but is inferred.

This notion that the angular rate appearing in the centrifugal force is inferred in the co-rotating frame because there is no way to directly observe it, seems to strike a nerve, and has been reverted.

In the co-rotating frame the above-quoted language points out that the co-rotating observer is forced to add the centrifugal term to the force inventory in order to get agreement with observation of the motion. This point is central to all discussion of fictitious force. However, instead of allowing this language, the present form of the article insists on viewing the centrifugal force as simply a transference of a term from one side of the equation to the other ("This radial equation can be rearranged"), a rather pallid and purely mathematical stance. In fact, in the co-rotating frame there is no awareness of the centrifugal term allowing "rearrangement" of the equation. Rather, the centrifugal force is discovered to be necessary and then added to the force inventory.

The argument made by Dicklyon for reversion is as follows:

I objected to this interpretation because it didn't relate at all to the sources that we were working from (Taylor, Goldstein, etc.), in which the rotation rate of the frame is known, since they start from the equations of motion in the inertial frame.

The supposed narrow view of the cited sources as little more than mathematical manipulation does not constitute a reason to avoid pointing out the more physical implications of these manipulations. The physical consequences have a centuries-old history and form the underlying notion of centrifugal force, as pointed out by a footnote (also reverted) drawing attention to the bucket argument and the rotating spheres argument. The "simple algebra" of "rearranging' terms in an equation hardly need be presented alone like a grade-school algebra problem without reference to the physical argument. Brews ohare ( talk) 05:07, 10 May 2009 (UTC)

According to what source does this have anything to do with the "underlying notion of centrifugal force"? I think you've got it all wrong; the physics in this section isn't much; we're showing the math to get from an inertial frame to a co-rotating frame, and how the equations change. The other arguments are well covered in the sections on those other topics, are they not? Why repeat them here, when all we're trying to do is show where the 1D equation for orbits comes from? Recall that the reason for this section is to show that Goldstein's formula trivial integrates with the point of view of centrifugal force as a fictitious force in a rotating frame, which is the topic of the article. There's no need to over-complicate it beyond that. In other words, there is good reason to avoid pointing out stuff that's largely irrelevant to the example and is well covered (presumably) elsewhere. Dicklyon ( talk) 05:32, 10 May 2009 (UTC)

Dick, there is no 1-D equation for planetary orbits. If the equation were 1-D then we couldn't have hyperbolic, parabolic, or elliptical solutions. We have a radial equation in one variable, but it is linked to an angular momentum constant. You have been repeatedly making this error just because Goldstein said that the radial equation is the same equation as would arise in the equivalent 1-D problem. Goldstein used the equivalent 1-D problem to do a qualitative treatment of the issue. A qualitative treatment is less rigorous than a full analysis. In the qualitative treatment, Goldstein looks at the reversal threshold/stability node where the outward inverse cube law crosses over with the inward inverse square law. You need to be aware of this mistake which you keep making. David Tombe ( talk) 17:04, 10 May 2009 (UTC)

The differential equation for r, with r double dot being proportional to a difference of inverse-square and inverse-cube terms, is the one-dimensional equation that the sources speak of. Using conservation of L, it's trivial to get back to the theta dot given a solution for r, and from there the x and y, in the 2D plane of motion, which is where you'll see those conic sections. We could add a few words to clarify if you think it's necessary. It's not clear what mistake you think I'm making. Dicklyon ( talk) 17:30, 10 May 2009 (UTC)

No Dick, it's a two dimensional equation no matter what way we write the centrifugal force term. If it were a one dimensional equation, then it could not solve to yield an elliptical orbit. When we write the equation using only one variable, the equation is then the same as the equation for the equivalent one dimensional problem. David Tombe ( talk) 01:06, 11 May 2009 (UTC)