Talk:Centrifugal force (rotating reference frame)/Archives 12

Some math details

Brews, because the derivation begins before you actually get to the expression,

${\frac {d\mathbf {u} _{j}(t)}{dt}}={\boldsymbol {\Omega }}\times \mathbf {u} _{j}(t)\ ,$

And that part of the derivation is not included in the article. If you begin the derivation from the very beginning you will see the point that I have been making.

Have a look at this web link. It shows the vector triangle in question,

[1]

David Tombe (talk) 15:29, 1 November 2008 (UTC)

Kwon's notes seem to be a detailed discussion of the derivation of, for example, his Eqs 8 and 9, which appear at first reading to be the same as

${\frac {d}{dt}}{\boldsymbol {Q}}=\left[{\frac {d}{dt}}{\boldsymbol {Q}}\right]+{\boldsymbol {\Omega \times Q}}\ ,$

from the Centrifugal force (rotating reference frame) article. Are you suggesting that there is some difference in results or in interpretation of the results in Kwon's notes? If so, perhaps you could point out just what I should be noticing?

Please excuse an interruption in replies from me, as I am away for a few days. Brews ohare (talk) 17:01, 1 November 2008 (UTC)

Brews, from the web link that I showed you, you should be able to see that the angular velocity is that of the fixed point in the rotating frame. Hence, these transformation equations only apply to co-rotating objects. David Tombe (talk) 20:21, 1 November 2008 (UTC)

It's not clear to me what the Kwon page is about, but in th analyses we've been talking about, the tranformations and fictitious forces apply to any object, whether co-rotating or not; but for objects that are not stationary in the rotating frame, you need to also include the Coriolis force, which is a vector proportional to the in-frame velocity of the object. That is, the transformation is conventionally resolved into two named parts, one of which depends only on the object's position (the centrifugal force), and other of which depends also on the object's velocity. Dicklyon (talk) 02:42, 2 November 2008 (UTC)

Yes Dick, we know all that. But it's wrong. It's wrong on quite a few counts. Coriolis force is never radial. Also, a circular motion requires a balance between centrifugal force and centripetal force, and not for the centrifugal force to be overridden by the centripetal force. But most important of all, you can see from the web link that I supplied above, that the angular velocity in the rotating frame transformation equations is that of a fixed point in the rotating frame. So the explanation which you have just given, which does unfortunately appear in some texbooks, should not be a basis for allowing this supposed centrifugal force on non-rotating objects to dominate the article. The article should be balanced. We don't need rotating frames of reference to be involved in order to have centrifugal force, and so the title and the introduction should not mention rotating frames of reference. That should be dealt with in a section within the article. David Tombe (talk) 14:35, 3 November 2008 (UTC)

If you restrict Coriolis force to be radial it won't work. But if you take the equations as written, they do work and they do explain the examples you've discussed so far, like the planets, the stationary object that appear to be rotating when viewed in a rotating frame, etc. Circular motion only requires a centripetal force. Try it. Dicklyon (talk) 02:50, 4 November 2008 (UTC)

You say: Yes Dick, we know all that. But it's wrong. It's wrong on quite a few counts. Given that these forces and effects appear from a deductive proof that makes no assumptions except Newtonian mechanics, you'll presumably be able to come up with a counterexample; preferably a numerical one, that we can just plug into the equations, and we can go 'oh yeah, how could we be so stupid'. Otherwise, I can continue to treat this as the bunch of useless hot air that I know it is.- (User) Wolfkeeper (Talk) 16:06, 3 November 2008 (UTC)

David: You say: you should be able to see that the angular velocity is that of the fixed point in rotating frame Kwon says in connection with his Figure 2 that ω is the rate of rotation of the frame. That is, his equations are identical with those in the Wiki article and with those in Arnol'd, Taylor, Gregory and Landau & Lifshitz. Brews ohare (talk) 01:24, 4 November 2008 (UTC)

Brews, Yes exactly. The angular velocity ω (omega) is the rate of rotation of the frame. Hence the centrifugal force term in the maths, which utilizes this angular velocity term, can only apply to particles that are co-rotating with the frame. It is totally meaningless to apply these equations to objects that are not co-rotating. Sure you can see that a non-rotating bucket of water doesn't have a parabolic surface, and as such there is no centrifugal force active in it, no matter how we look at it. Nobody was arguing with the equations. The argument was about the fact that the equations were being applied to situations outside of their jurisdiction. David Tombe (talk) 02:06, 4 November 2008 (UTC)

No, read again; the fictitious centrifugal forces applies to all objects, whether co-rotating or not; but the ones not co-rotating also feel the Coriolis force, which can't be ignored if you want a full picture. Dicklyon (talk) 02:50, 4 November 2008 (UTC)

David: Dick has this correct. Please read the section Polar coordinates referred to earlier for more details. Brews ohare (talk) 03:42, 4 November 2008 (UTC)

I agree also. Do feel free David to attempt to point to any part in any of the derivations where you think that this is no longer true.- (User) Wolfkeeper (Talk) 03:46, 4 November 2008 (UTC)

Outward centrifugal force

David: You say: The radial planetary orbital equation contains an outward centrifugal force term. The Wiki article outlines the solution. The "outward centrifugal force" appears only in the "effective one-dimensional" potential in the one-D solution. In the stage before this formulation of the problem, a single-particle in planar orbit subject to a central force is used. In this stage, the particle is subject to the radially inward gravitational force from the potential, and the acceleration in the two-dimensional problem is:

${\ddot {r}}-r{\dot {\theta }}^{2}=-GM{\frac {1}{r^{2}}}\ .$

In this formulation, the second term on the left side is simply one of two terms in the acceleration in polar coordinates. It is not an impressed force; it is part of the mathematics of derivatives in polar coordinates. The interpretation as a "centrifugal" part of the potential in the one-dimensional version is a "fiction", and that is why the One-D problem itself is called "fictitious": it pretends that ${\ddot {r}}$ is an "acceleration". It isn't an acceleration except in One-D. In Two-D the radial acceleration is ${\ddot {r}}-r{\dot {\theta }}^{2}$. Brews ohare (talk) 16:03, 5 November 2008 (UTC)

A different approach is that of Hildebrand, who introduces the "generalized coordinates" of the Lagrangian. One of these "generalized coordinates" is r and one of the "generalized forces" is $m{\ddot {r}}$. Hildebrand then calls $mr{\dot {\theta }}^{2}$ the "centrifugal force", but it actually is the "generalized centrifugal force", not the centrifugal force of this Wiki article and not the centrifugal force of Newtonian vector mechanics. (Hildebrand uses the words "generalized Coriolis force" but for some unknown reason does not add "generalized" in the case of centrifugal force.) I have tried repeatedly to get you to look at this formulation. If you were to adopt it, the entire controversy would disappear because you are talking about a different formulation. Brews ohare (talk) 16:44, 5 November 2008 (UTC)

As an aside, the idea of going to arbitrary coordinate systems might throw your intuition a bit of a curve, eh? If one were to go to spherical-polar coordinates (r, θ, φ), for example, the generalized force corresponding to $m{\ddot {r}}$ becomes $mr{\dot {\theta }}^{2}\sin ^{2}\phi +mr{\dot {\phi }}^{2}$. See Ginsberg. The actual motion need not change, only the coordinate system used to describe it. But the "generalized forces" do change. On the other hand, the Newtonian vector mechanics approach is coordinate-system independent. Brews ohare (talk) 17:44, 5 November 2008 (UTC)

Brews, the term $mr{\dot {\theta }}^{2}$ in the radial planetary orbital equation IS the centrifugal force. It is an outward force that is induced by tangential velocity. I thought that by now you might have seen the connection to the Lorentz Force and velocity dependent induction. But you seem to be more intent on talking yourself into believing that it isn't for real. I can't see how labouring the fact that we have a one dimensional equation in r somehow negates the underlying reality of the situation. Centrifugal force is a real outward inverse cube law force and in conjunction with the attractive inverse square law force it forms the basis of the interatomic force/potential energy graphs which are in turn the same as the one dimensional potential energy graph for planetary orbits as shown in Goldstein. The two power laws create a stability node and this is probably the basis of Boscovich's force law. David Tombe (talk) 03:04, 6 November 2008 (UTC)

David: This is your view, but not supported. I have raised several points; you have ignored them. Brews ohare (talk) 03:51, 6 November 2008 (UTC)

Coriolis force

David: You say: The tangential planetary orbital equation equates the sum of the Coriolis force and the angular force (Euler force) to zero, and is hence equivalent to Kepler's second law.

As described in the Wiki article, in an inertial frame of reference using polar coordinates, the equation for the acceleration is:

${\frac {d^{2}}{dt^{2}}}{\boldsymbol {r}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+{\frac {1}{r}}\quad {\dot {\overbrace {r^{2}{\dot {\theta }}} }}\quad {\hat {\boldsymbol {\theta }}}\ ,$

with the azimuthal component:

$(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }})={\frac {1}{r}}\quad {\dot {\overbrace {r^{2}{\dot {\theta }}} }}\quad \ .$

If the first term is called the "Euler force" (per unit mass) and the second the "Coriolis force", your remark that the sum is zero is valid in a central force problem where there is zero azimuthal impressed force. The zero sum is a statement of conservation of angular momentum. This terminology is in common use in discussion of polar coordinates. For example, see Shankar. The angular rate ${\dot {\theta }}$ is the angular rate of the particle in the inertial frame of reference.

However, this terminology is not that of Newtonian vector mechanics, which defines both the Euler and Coriolis forces as fictitious forces that are zero in inertial frames; the angular rate in Newtonian vector mechanics is the rate of rotation Ω of a frame of observation.

You are arguing at cross-purposes here by refusing to realize that multiple terminologies are in use: you are using the "polar-coordinate" terminology, while the article is using the Newtonian vector mechanics terminology. The first is entirely within the framework of a stationary, inertial frame. The second is concerned with a rotating frame. The first refers to the angular rate ${\dot {\theta }}$ of the moving particle in the inertial frame. The second refers to the angular rate of rotation Ω of a non-inertial frame. Brews ohare (talk) 18:53, 5 November 2008 (UTC)

Brews, the cross purposes is between you and Dick Lyon. While Dick is trying to deny what you have written above, you are accepting it but trying to make yourself believe that it is a completely separate topic from the centrifugal force and the Coriolis force as occurs in rotating frames of reference.

In actual fact, in co-rotating situations, it is all one single topic. In non-co-rotating situations, the application of the rotating frame transformation equations is an absurdity.David Tombe (talk) 03:12, 6 November 2008 (UTC)

David: This is personal opinion, not a reasoned response. You cannot circumvent the point that Ω ≠ ${\dot {\theta }}$ . Brews ohare (talk) 03:48, 6 November 2008 (UTC)

Brews, You are trying to say that there are two different concepts that both bear the name of centrifugal force, and which both use the same mathematical formula. You are basing this on the fact that you believe that the centrifugal force in rotating frames can apply to objects that are not co-rotating, whereas the centrifugal force in planetary orbits only applies to the actual rotation of the object in question. You are saying that the centrifugal force in rotating frames has a versatility which allows it to be associated with non-rotating objects.

I have demonstrated to you that such is not the case. The derivation of centrifugal force in both cases is identical in principle when we look at it in depth, and in both cases, it only applies to objects that possess actual rotation.

Centrifugal force cannot under any circumstances be associated with objects that are not in a state of actual rotation. A stationary bucket of water shows no parabolic surface, hence there is no centrifugal force acting and there is no centrifugal potential energy.

You cannot claim that the centrifugal force in this case is being cancelled by a radial Coriolis force, because the concept of a radial Coriolis force is an absurdity. Besides that, supposing we did have a radial Coriolis force. It would need to be a tension force to cancel out with the centrifugal pressure in the water. If that were to be so, we would be able to derive a Coriolis potential energy. And we all know that that cannot be done in the circumstances.

You need to stand back and look at all this from a distance. At the moment you are blinkered with the maths, and have lost sight of the wider physical reality.

You are operating under the erroneous belief that centrifugal force is only an illusion as observed from a rotating frame of reference. Yet you are also aware that this supposed illusion can have real physical effects. David Tombe (talk) 12:14, 6 November 2008 (UTC)

David: I will respond to several of your remarks above. I'll put your sentences in italics, followed by my observations.

You are trying to say that there are two different concepts that both bear the name of centrifugal force, and which both use the same mathematical formula.

Yes, there are two concepts. No, they do not use the same formula.

One concept stems from a Lagrangian approach. I call it the "generalized centrifugal force". An example of this method appears in the problem of a single particle in a central potential expressed in polar coordinates (r, θ). As we know this problem is equivalent to a fictitious One-D problem involving the centrifugal and impressed force. This problem also can be solved in Two-D, where the forces involved are only the centripetal force as provided by the central field, and no other forces at all. Within the Lagrangian approach, however, one ends up with the One-D problem by direct mathematics, and within the One-D problem one of the forces is called the "centrifugal force" by analogy with other problems. The analogy involves interpreting the angular rate ${\dot {\theta }}$ in this formulation as playing the same role as the angular rate Ω of a rotating frame in the Newtonian vector mechanics formulation. Of course, terminology is arbitrary, and equating ${\ddot {r}}$ to a "generalized force" is OK, but you cannot expect (and it does not) behave like Newtonian centrifugal force. In particular, it does not vanish in an inertial frame.

The second concept is the Newtonian centrifugal force. This idea is introduced as a coordinate transformation between a stationary and a rotating frame. It has application only when a rotating frame is involved. It does not apply when only an inertial frame is used, as in the previous example of a particle in a central force. This view is pointed out in innumerable references on Classical Mechanics, as you know. Your reaction to this fact is to say that these transformations are limited, and that these authors do not realize this fact and either apply the results incorrectly, or accidentally apply them in cases where they just happen to work. I'd say this position of yours is highly unlikely, and unsupported. That makes it ineligible in a Wiki article that presents common practice. Brews ohare (talk) 16:29, 6 November 2008 (UTC)

I have demonstrated to you that such is not the case. The derivation of centrifugal force in both cases is identical in principle when we look at it in depth, and in both cases, it only applies to objects that possess actual rotation.

This statement is not valid. No demonstration has been presented, no look in depth. Some vague suggestions of yours about limits and their consequences have never been spelled out, and responses to this suggestion have been ignored by you. Brews ohare (talk) 16:29, 6 November 2008 (UTC)

Centrifugal force cannot under any circumstances be associated with objects that are not in a state of actual rotation. A stationary bucket of water shows no parabolic surface, hence there is no centrifugal force acting and there is no centrifugal potential energy.

You continue to equate a parabolic surface to "centrifugal force". As pointed out before, and ignored by you, a parabolic surface indicates rotation. The explanation of the surface, as opposed to the observation of the parabolic surface, takes several forms. One form, that applicable to an inertial observer, involves centripetal force. Another form, that applicable to a rotating observer, involves centrifugal force. There is no existence of centrifugal force in inertial frames (using the Newtonian terminology).

You cannot claim that the centrifugal force in this case is being cancelled by a radial Coriolis force.

I guess we could ask about the non-rotating water with a flat surface as seen in a rotating frame, although that is not discussed in the article on rotating bucket. In this case, of course, the water appears to the rotating observer to be rotating, although the flat surface is evidence that the water is not rotating wrt absolute space. The rotating observer can do experiments that show the existence of centrifugal force, and therefore must conclude that it is acting upon the water too. To explain the lack of parabolic surface, this observer must introduce Coriolis force to counter the centrifugal force. Your arguments about what is possible and impossible for Coriolis forces is countered here by application of the vector formula for Coriolis force, using the apparent velocity of the water, which works out as perfectly canceling the centrifugal force in this example, and overcompensating to provide the necessary centripetal force to maintain the water in circular motion. That is, because of the Coriolis force (in contrast to the situation with truly rotating water in an inertial frame) it is unnecessary for the water to pile up in order to provide centripetal force. This situation also comes up in the case of rotating spheres that are stationary in the inertial frame, but appear to be rotating in a rotating frame. The complete argument and references are in rotating spheres. We don't need to go into all that to find differences of opinion, unless you find that example clarifies matters. Adding Coriolis force to the mix seems to me to make things more complex, not less. Brews ohare (talk) 16:29, 6 November 2008 (UTC)

At the moment you are blinkered with the maths, and have lost sight of the wider physical reality.

I'd say instead that "you are blinkered by a strong intuition, and lost sight of the math." As you know, intuition is a strong but fallible force. One cannot proceed without it, but one also cannot trust it without rigorous support. Brews ohare (talk) 16:29, 6 November 2008 (UTC)

Brews, How can you possibly look at the equation, ${\ddot {r}}-r{\dot {\theta }}^{2}=-GM{\frac {1}{r^{2}}}\$ and say that the second term on the left hand side is not centrifugal force? Initially, I couldn't even get any recognition of the validity of this equation at all. It was systematically denied and it was demanded that I produce sources. I produced a gold standard source and it was still denied. Now that we have reached the stage were it is no longer possible to deny it, thanks largely to the fact that it has long been acknowledged on the Kepler's laws of planetary motion article, you are now trying to deny that centrifugal force is involved in the equation. Centrifugal force is an inverse cube law repulsive force. This can be shown by substituting the Kepler areal constant into the centrifugal force term in the equation. It provides the centrifugal repulsive barrier that Goldstein refers to on page 78. It works in tandem with the inverse square law attractive force to produce stability nodes and to hence avoid any problems with Earnshaw's theorem. It is a conservative force which has an associated potential energy. You have even given us all a satisfactory qualitative treatment of centrifugal potential energy in a rotating bucket of water. Yet you think that in the inertial frame of reference we must drop this explanation and instead use centripetal force which is a completely different concept? What you are saying does not make any sense.David Tombe (talk) 18:31, 7 November 2008 (UTC)

David, your equation from Goldstein is for the case of a reference=frame axis r aligned with a planet, co-rotating at a non-constant rate. In this frame of reference, the acceleration (r double dot) is subject to a balance between the gravitation attraction (right hand side) and the second term on the left, which you call centrifugal force. This centrifugal force depends entirely on the rotating frame; there is no such actual force in an inertial frame, but rather only a gravitational acceleration that bends the planet's orbit to the right shape and speed, right? I'm a bit unclear whether sources agree that this "fictitious force" that depends on the rotating frame is called centrifugal even in the case when the rotation is not uniform, but I'm willing to accept that if Goldstein or someone says so. I don't think that's the point of disconnect between you and the rest of us. I'm not surprised that this formulation was denied before you produced sources, since it differs from the more common treatment of centrifugal and Coriolis forces in uniformly rotating frames; that why we need to cite sources, so we can understand where each other are coming from; now that we understand it, I don't see what the problem is. Dicklyon (talk) 19:08, 7 November 2008 (UTC)

Dick, First of all, there is no balance between the centrifugal force and the gravitational force, except in the special case of circular orbits. Do you even comprehend the differential equation in question? In elliptical orbits, the two forces alternate between which one of the two has got the greater magnitude, and in parabolic and hyperbolic orbits, the centrifugal force is always greater than the gravitational force. Secondly, there is no rotating frame of reference involved in the analysis. Show me a source which involves rotating frames of reference in relation to central force theory. The angular velocity in question is relative to the inertial frame. And thirdly, yes, this treatment does indeed differ from the treatment of centrifugal force and Coriolis force in uniformly rotating frames, which is exactly what the problem is. The problem is compounded by the fact that most of the editors here only seem to be familiar with the latter treatment. However, anybody who has done applied maths at university should be familiar with the former treatment as well. It seems to me that you have only come across central force theory for the first time, here on wikipedia, since this debate began. But rather than learning from it, it seems to me that you are trying to sweep it under the carpet. First you deny it. Then you try to reinterprate it within the context in which you are already familiar with the concepts of centrifugal force and Coriolis force. David Tombe (talk) 04:22, 8 November 2008 (UTC)

David: You say:

How can you possibly look at the equation,

${\ddot {r}}-r{\dot {\theta }}^{2}=-GM{\frac {1}{r^{2}}}\$

and say that the second term on the left hand side is not centrifugal force?

The answer is as follows. This equation applies in an inertial frame of reference. There is no centrifugal force in an inertial frame (Newtonian terminology). In Newtonian mechanics, this term is mathematically part of the radial acceleration, and is not interpreted as a force, neither real nor fictitious. In this inertial frame, the only force present is the gravitational force on the right-hand side, which provides the centripetal force for the orbit. The term you refer to as "centrifugal force" nonetheless is given that name by a select group of scholars in a very narrow context, not applicable throughout the entire sphere of classical mechanics. This use of the term "centrifugal force" is incompatible with Newtonian centrifugal force, because (as is evident in this example) it is non-zero in an inertial frame of reference.

To elaborate still further, there is no rotating frame here. Therefore there is no Ω. The angular rate in this expression is that of the particle itself, as seen from the inertial frame, not the angular rate of a frame. Newtonian centrifugal force is zero when the frame Ω = 0. Brews ohare (talk) 21:36, 7 November 2008 (UTC)

Apparently Brews and I are interpreting the equation differently. My interpretation is that it IS in a rotating frame, co-rotating with the planet. Am I wrong? Dicklyon (talk) 02:46, 8 November 2008 (UTC)

Dick, you were the one that insisted upon sources. I gave you a source and you totally denied the contents. That equation is in Goldstein and there is absolutely no mention of rotating frames of reference in the context. He uses plane polar coordinates referenced to the inertial frame. The planets have an absolute angular velocity relative to the inertial frame. That's what keeps them up. Rotating frames of reference have got absolutely nothing to do with the issue. Anyway, your own position is not even consistent. The last time you intervened here, you were trying to tell me that it was 'reactive centrifugal force'. It seems that you will look at this equation in any possible way at all other than recognizing the fact that centrifugal force occurs as an outward radial force without the involvement of rotating frames of rerefence. David Tombe (talk) 04:01, 8 November 2008 (UTC)

I don't think I'm denying anything that Goldstein says, but maybe the way I describe my interpretation is questionable. Using the r coordinate is equivalent to converting to a 1D problem by rotating the frame to keep the planet on a certain axis from the sun. If you take the gravitational force on the right, and look at what it does to r, you're rotating your frame to make it a 1D problem. If you take the same gravitational force and look at what happens to the complete motion in 2D inertial space, you find that the trajectory is curved, with an acceleration corresponding to the gravitation force alone; there's no centrifugal force in 2D inertial space, but there is in r since it's a coordinate of a rotating space. Do you have a different interpretation? Dicklyon (talk) 06:35, 8 November 2008 (UTC)

Dick: There is a missing ingredient in your description. First the azimuthal equation is solved (the one equivalent to conservation of angular momentum). That allows ${\dot {\theta }}$ to be eliminated from the radial equation and an equation only in r is left. That is the co-rotational version that involves no angular dependence. Brews ohare (talk) 06:40, 8 November 2008 (UTC

The ${\dot {\theta }}$ of the inertial system is just the omega of the co-rotating frame, right? The only angular dependence is thus this rotation rate? Dicklyon (talk) 07:00, 8 November 2008 (UTC)

Yes. The r-dependent centrifugal force is dictated by the angular momentum substituted into the radial equation from the solution of the azimuthal equation. Brews ohare (talk) 10:51, 8 November 2008 (UTC)

Dick, You said above that This centrifugal force depends entirely on the rotating frame. This is where you are quite wrong. The centrifugal force in planetary orbits depends on the actual angular velocity of the planets relative to the inertial frame. Nowhere are rotating frames of reference ever mentioned in central force orbital theory. The planetary orbits topic is an unambiguous example of centrifugal force existing without the need to involve rotating frames of reference. It is what the edit war has been all about. There has been a systematic denial of the fact that centrifugal force can exist outside of the context of rotating frames of reference. But now you know that this denial has been totally unfounded. You have seen with your own eyes, a textbook, which is one of the most well known textbooks used in modern university applied maths courses. You have seen that Goldstein deals with centrifugal force in planetary orbits without any mention of rotating frames of reference. There is nothing further to argue about the matter. The introduction to the main article is wrong because it implies that centrifugal force is something that can only exist in rotating frames of reference. You have demanded sources to prove that that is not true. You have seen the sources. So if you wish to leave the introduction as it stands, then you are knowingly subscribing to wrong information simply because you have only just learned about planetary orbital theory, and it hasn't yet sunk in. David Tombe (talk) 10:52, 8 November 2008 (UTC)

David, "the actual angular velocity of the planets relative to the inertial frame" is the same as the frame rotation rate when you're in the co-rotating r-only frame. The systematic denial that you refer to is based on real physics and the overwhelming majority of sources, including the most recent edition of Goldstein. Goldstein never said there's centrifugal force in an inertial frame; that's your own misinterpretation. When he reduced the 2D problem to a 1D r-only problem, he is in a rotating frame of reference, even though he didn't make the sufficiently clear (to you) in earlier editions. You have not produced sources that explicitly back up what you are saying, nor even any additional sources that leave it open to your misinterpretation. Give it up. Dicklyon (talk) 17:03, 9 November 2008 (UTC)

Dick, you are once again trying to deny the contents of Goldstein. There is nothing in Goldstein about rotating frames of reference in relation to planetary orbits. If you think otherwise, let's have the quote and the page number. David Tombe (talk) 21:27, 9 November 2008 (UTC)

The quote from Goldstein above (search for "The full quote of Goldstein on page 76") says that the additional term involving ${\dot {\theta }}$ comes from analysis in a one-D frame co-rotating with the planet, does it not? Dicklyon (talk) 21:37, 9 November 2008 (UTC)

No it doesn't say anything of the sort.David Tombe (talk) 21:42, 9 November 2008 (UTC)

Here is the exact quote,

The equation of motion in r, with ${\dot {\theta }}$ expressed in terms of l, Eq. (3.12), involves only r and its derivatives. It is the same equation as would be obtained for a fictitious one-dimensional problem in which a particle of mass m is subject to a force $f'=f+{\frac {l^{2}}{mr^{3}}}$. The significance of the additional term is clear if it is written as $mr{\dot {\theta }}^{2}=mv_{\theta }^{2}/r$, which is the familiar centrifugal force.

David Tombe (talk) 21:47, 9 November 2008 (UTC)

That's what it says, I agree. The r is the coordinate in a one-D co-rotating system, which is the only system in which r-double-dot can be interpreted as an acceleration. Here is a book that should help you understand what it means. Dicklyon (talk) 21:52, 9 November 2008 (UTC)

Dick, I'll repeat again. Goldstein doesn't mention rotating frames of reference in relation to planetary orbits, and neither does any textbook. He gives the radial planetary orbital equation. He then substitutes the Keplerian areal constant into the centrifugal force term and reduces it to a one-dimensional equation in r, in which the centrifugal force is a repulsive inverse cube law force. Any reference to co-rotating frames is only in your imagination. We do not need to involve rotating frames of reference, and the textbooks don't do it. David Tombe (talk) 11:36, 10 November 2008 (UTC)

Co-rotating frame

Perhaps it would help to look at the central force problem in a co-rotating frame. In that frame the particle is stationary. Because the frame is co-rotating, the frame rotation Ω is the same as the particle's angular rotation:

$\Omega ={\dot {\theta }}\ .$

However, in this co-rotating frame experiments show there is a radially outward centrifugal force always present of value:

$F_{Cfgl}=\Omega ^{2}r\ .$

It is apparent to the observer in the co-rotating frame that their observed equilibrium requires that their ubiquitous outward centrifugal force is balanced by the gravitational attraction. It also is evident to this observer that they are not rotating ${\dot {\theta }}=0$ and not accelerating radially ${\ddot {r}}=0$. Consequently their Newton's law formulation is:

${\ddot {r}}-r{\dot {\theta }}^{2}=0=r\Omega ^{2}-GM{\frac {1}{r^{2}}}\ .$

This appears to be exactly what you are looking for David.

In a more general motion, the same ideas can be applied using an arc-length coordinate system locating the particle by the distance s along its trajectory. In that case, the axis of rotation of the co-rotating frame is located at the center of curvature of the path, and the radial distance is the instantaneous radius of curvature ρ. The angular rate of rotation of the frame matches the instantaneous speed of the particle on its path, Ω = v / ρ . Again the particle appears to be in equilibrium, and again a centrifugal force ρΩ² is present that exactly counters the kinematically required centripetal force, which must be provided by gravity or some other real force. Brews ohare (talk) 01:09, 8 November 2008 (UTC)

Brews, One minute you are trying to tell us all that the centrifugal force in planetary orbits is a different kind of centrifugal force than the one in rotating frames of reference. But now you have attempted to bring it into the context of rotating frames of reference. Rotating frames of reference don't come into this at all. This is about absolute angular velocity relative to the background stars. That's what keeps the planets up. It would be a very awkward rotating frame of reference that we would need to use here, bearing in mind the variable angular acceleration of the frame. It is simply not done. Can you provide me with a source or a citation which deals with planetary orbits using rotating frames of reference? David Tombe (talk) 04:06, 8 November 2008 (UTC)

David: So you turn down this olive branch, eh? You are unwilling to accept that there are two versions of centrifugal force. You want only one. I am much more flexible. That is why you find me seemingly contradicting myself. For the Newtonian view says there is no centrifugal force without a rotating frame, while the "polar coordinate" view says there is always a centrifugal force, and no rotating frame is needed. Obviously, these viewpoints clash. It is not a case of being on both sides of the issue. It is a case of different definitions. The Newtonian side says the equation is:

${\ddot {r}}-r{\dot {\theta }}^{2}=-GM{\frac {1}{r^{2}}}\ ,$

and the second term on the left is part of the mathematical expression for acceleration in polar coordinates. The polar coordinate side says the equation is:

${\ddot {r}}=r{\dot {\theta }}^{2}-GM{\frac {1}{r^{2}}}\ ,$

and the first term on the right is the centrifugal force. Obviously, we all get the same answer. Obviously our language describing the situation is different. Clearly the Newtonian interpretation says there is no centrifugal force in inertial frames. Clearly the polar coordinate view says there is centrifugal force in inertial frames.

So what? For the planetary problem, who cares? On the other hand if you stray outside of this narrow problem and ask about the historical development of general relativity and how gravitation came to be thought of as a fictitious force, the polar coordinate viewpoint is unintelligible. If you read textbooks about Foucault's pendulum, or meteorology books about weather patterns, or discussions of balls on carousels, or about gunnery and sight corrections, the polar-coordinate view is unintelligible. By unintelligible I mean the discussions of these topics are based upon the Newtonian view and the notions of inertial and non-inertial frames, not the other view, so the reader with the polar coordinate view alone is lost. Brews ohare (talk) 06:00, 8 November 2008 (UTC)

First of all, planetary orbital theory encompasses the basic principles of all classical physics. So it's not just a matter of saying 'who cares?. This article is about centrifugal force. There is no better illustration of centrifugal force, that shouldn't encounter controversy, than planetary orbital theory.

I could of course bring in Maxwell's ideas on magnetic repulsion, but that would lead to accusations of original research, since it is not modern textbook material.

But unfortunately, I have been accused of original research anyway, merely by trying to bring your attention to planetary orbital theory as it is being taught right now in the universities.

It is all about two equations. There is a radial equation and a tangential equation. The four terms between them all appear in equation (5) in Maxwell's 1861 paper in connection with magnetism.

You can't get a better medium than planetary orbits within which to introduce centrifugal force.

Centrifugal force is the inverse cube law repulsive force which exists between any two objects that possess mutual tangential velocity relative to the inertial frame of reference. It is the basis of stability in matter. If we were to depend soley on the inverse square law and electrostatics for the stability of matter, everything would collapse as per Earnshaw's theorem. We need that inverse cube law repulsive force in order to obtain the stability nodes on the combined force graph. You can see it in Goldstein in the planetary orbit section, and you can see it in physics books in inter-atomic force graphs.

Unfortunately however, centrifugal force has got mixed up with the concept of a rotating frame of reference. That wouldn't be so much of a problem if we were focused on co-rotating situations. But unfortunately the maths has been allowed to become unanchored from its sure physical foundations, and this has led to an absurdity which has crept into a few textbooks. That absurdity has been allowed to dominate the centrifugal force article here.

The Coriolis force cannot swing into the radial direction. How can you take terms from Kepler's second law and put them into Kepler's first law?

And you've now finally acknowledged the radial equation. You can see that we need to balance the centrifugal force with the centripetal force in order to get a circular motion. Yet the absurdity in question manages to produce a circular motion out of a centripetal force alone.

Nowhere is this absurdity more absurd than in matters to do with Coriolis force. There is no Coriolis force in the gravitational field, apart from that subtle one which is always mutually cancelling with the Euler force in elliptical orbits.

You cannot create a Coriolis force by observing something from a rotating frame of reference.

Modern treatment of Coriolis force is a nonsense. The cyclones involve a much more complicated explanation than anything you will find in a modern textbook about rotating frames of reference. They involve hydrodynamics and inter-molecular forces.

Anyway, back to the point. You have at least realized that something doesn't tie up as between the centrifugal force in planetary orbits and the centrifugal force in rotating frames. But you deal with this dilemma by splitting the topic. I deal with it by pointing out that the extrapolation of rotating frame transformation equations to objects that are stationary in the inertial frame is a total nonsense.

Unfortunately, there are others who don't even see the dilemma and who have been trying to paper over cracks.

I'm not trying to be funny here but the ridiculousness of your own position is very much caracaturized by your presentation above of two identical equations which you are suggesting are obviously different, simply on the basis that the controversial term in question has been shifted to the other side.David Tombe (talk) 11:19, 8 November 2008 (UTC)

Hi David: Well, try or not you have succeeded in being funny. BTW, my point is that the two equations are obviously the same. However, the verbal description is different. The controversy is over nomenclature, which happens to correspond to switching a term from the acceleration side to the force side of the equation, thereby reclassifying a kinematic term as a fictitious force. Mathematically a "so what", but in terms of the wider context, including the ideas of inertial vs. non-inertial frames, a headache. To repeat myself, the textbook discussions of a wide variety of topics from carousels to meteorology are based upon the Newtonian view and the notions of inertial and non-inertial frames (that is, leaving the kinematic term on the acceleration side, and saying there is no centrifugal force in inertial frames), not upon the polar-coordinate, planetary-motion view (that is, taking the kinematic term to the force side and saying there absolutely is so too centrifugal force in inertial frames). Brews ohare (talk) 12:58, 8 November 2008 (UTC)

Brews, In Maxwell's vortex sea, the vortices press against each other in their equatorial planes due to centrifugal force. There are therefore many rotating frames of reference for the many vortices. Yet we can observe the centrifugal pressure by being in none of these rotating frames of reference.

Whoever first got the idea that one has to be in a rotating frame of reference to observe centrifugal force?

And just to add to the mess, Goldstein gives a Lagrangian treatment of the Lorentz force. The vXB term is clearly the Coriolis force. You may not believe that but it doesn't matter, because even if it is not the Coriolis force, it has the same mathematical form as the Coriolis force and therefore it cannot have an associated potential energy. Yet Goldstein manages to produce a potential energy for it. He produces (A.v) which is of course the potential energy for centrifugal force. He has inadvertently brought back the missing fourth term of the Lorentz force which appeared in Maxwell's 1861 paper, but he has linked it to the vXB term. Interestingly, Maxwell himself used the Coriolis force term to apply to force on a current carrying wire, but proceeded to give a physical explanation using centrifugal force, even though he already had a centrifugal force term in his equation. But he had already used that centrifugal force term to explain paramagnetic attraction and diamagnetic repulsion.

Lagrangian is fine as regards balancing energy. But it totally loses track of cause.

The point is that if you try to take all the modern textbooks literally, you will end up totally confused. All your writings above about name changes and terminologies is a product of that confusion.

I'm trying to cut through all that confusion and point out to you that all of classical physics is contained in those two planetary equations. (1) the radial equation, and (2) the tangential equation.

When you comprehend that, then you will be able to write a simplied article on centrifugal force in a coherent manner, using prose, and with a minimal of mathematics, and without contradicting the textbooks. It's a question of ignoring fringe theories which you know to be wrong. Concentrate on the key points for the benefit of the lay reader. David Tombe (talk) 08:48, 9 November 2008 (UTC)

David: You have presented items you mentioned before, and haven't advanced the argument by engaging any of the points raised. I guess the short summary of your view is your statement: "Modern treatment of Coriolis force is a nonsense.". I guess we'll just have to call it quits. Brews ohare (talk) 15:56, 9 November 2008 (UTC)

Brews, No. Your approach to centrifugal force, as applied to particles that are stationary in the inertial frame, totally collapses when you come face to face with the radial equation,

${\ddot {r}}=r{\dot {\theta }}^{2}-GM{\frac {1}{r^{2}}}\ ,$

Notice that we sum the centrifugal force and the centripetal force and that we get a circular motion only when the two are balanced? However, in your artificial circle scenario when the stationary particle is viewed from the rotating frame, you advocate a net centripetal force. It doesn't make sense. How can we get a circular motion with a net centripetal force? That's not to mention the fact that the radial centripetal force in question has been supplied by a Coriolis force, and you know that Coriolis force is a tangential force. There is an absurdity in some of the textbooks that is best ommitted from this article. Restrict rotating frames of reference to a single section in the article and generalize the title and introduction. That's what I suggest. And get rid of all the forked articles such as 'reactive centrifugal force' because that effect is already catered for in this article. David Tombe (talk) 16:36, 9 November 2008 (UTC)

David: Your comments indicate a complete misreading of what has been said. Please review the matter. Brews ohare (talk) 16:49, 9 November 2008 (UTC)

David, you ask "How can we get a circular motion with a net centripetal force?" Huh? Isn't that that classical condition for circular motion? Centripetal forces causes a particle's motion to accelerate sideways into a circle. What's more mysterious to me is why even Brews would mention "the polar-coordinate, planetary-motion view (that is, taking the kinematic term to the force side and saying there absolutely is so too centrifugal force in inertial frames)" as if the equations for r represent accelerations in an inertial frame. They don't. To interpret r-double-dot as an acceleration, you have to be in the co-rotating frame. Dicklyon (talk) 16:57, 9 November 2008 (UTC)

Dick: You say: What's more mysterious to me is why even Brews would mention "the polar-coordinate, planetary-motion view. Your comment is perfectly valid in the context of Newtonian vector mechanics, and is the traditional view throughout most of Classical Mechanics. Unfortunately, this view is not universal, and in the niche areas of planetary motion, polar coordinates, and design of robotic manipulators the contrary view is explicitly stated and used, introducing havoc when the two groups try to talk to each other. See Hildebrand, for an example. Brews ohare (talk) 18:19, 9 November 2008 (UTC)

Dick, the planetary orbital equation a few lines above totally contradicts what you have just said. It illustrates that when the second time derivative of the radial distance is zero, as is the case in circular motion, then the centrifugal force must be balanced with the centripetal force. We cannot have a circular motion with a centripetal force alone. I'll write it out again,

${\ddot {r}}=r{\dot {\theta }}^{2}-GM{\frac {1}{r^{2}}}\$

It was very hard for me to clearly make this point before when this equation was being systematically denied. But it is well known to people who have done advanced classical mechanics courses, and thanks to the Kepler article it is now out in the open. It took the Kepler article rather than Goldstein to get it acknowedged. In Goldstein, it appears in an even more general form at either 3.11 or 3.12 with a general centripetal force. It should also be clear by now that Coriolis force appears in the tangential equation and not in the radial equation.

How am I supposed to have an argument with Brews ohare if you keep trying to deny what the argument is about. There are two conflicting positions on centrifugal force. Brews wants to resolve it by having two centrifugal forces. I want to resolve it by playing down rotating frames of reference and totally ignoring cases of non-co-rotation. David Tombe (talk) 21:02, 9 November 2008 (UTC)

No, David, the equation is completely consistent with what I said. The radius r represents position in a co-rotating frame, so r-double-dot is acceleration in such a frame; it is not acceleration in an inertial frame. The Coriolis force has a radial component when the object is not co-rotating wiht the frame. I'm not sure what two conflicing positions on centrifugal force you refer to. As far as I can tell, Brews and I and all the sources see centrifugal force as a fictitious force due to frame rotation, applying to all objects. Only you have the strange idea that there is centrifugal force in inertial frames; certainly your source Goldstein has no such implication. Your misinterpretation of r-double-dot as acceleration is leading into a strange state. Dicklyon (talk) 21:07, 9 November 2008 (UTC)

Dick, there is nothing in Goldstein that links rotating frames of reference to planetary orbits. If you believe otherwise, let's have the quote and the page number. In fact, let's have the quote and page number from any textbook that deals with planetary orbits in rotating frames of reference. And anyway, your argument above about co-rotation is irrelevant. Let's look at it from a co-rotating frame of reference for the sake of argument. If r is to remain fixed, as in circular motion, we still need to balance the centrifugal force with the centripetal force. And let's see you deriving a radial Coriolis force. David Tombe (talk) 21:38, 9 November 2008 (UTC)

I don't know how you can read Goldstein and not see what he is saying in the passge you quote. He gets a equation for r-double-dot. He shows that it is the same of the equation for the acceleration of position in the 1D frame that co-rotates with the planet. He doesn't imply or suggest that it can be an acceleration in any other way. As for deriving a radial Coriolis force, no, I'm not going to try it, but it's been done plenty, and the answer is that in the co-rotating system that component is zero; you only get a radial Coriolis force for objects rotating faster or slower than the rotating frame, as is obvious from the cross product in the Coriolis force equation. Dicklyon (talk) 08:01, 10 November 2008 (UTC)

Dick, Goldstein doesn't mention frames co-rotating with the planet. That is purely in your own imagination. He reduces the radial planetary orbital equation to a one dimensional equation in r by substituting Kepler's areal constant into the centrifugal force term. This makes the centrifugal force term into an inverse cube law repulsive force. He then states that this is equivalent to the fictitious one-dimensional problem. Nowhere does he mention rotating frames of reference. They are not needed, and the textbooks don't use them in planetary orbital theory. No textbooks use rotating frames of reference in planetary orbital theory. In planetary orbital theory, the centrifugal force is a real outward inverse cube law force that keeps the planets up. David Tombe (talk) 11:42, 10 November 2008 (UTC)

And as regards Coriolis force, the polar coordinate derivation leaves the final result such that Coriolis force is unequivocally a tangential force. The 'rotating frames' derivation leaves the final result such that this fact is not overtly specified. But the derivation itself insists that it can only apply to radial velocities, and hence the Coriolis force must only be tangential. You have just looked at the cross product term in the final result of the latter derivation and concluded that there are no restrictions on the direction of v. You have totally ignored the derivation. David Tombe (talk) 11:47, 10 November 2008 (UTC)

More for Tombe

It illustrates that when the second time derivative of the radial distance is zero, as is the case in circular motion, then the centrifugal force must be balanced with the centripetal force. We cannot have a circular motion with a centripetal force alone. I'll write it out again,

${\ddot {r}}=r{\dot {\theta }}^{2}-GM{\frac {1}{r^{2}}}\$

David Tombe (talk) 21:38, 9 November 2008 (UTC)

David: If you call the first term on the right "centrifugal force" your statement is irrefutable. However, there are two schools of thought on using this name for the term on the right. You belong to this school, along with Hildebrand and some others, but it is not the school of Arnol'd, or Taylor, or Gregory, or Landau & Lifshitz. They say that the first term on the right side is part of the mathematical expression for ${\boldsymbol {\ddot {r}}}$, and should not be interpreted as centrifugal force because the equation applies in an inertial frame of reference and an inertial frame by definition has no centrifugal force. BY definition. Not by intuition, Not by argument, Not by observation. By DEFINITION. It is part of the framework of discourse, like deciding that ten rounds is the total number of rounds there will be in a boxing match, or that a pound will be 16 oz. or that electrons will have negative charge. Can you deal with this? They use a different terminology than do you. I repeat: These authors use a different definition. Their definition is not the same as yours. Your definition is different from theirs. Brews ohare (talk) 00:28, 10 November 2008 (UTC)

Brews, that may be right, but it goes deeper than that. One can call it centrifugal force, or not, but that doesn't change what it is. Interpreting it as a force (force over mass in this case), so that r-double-dot is an acceleration, is just equivalent to saying that it's the equation of motion of a 1D system with position r. That's what Goldstein said, and that's the context in which it's the conventional "centrifugal force", a "fictitious force" in a frame co-rotating with the planet. In an inertial frame, it doesn't really matter what you call it, but it's clearly not a force, and r-double-dot is clearly not an acceleration. The only force in the inertial frame is the centripetal term. That's why the planet's path bends to keep constant r. When you say "an inertial frame by definition has no centrifugal force", that's true, by the usual definition of the centifugal force, a fictitious force in a rotating system. But David argues that there is another definition, by which a centrifugal forces does exist in the inertial system. He is wrong there, and that's not simply a matter of definition, right? That is, there's no such force as he says, since that term in the equation is not a force unless r-double-dot is an acceleration, and that means the system is co-rotating with the planet. Do you agree? Dicklyon (talk) 07:56, 10 November 2008 (UTC)

Brews, It doesn't matter if we call the $r{\dot {\theta }}^{2}$ term Alfred. It represents the force which keeps the planets up. It's an inverse cube law repulsive force. No matter how you try to write the radial planetary orbital equation, you will always be stuck with Alfred. If you work in reverse and obtain the radial equation by differentiating a conic section, you will still get the attractive inverse square law of gravity and Alfred. Alfred is an integral part of the planetary show. David Tombe (talk) 11:54, 10 November 2008 (UTC)

Dick says: "It goes deeper than that."

I believe the rest of your discussion amounts to explanation of the standard Classical Mechanics Newtonian viewpoint, and if one adopts that view, there is no issue at all. The role of centrifugal force is limited to rotating frames, just as you describe. However, you seem to feel that David has a different take on the matter, and he does. However, he is not alone, and I have mentioned Hildebrand as an exemplar of a different view in which ${\ddot {r}}$ is an "acceleration", and not ${\ddot {\boldsymbol {r}}}$. A careful reading of Hildebrand suggests that he might agree that the resulting interpretation of $r{\dot {\theta }}^{2}$ should be as a "generalized centrifugal force", in the sense of a Lagrangian formulation in terms of generalized coordinates. He is just being a bit sloppy leaving out the adjective "generalized". I have suggested that there is a group of scholars out there that participate in this approach, and that it conflicts with the Newtonian viewpoint (unless the adjective "generalized" is implied, so a distinction is made between the two different usages for "centrifugal force"). Do you agree? I wonder if you took a look at overview?

That does not mean that these authors would subscribe to all David's intuitions about the role and properties of this $r{\dot {\theta }}^{2}$ centrifugal force. Brews ohare (talk) 16:26, 10 November 2008 (UTC)

David says "It doesn't matter if we call the $r{\dot {\theta }}^{2}$ term Alfred. It represents the force which keeps the planets up."

Calling the term Alfred would have one advantage: if everybody agreed to do that, then everybody would use the same terminology. Of course, the whole notion of inertial frames won't go away, and we'll have to decide whether Alfred is part of their definition or not. I suspect that you personally will have no difficulty with this stage of agreement, because you absolutely do not believe that inertial frames are any different from non-inertial frames. Am I right about this? Does Alfred have anything to do with inertial frames?

The notion that centrifugal force exists "in reality" outside of any choice of frame of reference, is a peculiar hang-up of yours, David, which you appear unwilling to relinquish. It is your own private belief, but from the Newtonian classical mechanics viewpoint (in an inertial frame) circular motion requires an inward force (e.g. gravity) to supply the kinematic centripetal force, and the planets will not fall into the sun without centrifugal force to counteract gravity. All this is explained in the ball on a string example.

There are at least three ways to explain this view of yours:

1. It is based upon your intuitive adoption of a frame of reference attached to the individual planet, in which the planet appears to be in equilibrium only because the centrifugal and gravitational forces balance. This point of view is so natural to you that you are actually unaware that it is only the view from a particular reference frame.
2. You subscribe to the Hildebrand formulation that $r{\dot {\theta }}^{2}$ is "centrifugal force", and it exists in inertial frames.
3. You believe in a personal mixture of the two views that is internally logically inconsistent. Brews ohare (talk) 16:26, 10 November 2008 (UTC)

Brews, my position is that centrifugal force exists on an object when the angular velocity term in $r{\dot {\theta }}^{2}$ is measured relative to the inertial frame of reference. As far as I am concerned, there is no need to bring rotating frames of reference into the matter at all. If there were four objects all moving in the same plane, paired into two pairs of two, with each pair undergoing a closed orbit, I would expect there to be a mutual centrifugal repulsive force acting between any pair within the four, and not just between the ones that are paired into the mutual orbits. There could be no rotating frame of reference that would be suitable for analyzing this situation. Ideally I would expect centrifugal force to cause the two orbital systems to repel each other in the absence of any intervening vortices. Have you ever thought about extrapolating centrifugal force to the four body problem? I would say that centrifugal force doesn't only keep the planets up. I'd say that it stops matter from collapsing. What do you think stops matter from collapsing? It can't be the electrostic repulsive force because that is an inverse square law force and it would fail to produce stability nodes under Earnshaw's Theorem. We need the inverse cube law centrifugal repulsive force for universal stability. And just as you say, this take on centrifugal force is at total variance with the so-called centrifugal force that is said to act on objects that are stationary in the inertial frame, as viewed from a rotating frame. The latter is not even a fictitious effect. It is a bogus effect. And that's where you and I differ. I want to scrap the concept. It just totally clouds the real issue of what centrifugal force is all about and its importance in planetary orbits, electromagnetism, and atomic and molecular stability. David Tombe (talk) 19:05, 10 November 2008 (UTC)

David: You say: Brews, my position is that centrifugal force exists on an object when the angular velocity term in rǿ² is measured relative to the inertial frame of reference. The term rǿ² is non-zero for any particle with other than a purely radial motion. If I observe a particle from two frames that differ only in that their origins are displaced, and otherwise identical, their (r, ø) coordinates will differ and so will their rǿ² forces. That is a bit odd, eh, as normally forces are frame origin independent.

This is a real difficulty: ${\ddot {\boldsymbol {r}}}$ transforms like a vector, but ${\ddot {r}}$ doesn't. In Newton's law ${\boldsymbol {F}}=m{\boldsymbol {\ddot {r}}}$ the right side transforms like a vector, and so do all the real forces on the left side. If you split off only part of ${\boldsymbol {\ddot {r}}}$ like rǿ² and put it on the force side, neither side is a vector any more.

This weird transformation behavior is one problem. Here is another one: ostensibly an inertial frame has only real forces from identifiable physical sources. What physical body originates rǿ²? Are we to imagine some real force field that acts on moving objects with a rǿ² force? Neither rǿ²-fields nor vortexes are part of the standard model, for example. Brews ohare (talk) 20:08, 10 November 2008 (UTC)

Brews, Yes it means that a particle will have a different centrifugal force in relation to every other particle, just as it will have a different gravitational attraction relative to every other particle. That shouldn't be a problem. Your question is very interesting. Is Alfred a fundamental physical interaction or is it a mathematical nicety? Well of course, rotating frames well and truly go out the window when we start looking at multi-particle systems, so we need to consider something more fundamental for the purposes of how we measure the angular velocity such as to make the centrifugal force kick off. Consider another dilemma. When the centrifugal force and the graviational force are balanced in the radial equation, r remains constant. We get circular motion. But in the tangential direction, when the two forces are balanced, as they normally are even in elliptical motion, there is still motion in the tangential direction. In fact, there is still angular acceleration in the tangential direction in elliptical orbits. So we need to ask ourselves if there is in fact something moving radially even in a circular gravitational orbit that makes the situation equivalent to the tangential situation? Have you checked 'the cause of centrifugal force' on google? Planetary orbital theory as in the textbooks is only the tip of the iceberg. But at least it deals with centrifugal force in the absence of rotating frames of reference. That's one of the things that makes it so interesting. It also teaches us that centrifugal force and centripetal force must be balanced in circular motion. That pulls the rug from beneath the absurdities that have been taught in some textbooks in more recent years as a result of over-extrapolation of the rotating frame transformation equations. When you realize that, you will lose all enthusiasm about putting such absurdities into the main article. I meant to add regarding your last question Are we to imagine some real force field that acts on moving objects with a rǿ² force?, that in Maxwell's 1861 paper he has this force essentially listed as a component of the Lorentz force and as a real hydrodynamical force. It is the missing fourth term of the Lorentz force. Can't you see that magnetism involves velocity induced forces? David Tombe (talk) 21:08, 10 November 2008 (UTC)

David: You haven't addressed the incorrect transformation properties of rǿ². Very likely that drawback means that no satisfactory real force field can be constructed with a real source. Brews ohare (talk) 21:32, 10 November 2008 (UTC)

You add: "Well of course, rotating frames well and truly go out the window when we start looking at multi-particle systems, so we need to consider something more fundamental for the purposes of how we measure the angular velocity such as to make the centrifugal force kick off." That is not so clear to me. For if we take any one particle of the group and look at its trajectory, we can sit on a frame attached to that particle and find its instantaneous center of curvature about which it rotates with its instantaneous radius of curvature ρ. In that frame, where the particle is at rest, the instantaneous centrifugal v² / ρ must equal the attractive force drawing the particle toward its center of curvature. Brews ohare (talk) 21:39, 10 November 2008 (UTC)

Brews, You understand the two body problem and you can see that we don't need to involve any rotating frames of reference to complicate the issue, even though we can if we want. But no textbooks do so. I was pointing you to the four body problem simply to highlight the ridiculousness of trying to involve rotating frames of reference. All the mutual centrifugal forces involved will be based on the mutual tangential velocities and the mutual distances between an pair of particles. Consider two magnetic north poles facing each other. The magnetic field lines spread outwards and away from each other. Maxwell had those lines of force as the axes of tiny vortices. The magnetic repulsion was explained by centrifugal force acting between the field lines. Planetary orbital theory is the stepping stone to electromagnetism. Faraday's law is where Kepler's second law breaks down. But you'll never understand any of this if you continue to be blinkered by rotating frames of reference. Now let's get back to the point. You say that there are two centrifugal forces. I say that one of those centrifugal forces is an absurdity. In your demonstration of that absurdity, you draw attention to an illusional circular motion which involves only a centripetal force. Yet you know from looking at the radial equation that a circular motion requires a balance between centripetal force and centrifugal force. You cannot wriggle out of this by playing on words. For circular motion, we need r double dot to be zero. That only happens when the centripetal force is equal and opposite to the centrifugal force, or to Alfred if your not happy using the term centrifugal force. These modern examples that appear in some textbooks are a product of the modern education system which teaches centripetal force in high schools in terms of a vector diagram around a displacement along the arc of a circle. This treatment totally neglects the already existing centrifugal force, as referenced to the same origin, that would result in straight line motion if the centripetal force weren't acting. This neglect has left a whole generation of scientists walking around trying to tell us all that only a centripetal force is involved in circular motion. They say that the centripetal force causes the direction to change and that no centrifugal force is involved. They are wrong because the centripetal force is curving the straight path that would be determined by the centrifugal force acting alone. When the two forces are equal and opposite, we get circular motion. So it's up to you. Do you want to balance the article by introducing centrifugal force as being an effect of rotation? Or do you wish to cloud the whole issue by introducing it as being an illusion that is associated with observing things from a rotating frame of reference?David Tombe (talk) 15:07, 11 November 2008 (UTC)

David: Non-responsive: You haven't addressed the incorrect transformation properties of rǿ². Also, you say Yet you know from looking at the radial equation that a circular motion requires a balance between centripetal force and centrifugal force. This statement is valid only with your definition of rǿ² as centrifugal force. Insisting on this interpretation is not engaging the discussion. Brews ohare (talk) 19:32, 11 November 2008 (UTC)

Brews, You say, This statement is valid only with your definition of rǿ² as centrifugal force. Goldstein calls it centrifugal force because it IS centrifugal force. It has the formula for centrifugal force and it is a radially outward acting force that occurs in conjunction with rotation. You can't get more centrifugal force than that. On your other point, one particle can have many different centrifugal forces acting on it in relation to every other particle in the universe, just as one particle has a gravitational pull from every other particle in the universe. There is no need to complicate the issue with all your considerations above about transformations and observing the effect from different perspectives. There will be a centrifugal force between every pair of particles and r will be the distance between the particles, and the angular velocity will be relative to the background stars. You are strangling your own understanding of this issue by clouding it all up with rotating frames of reference. David Tombe (talk) 02:51, 12 November 2008 (UTC)

Transformation properties

Brews, I don't know what incorrect transformation properties that you are talking about. I never mentioned any transformation properties. I don't even advocate transformations. All I said was that you'd have a hard job analyzing a four body problem in a rotating frame. All the centrifugal forces involved between all the mutual pairs arise from the mutual tangential speeds relative to the inertial frame. As for the issue of balancing centrifugal force and centripetal force in circular motion, it is nothing to do with any definitions of mine. The radial equation for a central force gravity problem is,

${\ddot {r}}=r{\dot {\theta }}^{2}-GM{\frac {1}{r^{2}}}\$

Nobody is disputing this equation any longer. They tried to dispute it in the past. It was deleted from the main article in July as being unsourced and I was accused of disruptive editing for having put it in. But now we have all finally agreed that this equation is in Goldstein and that it is already listed on the Kepler's law page. So I would hope that we won't once again go back to disputing it. This equation means that if r double dot is zero as is the case in circular motion, then the centripetal force and the centrifugal force must be balanced. Playing around with names and terminologies won't change this fact. I've noted how some have tried to wriggle out of this reality by arguing that r double dot is not the radial acceleration. So what? Call it something else then. Call it Peter. But when Peter is zero, then we have circular motion and the centrifugal force will balance with the centripetal force. The application of rotating frame transformation equations to particles that are stationary in the inertial frame utilizes a blatant violation of this fact. It tries to justify a circular motion on the basis of a centripetal force alone. And once you realize the absurdity of doing so, and also the absurdity of having a radial Coriolis force, then you will soon lose your enthusiasm for writing about such absurdities. The textbooks can be wrong. And this is one issue about which some textbooks are demonstratably very wrong. So why make this fringe theory the flagship of the article? David Tombe (talk) 21:18, 11 November 2008 (UTC)

David: The transformation properties I am referring to are the vector properties of physical forces. You may recall that in Newtonian vector mechanics, forces, accelerations, velocities and displacements all are vectors. Among other things, their vector nature implies certain transformation properties upon switching coordinate systems. It doesn't matter if you actually need to switch systems in order to analyze a problem; the transformation properties must hold in principle under switching. The so-called "acceleration" rǿ² in the radial direction does not transform properly. To obtain a properly transforming quantity you need all of ${\ddot {\boldsymbol {r}}}$, in particular, all of its radial component ${\ddot {r}}-r{\ddot {\theta }}^{2}$, not just part of it. Brews ohare (talk) 22:18, 11 November 2008 (UTC)

Here is an illustration: Suppose in frame S the particle moves radially away from the origin at a constant velocity. The force on the particle is zero by Newton's first law. Now we look at the same thing from frame S' , which is the same, but displaced in origin. In S' the particle still is in straight line motion at constant speed, so again the force is zero.

What if we use polar coordinates in the two frames? In frame S the radial motion is constant and there is no angular motion. Hence, the acceleration is:

${\boldsymbol {a}}=\left({\ddot {r}}-r{\dot {\theta }}^{2}\right){\hat {\boldsymbol {r}}}+\left(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}\right){\hat {\boldsymbol {\theta }}}=0\ ,$

and each term individually is zero because ${\dot {\theta }}=0,\ {\ddot {\theta }}=0$ and ${\ddot {r}}=0$. There is no force, including no $r{\dot {\theta }}^{2}$ "force".

In frame S' , however, we have:

${\boldsymbol {a}}'=\left({\ddot {r}}'-r'{\dot {\theta }}'^{2}\right){\hat {\boldsymbol {r}}}'+\left(r'{\ddot {\theta }}'+2{\dot {r}}'{\dot {\theta }}'\right){\hat {\boldsymbol {\theta }}}'\$

In this case the azimuthal term is zero, being the rate of change of angular momentum. To obtain zero acceleration in the radial direction, however, we require:

${\ddot {r}}'=r'{\dot {\theta }}'^{2}\ .$

The right-hand side is non-zero, inasmuch as neither r' nor ${\dot {\theta }}'$ is zero. That is, we cannot obtain zero force if we retain only ${\ddot {r}}$ as the acceleration; we need both terms.

Aha, I imagine you saying, of course we need the centrifugal force $r{\dot {\theta }}^{2}$ to balance the ${\ddot {r}}$ force.

I'd say it is nuts to say a physically real centrifugal force is zero in one frame S, but non-zero in another S' identical, but a few feet away, wouldn't you? After all, if we take $r{\dot {\theta }}^{2}$ as "centrifugal force", it does not have a universal significance: it is unphysical. Even for exactly the same particle behavior the expression $r{\dot {\theta }}^{2}$ is different in every frame of reference, no matter how trivial the distinction between frames. It's zero in S and non-zero in S' . Beyond this problem, the centrifugal force identified with $r{\dot {\theta }}^{2}$ in polar coordinates is zero in Cartesian coordinates. Assuming agreement :-) about the absurdity of this situation, one must say that $r{\dot {\theta }}^{2}$ is not centrifugal force, but simply one of two terms in the acceleration. This last view as two terms in the acceleration is frame-independent: there is zero centrifugal force in any and every inertial frame. It also is coordinate-system independent: we can use Cartesian, polar or any curvilinear system: they all produce zero. Brews ohare (talk) 00:40, 12 November 2008 (UTC)

Brews, You say, This statement is valid only with your definition of rǿ² as centrifugal force. Goldstein calls it centrifugal force because it IS centrifugal force. It has the formula for centrifugal force and it is a radially outward acting force that occurs in conjunction with rotation. You can't get more centrifugal force than that. On your other point, one particle can have many different centrifugal forces acting on it in relation to every other particle in the universe, just as one particle has a gravitational pull from every other particle in the universe. There is no need to complicate the issue with all your considerations above about transformations and observing the effect from different perspectives. There will be a centrifugal force between every pair of particles and r will be the distance between the particles, and the angular velocity will be relative to the background stars. You are strangling your own understanding of this issue by clouding it all up with rotating frames of reference. David Tombe (talk) 02:51, 12 November 2008 (UTC)

Sorry, David. I think you completely misread Goldstein, who brings up the centrifugal force argument only in the "fictitious one dimensional problem". Here's a quote from p. 64, first edition in the subsection 3.3: The equivalent one-dimensional problem.

"It is the same equation as would be obtained for a fictitious one-dimensional problem in which a particle of mass m is subject to a force f + ℓ ² / m r³. (Eq. 3.22) The significance of the additional term is clear if it is written as mrǿ² = mv² / r, which is the familiar centrifugal force."

You also ignore his discussion on p. 135, Section 4.9 which is the same stuff found in all the other authors, leading to:

$F_{Cfgl}=-m{\boldsymbol {\omega \times }}\left({\boldsymbol {\omega \times r}}\right),$

identified by Goldstein following his Eq. (4-107) as

"It will be recognized that this term is the familiar centrifugal force."

In other words, Goldstein has the exact same idea as Arnol'd and Landau & Lifshitz. You wish to dwell on the first quotation, which requires a bit more careful reading because there are three versions of the problem to keep straight (the two-body problem, the equivalent one-body version, and the fictitious One-D version), and to ignore totally the second quotation, which is extremely unambiguous. So we can add Goldstein to the list that includes Arnol'd and Landau & Lifshitz, all of which are very substantial sources with time-honored authority, and none of which support your interpretation.

The example above on radial motion in one frame as viewed from an adjacent frame shows that mrǿ² cannot be called "centrifugal force". It does not behave like the radial component of a force unless coupled with the ${\ddot {r}}$ term. You have simply blown off this example with no direct discussion. That is unfortunate, as it is a definitive counterexample discrediting the mrǿ² = centrifugal force argument, and settling this matter once and for all. Brews ohare (talk) 04:42, 12 November 2008 (UTC)

Brews, I certainly did not misread Goldstein. When Goldstein says "It is the same equation as would be obtained for a fictitious one-dimensional problem in which a particle of mass m is subject to a force f + ℓ ² / m r³. (Eq. 3.22) The significance of the additional term is clear if it is written as mrǿ² = mv² / r, which is the familiar centrifugal force.", how on Earth do you read into it that the equation in question is the fictitious one dimensional problem? The equation in question is the same as would be obtained for a fictitious one-dimensional problem if the centrifugal force term mrǿ² = mv² / r is written as an inverse cube law force in r. There is a serious problem with the comprehension of plain English in this issue. At any rate, this is totally irrelevant. This is a 'play on words' sidetrack. The equation in question is,

${\ddot {r}}=r{\dot {\theta }}^{2}-GM{\frac {1}{r^{2}}}\$

Let's call the left hand side Peter. And let's call the first term on the right hand side Alfred. For circular motion, Peter must equal zero. This means that the centripetal force must be equal and opposite to Alfred. So you can't have a circular motion without Alfred. Yet when you are trying to justify the absurdity of extrapolating the rotating frame transformation equations to particles that are stationary in the inertial frame, as viewed from the rotating frame, you end up trying to justify a circular motion with only a centripetal force and no Alfred. That extrapolation is nonsense on a number of grounds. And if you can't see the other grounds in the derivation, then you have at least seen this one. There is absolutely no counter argument. A circular motion needs a force to counter balance the centripetal force and so the rotating frame transformation equations can only apply to co-rotating situations, and that makes them exactly equivalent to the plane polar coordinate versions. There is only one universal centrifugal force. There is no separate centrifugal force called 'reactive centrifugal force'. The effect in question is the result of centrifugal force acting on object A and object A transmitting the effect to object B. David Tombe (talk) 08:36, 12 November 2008 (UTC)

David, re "how on Earth do you read into it that the equation in question is the fictitious one dimensional problem?", it's because in order to interpret r-double-dot as an acceleration and the other terms as forces, you have to interpret the equation as if it represents a 1D mechanical problem, that is, take r as the one dimension in a system rotating with the planet. That's the only context in which Goldstein refers to the term as a force. It's "fictitious" in that it is not an inertial system; more commonly the force is given the adjective "fictitious", as it's a force that doesn't exist in an inertial system, but rather is induced by viewing the system from a rotating coordinate system. Your old edition of Goldstein is perhaps less clear on this than most sources, but it's still the only possible interpretation. Dicklyon (talk) 16:20, 12 November 2008 (UTC)

Dick, Terminologies are totally irrelevant here. We now all know that the equation,

${\ddot {r}}=r{\dot {\theta }}^{2}-GM{\frac {1}{r^{2}}}\$

holds. You are learning. I'm not interested in terminologies. If there is a circular motion, then the left hand side will be zero. That means that the first term on the right hand side will have to be equal and opposite to the centripetal force. Hence circular motion cannot exist on a centripetal force alone. That is all there is to it. If you want to deny this and carry on applying the rotating frame transformation equations to particles that are stationary in the inertial frame, then that is up to you. But it is total nonsense. David Tombe (talk) 00:06, 13 November 2008 (UTC)

Request for link

David, you say re Goldstein's planetary radius equation "It was deleted from the main article in July as being unsourced and I was accused of disruptive editing for having put it in." I can't find it in July; maybe you got the month wrong. I'd like to understand what you're saying happened, as I'm a relative new-comer here, but I can't find it. Please provide a link to a version that has the equation, so I can examine the history around that time. Dicklyon (talk) 01:52, 12 November 2008 (UTC)

Dick, It doesn't matter now. That's in the past. The point is that we now know that the equation in question is in Goldstein. It also conveniently happens to be on the Kepler's law page.

But since you asked, I added this section to the main article on 29th May,

== The Planetary Orbital Equation ==

-

Centrifugal force can be understood in its most general form in conjunction with planetary orbits. The general equation for any central force motion is a scalar equation and it takes the form,

Applied centripetal force + centrifugal force = m$d^{2}r/dt^{2}$

where r is the radial length. In the case of planetary orbits, the centripetal force is the inward acting force of gravity.

The centrifugal force and the centripetal force that act on the same body are not necessarily balanced, and they should not be considered as an action-reaction pair. When they are balanced, we will have a circular motion, but even then the centrifugal force and the centripetal force should not be considered as an action-reaction pair because this is just a particular situation.

Newton's third law of motion is satisfied across two interacting bodies. For example, in the case of the Earth and the Moon, the centripetal force (gravity) that acts on the Earth is balanced by an equal and opposite centripetal force acting on the Moon. Likewise, the outward centrifugal force acting on the Moon is balanced by an equal and opposite centrifugal force acting on the Earth.

The reaction to a centripetal force across two bodies is sometimes called a reactive centrifugal force. The centripetal force and the reactive centrifugal force are always equal and opposite and they form an action-reaction pair. The reactive centrifugal force is essentially the centripetal force from the perspective of the other body.

In the gravity orbit, in the general case when the centripetal force and the centrifugal force are not balanced, the term $d^{2}r/dt^{2}$ will be non-zero and we will have an elliptical, parabolic, or hyperbolic orbit.

Reading this now, I would perhaps write it slightly differently, leaving out the references to 'reactive centrifugal force'. Neverthless, this section was almost instantly deleted on the grounds that it was unsourced. Soon after that, I was blocked from editing for a month when I called somebody a wikistalker, after they deleted a perfectly good section which I introduced which involved a marble rolling in a radial groove in a rotating turntable.

On 23rd July, I introduced a summary of the above section to the introduction and it was deleted by Wolfkeeper within seconds, before I even had a chance to put the source in. The interchange is all on record.

As you are now aware, the contents are factually correct. The radial equation does indeed appear in Goldstein at 3.11 or 3.12. Any attempts to mention Goldstein at the time were simply brushed aside. There is no need to maintain any pretence that the deletion of that edit was done out of any concern for accuracy. David Tombe (talk) 02:39, 12 November 2008 (UTC)

I see; but not what you mean by "factually correct". And I see why it was removed, which obviously had nothing to do with Goldstein, as Goldstein was not mentioned there. What I wanted to see was whether you had ever actually added your equation from Goldstein, sourced, and if so what you had said about it, and why it was removed. But since you never added it, you're just making noise. What you did add is sort of equivalent, but it was unsourced and surrounded by largely uninterpretable prose, and with bizarre claims like "in its most general form" -- does Goldstein say that? Also, you piece on "marble rolling in a radial groove in a rotating turntable" did appear to be nonsense, with a creatively original interpretation of Coriolis force. If you really want to add your equation, you need to say something sensible about it, like Goldstein does. Instead of "The general equation for any central force motion is a scalar equation" you could explain how to convert to a 1D problem, and explain as Goldstein did that in this equation you can interpret the equation as forces and accelerations. Since the 1D system is rotating when viewed from an inertial frame, it's easy enough to integrate Goldstein's equation and words into the article. Dicklyon (talk) 09:00, 12 November 2008 (UTC)

Dick, I am not interested in your opinion about why it was removed. Your opinion was totally predictable. There is no need to have an inquisition into what happened in the past. What you wrote above was just a side track. Let's stick to the point. We now all know that the equation,

${\ddot {r}}=r{\dot {\theta }}^{2}-GM{\frac {1}{r^{2}}}\$

holds. You are learning. I'm not interested in terminologies. If there is a circular motion, then the left hand side will be zero. That means that the first term on the right hand side will have to be equal and opposite to the centripetal force. Hence circular motion cannot exist on a centripetal force alone. That is all there is to it. If you want to deny this and carry on applying the rotating frame transformation equations to particles that are stationary in the inertial frame, then that is up to you. But it is total nonsense. David Tombe (talk) 00:01, 13 November 2008 (UTC)

Co-rotation

Dick, you have got it all wrong. You are trying to account for a zero effect in terms of the sum of two mutually cancelling illusions, both of which are merely a product of faulty interpretation of mathematics. There is no centrifugal force acting on anything unless there is actual rotation occurring. The rotating bucket of water clearly demonstrates this very basic fact. The angular velocity term in the centrifugal force is for the frame of reference, which essentially means that it is for particles that are co-rotating with the frame, which essentially means that it is for the particles themselves. That is enshrined in the derivation. So at what point does the final result suddenly become liberated to apply to all particles, irrespective of their state of actual rotation? And as regards Coriolis force, the derivation means that the Coriolis force is a tangential force. It is impossible to have a Coriolis force accounting for a radially inward centripetal force. Finally, the central force equation tells us that in order to have a circular motion, the outward centrifugal force must be exactly balanced with the inward centripetal force. If we were to use the Coriolis force for the centripetal force, as you have suggested, it would be twice the size of the centrifugal force and so it would result in a net inward force. A rotating frame of reference is only what its name suggests. It is not a physical entity in its own right. Centrifugal force on an object is already either present or not. A rotating frame of reference merely causes a circular motion artifact to be imposed on the already existing motion. And that artifact cannot be described mathematically with a net inward radial force, never mind a net radial inward force supposedly being supplied by a tangential Coriolis force. You have got it all very badly wrong. David Tombe (talk) 12:43, 4 November 2008 (UTC)

David, I'm just trying to explain what the sources say; if they have it all wrong, we still need to report it the way they do it. But it's not wrong. You are obviously focused on something quite different from the "fictitious force" called centrifugal force when you say of the rotating frame "It is not a physical entity in its own right. Centrifugal force on an object is already either present or not." Quite false; the centrifugal force we are speaking of is "fictitious"; it is NEVER actually present on an object in an inertial frame; it is entirely frame dependent. If you're speaking of something else, it's probably the reactive centrifugal force. Dicklyon (talk) 14:59, 4 November 2008 (UTC)

Dick, I'm not talking about something else. I'm talking about centrifugal force. And there is no longer any need to mention sources or citations because it is clear that you refuse to look at sources and citations that don't suit you. Centrifugal force is the outward radial force in planetary orbital theory. I have directed you to Goldstein's 'Classical Mechanics' where this is written in plain English, but you have somehow managed to take a different meaning out of it. As for this 'reactive centrifugal force' that you keep talking about, anybody with a full comprehension of the subject would understand that there only is one universal centrifugal force. Splitting the subject up is a clear sign of a lack of understanding. David Tombe (talk) 01:31, 5 November 2008 (UTC)

David, I've looked at Goldstein, and it's clear there that the outward centrifugal force is the fictitious force due to the way he rotates the frame to make a one-dimenional problem; just like all the other sources. In the corresponding inertial system, there is no outward force, and if you have a source that says there is, please mention it again; I don't think I could ignore such a thing, but if I've overlooked, I'd sure like to hear about it. Dicklyon (talk) 06:56, 5 November 2008 (UTC)

Dick, The Goldstein doesn't even mention frames of reference in relation to the planetary orbital equation. The equation in question is either 3.11 or 3.12. It is dealt with in plane polar coordinates referenced to the inertial frame. He makes the equation into a single variable equation in r by substituting Kepler's areal constant into the centrifugal force term. The centrifugal force then becomes an inverse cube law force. It is then that he states that it is equivalent to the fictitious one dimensional problem. And so it is. It is a one dimensional radial scalar differential equation in r. I can't see how you deduce that the outward centrifugal force is fictitious. But that doesn't matter anyway. The point here is that centrifugal force exists in the absence of a rotating frame of reference, and the angular velocity in the centrifugal force belongs to the particle. And you continue to ignore the statement that Goldstein made at the top of page 179, which I have quoated at least twice above. David Tombe (talk) 07:22, 5 November 2008 (UTC)

David: you are resorting to exhortation, with neither argument nor sources for support. It seems likely that you agree that the Wiki article is totally correct if one accepts the premise for the vector form of the various fictitious forces, namely:

${\boldsymbol {F}}-m{\frac {d{\boldsymbol {\Omega }}}{dt}}{\boldsymbol {\times r}}-2m{\boldsymbol {\Omega \times }}\left[{\frac {d\mathbf {r} }{dt}}\right]-m{\boldsymbol {\Omega \times }}({\boldsymbol {\Omega \times r}})$ $=m\left[{\frac {d^{2}{\boldsymbol {r}}}{dt^{2}}}\right]\ .$

Thus, to impact acceptance of the article, you must discredit this vector formulation, which you have attempted so far by attacking its interpretation, and suggesting its use is limited to the case of co-rotation. (You may recall Fugal also attempted to discredit it, suggesting that this equation applied only to Cartesian coordinates.) Next, I will critique your suggestion.

The derivation in Fictitious force uses three variables: (i) the vector x_A (t) between the origin of the inertial frame A and the moving object, (ii) the vector x_B (t) between the origin of the rotating frame B and the moving object, and (iii) the vector joining the origins of the two frames X_AB (t), which may move relative to one another. The vector x_B (t) between the origin of the rotating frame B and the moving point is free to take any form, and is not restricted to a co-rotating particle. If you think differently, one way to proceed is to show where in the derivation the co-rotation restriction snuck in, unannounced.

So far you have suggested that this restriction to co-rotation occurs with the introduction of Ω. As you have pointed out, the introduction of Ω occurs in the subsection Rotating systems with the introduction of the time dependence of the unit vectors in the rotating frame:

${\frac {d\mathbf {u} _{j}(t)}{dt}}={\boldsymbol {\Omega }}\times \mathbf {u} _{j}(t)\ ,$

There is no restriction upon x_B (t) introduced here, because the unit vectors apply to the frame, not the particle, and x_B (t) describes the location of the particle wrt the origin of the frame, irrespective of the directions of the frame's unit vectors. The location of the origin of the rotating frame also is unaffected by the frame's rotation: only the directions of the frame's axes change in time. The introduction of Ω above does not affect the frame's origin. Hence, independent of the above equation, the particle motion still is the same arbitrary function of time with the same arbitrary trajectory, but of course the coordinates of x_B (t) in the rotating system change in time, both because the particle moves, and because the axes change orientation in time. In particular, the particle can be chosen to co-rotate or not, as the occasion demands.Brews ohare (talk) 14:45, 4 November 2008 (UTC)

Brews, you have already answered the question for me. The angular velocity in the analysis applies to the frame. This means that it applies to fixed points in the frame and not to moving points in the frame. I can't see how you could interpret the meaning otherwise. David Tombe (talk) 01:26, 5 November 2008 (UTC)

David: You don't get the point here. Yes, the angular velocity belongs to the frame. But x_B (t) = r (t) belongs to the particle, which therefore can have any motion whatever, unrestricted by the selection of rotation by the frame. That means the relation

${\boldsymbol {F}}-m{\frac {d{\boldsymbol {\Omega }}}{dt}}{\boldsymbol {\times r}}-2m{\boldsymbol {\Omega \times }}\left[{\frac {d\mathbf {r} }{dt}}\right]-m{\boldsymbol {\Omega \times }}({\boldsymbol {\Omega \times r}})$ $=m\left[{\frac {d^{2}{\boldsymbol {r}}}{dt^{2}}}\right]\ .$

is unrestricted, and applies regardless of any general motion of the particle, not to only co-rotating particles. Please re-read what I have said above. You have not let it sink in. Brews ohare (talk) 06:28, 5 November 2008 (UTC)

Brews, I haven't overlooked anything. The velocity of the particle relative to the frame is free at the beginning of the derivation, and it doesn't possess any particular angular velocity at that stage. At the end of the derivation, after dr and dθ have gone to zero in the limit, the vector triangle shrinks to an infinitessimal right angle triangle and the particle will have been restricted to co-rotating radial motion. It is only under those conditions that the centrifugal force term and the Coriolis force term can take on their familiar mathematical form, while being restricted to the tangential and the radial directions respectively. David Tombe (talk) 07:32, 5 November 2008 (UTC)

The restriction of Coriolis force to the tangential direction is "familiar" because you are accustomed to Goldstein's treatment, in which the frame rotates to keep a planet a given direction, to reduce it to a 1D problem. The math actually works fine in the much more general situation, with Coriolis force not restricted in direction. Dicklyon (talk) 07:43, 5 November 2008 (UTC)

Dick, Goldstein doesn't mention rotating frames of reference. He deals with the planetary orbital equation in polar coordinates referenced to the inertial frame. The radial planetary orbital equation contains an outward centrifugal force term. The tangential planetary orbital equation equates the sum of the Coriolis force and the angular force (Euler force) to zero, and is hence equivalent to Kepler's second law. There is never a radial Coriolis force under any circumstances in the two body problem. And furthermore, a rotating frame of reference cannot even create the illusion of a Coriolis force because it cannot create a rotating radial motion. David Tombe (talk) 09:21, 5 November 2008 (UTC)

David: You say that a limiting procedure (which appears simply to relate to taking a derivative) in which dr and dθ go to zero introduces a restriction limiting the possible forms of r(t) in the equation:

${\boldsymbol {F}}-m{\frac {d{\boldsymbol {\Omega }}}{dt}}{\boldsymbol {\times r}}-2m{\boldsymbol {\Omega \times }}\left[{\frac {d\mathbf {r} }{dt}}\right]-m{\boldsymbol {\Omega \times }}({\boldsymbol {\Omega \times r}})$ $=m\left[{\frac {d^{2}{\boldsymbol {r}}}{dt^{2}}}\right]\ .$

I fail to see any part of the derivation where a restriction on the form of r(t) arises. Straightforward application of the chain-rule of differentiation is used, and it does not require any "special" treatment of limiting procedures beyond the customary derivative considerations. Explicit introduction of polar coordinates (r, θ) is not necessary at all. The functional form of r(t) is left entirely general, and that is the interpretation in all the references as well. Brews ohare (talk) 15:50, 5 November 2008 (UTC)

Brews, you must go to the root of the derivation, which is a vector triangle. See figure 1 in this web link, [2] The familiar expressions for centrifugal force and Coriolis force only arise when that triangle shrinks to zero in the limit as dr and dθ tend to zero. When that occurs, centrifugal force will be a radial force and Coriolis force will be a tangential force. The derivation involves exactly the same principles of vector calculus as are involved in the derivation of these same terms in polar coordinates. And in the latter derivation, the tangential nature of Coriolis force is overtly recognized in the final expression. It is all one single topic.David Tombe (talk) 02:51, 6 November 2008 (UTC)

This is a re-hash and not responsive. There is no need for polar coordinates, and the only principle involved is the chain rule of differentiation, which imposes no restriction whatsoever upon the functional form of r(t). Brews ohare (talk) 03:50, 6 November 2008 (UTC)

The maths does not stand on its own. It applies to a vector triangle in the limit as dr and dθ tend to zero, in which case the centrifugal force will be radial and the Coriolis force will be tangential. David Tombe (talk) 07:50, 13 November 2008 (UTC)

Radial motion

Here is an illustration showing the futility of using mrǿ² as a "force". Suppose in frame S a particle moves radially away from the origin at a constant velocity. The force on the particle is zero by Newton's first law. Now we look at the same thing from frame S' , which is the same, but displaced in origin. In S' the particle still is in straight line motion at constant speed, so again the force is zero.

Two coordinate systems differing by a displacement of origin. Radial motion with constant velocity v in one frame is not radial in the other frame. Angular rate ${\dot {\theta }}=0$ but ${\dot {\theta }}'\neq 0$

What if we use polar coordinates in the two frames? In frame S the radial motion is constant and there is no angular motion. Hence, the acceleration is:

${\boldsymbol {a}}=\left({\ddot {r}}-r{\dot {\theta }}^{2}\right){\hat {\boldsymbol {r}}}+\left(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}\right){\hat {\boldsymbol {\theta }}}=0\ ,$

and each term individually is zero because ${\dot {\theta }}=0,\ {\ddot {\theta }}=0$ and ${\ddot {r}}=0$. There is no force, including no $r{\dot {\theta }}^{2}$ "force". In frame S' , however, we have:

${\boldsymbol {a}}'=\left({\ddot {r}}'-r'{\dot {\theta }}'^{2}\right){\hat {\boldsymbol {r}}}'+\left(r'{\ddot {\theta }}'+2{\dot {r}}'{\dot {\theta }}'\right){\hat {\boldsymbol {\theta }}}'\$

In this case the azimuthal term is zero, being the rate of change of angular momentum. To obtain zero acceleration in the radial direction, however, we require:

${\ddot {r}}'=r'{\dot {\theta }}'^{2}\ .$

The right-hand side is non-zero, inasmuch as neither r' nor ${\dot {\theta }}'$ is zero. That is, we cannot obtain zero force (zero a') if we retain only ${\ddot {r}}$ as the acceleration; we need both terms.

Aha, I imagine you saying, of course we need the centrifugal force $r{\dot {\theta }}^{2}$ to balance the ${\ddot {r}}$ force. Woops! Now we need a ${\ddot {r}}$ force? Well, put aside the ${\ddot {r}}$ force, and focus on the centrifugal force for a moment.

How can a physically real centrifugal force be zero in one frame S, but non-zero in another S' identical, but a few feet away? Even for exactly the same particle behavior the expression $r{\dot {\theta }}^{2}$ is different in every frame of reference, no matter how trivial the distinction between frames. It's zero in S and non-zero in S' . In short, if we take $r{\dot {\theta }}^{2}$ as "centrifugal force", it does not have a universal significance: it is unphysical.

Beyond this problem, the real impressed net force is zero in Cartesian coordinates. (In a Cartesian system in S' , ${\ddot {x}}'={\ddot {y}}'=0$; there is no real impressed force in straight-line motion at constant speed). If we adopt polar coordinates, and wish to say that $r'{\dot {\theta }}'^{2}$ is "centrifugal force", and reinterpret ${\ddot {r}}'$ as "acceleration" (without dwelling upon justification), we have the oddity that straight-line motion at at constant speed requires a net force in polar coordinates, but not in Cartesian coordinates. This perplexity applies in frame S', but not in frame S.

Assuming agreement :-) about the absurdity of this situation, that a proper formulation of physics is geometry and coordinate-independent, one must say that $r{\dot {\theta }}^{2}$ is not centrifugal force, but simply one of two terms in the acceleration. This last view as two terms in the acceleration is frame-independent: there is zero centrifugal force in any and every inertial frame. It also is coordinate-system independent: we can use Cartesian, polar, or any other curvilinear system: they all produce zero. Brews ohare (talk) 00:40, 12 November 2008 (UTC)

The example above on radial motion in one frame as viewed from an adjacent frame shows that mrǿ² cannot be called "centrifugal force", real or fictitious, as it doesn't behave like the radial component of a force unless coupled with the ${\ddot {r}}$ term. This illustration is a definitive counterexample discrediting the mrǿ² = centrifugal force argument, and settling this matter once and for all. Brews ohare (talk) 04:42, 12 November 2008 (UTC)

How about adding such a discussion to one of the articles; possibilities are mechanics of planar particle motion, polar coordinates or this article. That would serve to show the invariance of the Newtonian approach and the expediency of the Hildebrand approach cooked up for mathematical convenience. Brews ohare (talk) 18:19, 12 November 2008 (UTC)

I put the argument in polar coordinates. Brews ohare (talk) 01:26, 13 November 2008 (UTC)

Is there an article in Wiki somewhere presenting the invariance of physical law under change of coordinate systems, for example, translated or rotated coordinate systems have the same laws? The articles Tensor field and Vector field seem to dwell on the mathematics, not mentioning the physical interpretation. Brews ohare (talk) 19:06, 12 November 2008 (UTC)

An example is General covariance, which mentions the "geometric, coordinate-independent formulation of physics". The article Lorentz covariance explains the math (but not the point), Galilean invariance refers to inertial frame (a bit overly technical, I'd say) Special relativity never makes the point. Any suggestions? Brews ohare (talk) 19:27, 12 November 2008 (UTC)

Brews, Once again, this is not about terminologies. We now all know that the equation,

${\ddot {r}}=r{\dot {\theta }}^{2}-GM{\frac {1}{r^{2}}}\$

holds. I'm not interested in terminologies. If there is a circular motion, then the left hand side will be zero. That means that the first term on the right hand side will have to be equal and opposite to the centripetal force. Hence circular motion cannot exist on a centripetal force alone. That is all there is to it. If you want to deny this and carry on applying the rotating frame transformation equations to particles that are stationary in the inertial frame, then that is up to you. But it is total nonsense. David Tombe (talk) 00:09, 13 November 2008 (UTC)

Apparently arguments that make sense to me can be dismissed by you as "total nonsense" without discussion. The above discussion does not involve "rotating frame transformations" nor "particles stationary in an inertial frame". I have engaged your brand of "total nonsense" with what seems to me to be sensible argument. You don't even read the argument, never mind reciprocate with discussion. But then, that's up to you. Brews ohare (talk) 00:38, 13 November 2008 (UTC)

Brews, Your argument above was unnecessary. The equation,

${\ddot {r}}=r{\dot {\theta }}^{2}-GM{\frac {1}{r^{2}}}\$

contains all that you need to know about centrifugal force for the purposes of an encyclopaedia article. Ever since this debate began in early 2007, there has been a systematic effort to deny that equation. Centrifugal force is a radially outward force that arises in connection with rotation. It's as simple as that. David Tombe (talk) 03:14, 13 November 2008 (UTC)

Centrifugal force is a radially outward force that arises in a rotating reference frame in connection with rotation. It's as simple as that. Brews ohare (talk) 03:49, 13 November 2008 (UTC)

BTW, your equation, whatever its content, does not address my above "unnecessary argument", which refers to a force-free motion. Brews ohare (talk) 03:53, 13 November 2008 (UTC)

Brews, centrifugal force is a radial effect. A rotating frame of reference cannot bring about a radial effect, even as an illusion. Any centrifugal forces as seen from a rotating frame of reference are already in existence to begin with. The rotating frame merely masks the cause, which is the actual rotation. The equation which I have given is the equation which explains what centrifugal force is all about. It uses polar coordinates referenced to the inertial frame of reference. And it is silly to argue about what the first term on the right hand side is. It is centrifugal force. Your argument was merely a decoy to avoid having to face up to the radial planetary orbital equation and the fact that centrifugal force is not something which depends on rotating frames of reference for its existence. David Tombe (talk) 04:54, 13 November 2008 (UTC)

David: Not pertinent to my argument, which does not use a rotating frame at all. It is not a decoy, but a very simple case of straight-line motion, and doesn't even need gravity. If your approach doesn't work here (it doesn't, I'd say), it cannot be expected to work at all. Brews ohare (talk) 06:54, 13 November 2008 (UTC)

Brews, My approach is the standard textbook planetary orbital approach. Who said anything about it not working? I'm trying to show you that centrifugal force exists independently of rotating frames of reference. David Tombe (talk) 07:11, 13 November 2008 (UTC)

Physically real centrifugal force

And Brews, Just in case you think I'm ignoring your argument, I'm not. You make out a case for a physically real centrifugal force having a different value in relation to two different origins. That is not a problem. In reality, a particle will have a different centrifugal force relative to every other particle in the universe and the effects will all add together. I've already tried to explain that to you. It means nothing to consider centrifugal force relative to an imaginary origin, just as it means nothing to consider the gravitational force relative to an imaginary origin. David Tombe (talk) 07:32, 13 November 2008 (UTC)

David: You aren't ignoring the argument, just not responding to it. It's like talking to a politician about raising taxes. He'll somehow make that a question about apple pie and motherhood.

Real forces transform like vectors. Your version of centrifugal "force" does not.

Your statement "In reality, a particle will have a different centrifugal force relative to every other particle in the universe and the effects will all add together." is completely off the wall. It sounds like you are saying bodies originate centrifugal force by virtue of their motion, and, I suppose, at a given point in space all these contributions to the net centrifugal force add vectorially? Where are your citations for this one? Brews ohare (talk) 16:00, 13 November 2008 (UTC)

I'm not sure what "physically real" is intended to mean here. Centrifugal force is a "fictitious force", in that it only comes up in rotating reference frames as a way to "pretend" that it's a normal reference frame. The "r" coordinate is the 1D coordinate of a frame rotating with a planet, which is why r-double-dot is related to a centrifugal force term, even though there's no such force on the planet. Dicklyon (talk) 16:19, 13 November 2008 (UTC)

My meaning of "physically real" is that for the rotating observer the centrifugal force is placed on the force side of Newton's law of motion and treated like a real force. That means it behaves like a vector, adds vectorially to other forces, and should the rotating observer switch origins or orientation of his coordinate system, the centrifugal force transforms like a vector. Brews ohare (talk) 16:22, 13 November 2008 (UTC)

OK, I agree that with that definition, the centrifugal force due to frame rotation is "physically real"; I still think "fictitious" is a more informative descriptor of it, and more common in the literature. Dicklyon (talk) 17:50, 13 November 2008 (UTC)

Hi Dick: Of course your are right about that. I didn't intend to introduce a new descriptor. My objective originally was to discredit the referral to $r{\dot {\theta }}^{2}$ (or ${\boldsymbol {\omega \times }}\left({\boldsymbol {\omega \times r}}\right)$ with ${\boldsymbol {\omega }}={\dot {\theta }}{\boldsymbol {\hat {k}}}$ ) as a "force" because it has incorrect transformation properties when the origin of coordinates is shifted. The value of ${\dot {\theta }}$ changes with the origin of coordinates, so an observer sees a different "force" at every moment if they walk across their rotating reference frame.. The standard force ${\boldsymbol {\Omega \times }}\left({\boldsymbol {\Omega \times r}}\right)$ transforms properly because Ω and r really are vectors. I moved this discussion to Change of origin. Brews ohare (talk) 18:10, 13 November 2008 (UTC)

Archiving about a megabyte

I've archived over 900K of what I claim is largely pointless drivel mostly two people that are treating the wikipedia talk page as a chat page; there's another 100+k that is less than a month old.

Over a megabyte...- (User) Wolfkeeper (Talk) 04:31, 13 November 2008 (UTC)

Deleting more chat here. This is not a forum for discussing centrifugal force, and unreferenced chat is just that. If people want to talk about information they want to add from reliable and verifiable sources, great. That's what this page is for. dougweller (talk) 10:39, 13 November 2008 (UTC)

Dougweller, you obviously haven't read the contents of what you deleted. It was exactly to do with improving the article. It was all about reducing the topic of centrifugal force to one single article. The discussion was about whether there was only one universal centrifugal force or whether there were two or three centrifugal forces. At the moment, there are too many forks. Regarding the note that you left on my talk page, did you leave a similar note to everybody else involved? David Tombe (talk) 10:47, 13 November 2008 (UTC)

This section is about archiving, it is not about forks, how many forces, etc. (although some spaces have made it a subjection, I'll change that). I'll warn them if I think there is a problem. In any case, if you want to argue for 1, 2 or 3 forces, you need sources, not just assertions. Good, mainstream sources. dougweller (talk) 10:57, 13 November 2008 (UTC)

Well I'll be only too glad to wait for these sources to be produced, saying that there is more than one kind of centrifugal force. Can you please meanwhile delete the forks and warn them that they will need good mainstream sources which state that there is more than one kind of centrifugal force? David Tombe (talk) 11:03, 13 November 2008 (UTC)

What bit of 'this section is about archiving, it is not about forks" etc. is giving you a problem? Anything else directly related to the article should be in another section. dougweller (talk) 11:50, 13 November 2008 (UTC)

I thought you were talking about the whole page originally. David Tombe (talk) 12:01, 13 November 2008 (UTC)

Re-unification of the article

Are there any good reliable sources which say that there is more than one kind of centrifugal force? If not, and I don't believe that there are any, I suggest that the forked articles Reactive centrifugal force and 'Centrifugal force in polar coordinates' should be merged into one single article on centrifugal force. David Tombe (talk) 12:04, 13 November 2008 (UTC)

Reactive centrifugal force is just the other end of centripetal force. As your equation shows, it is not the same as centrifugal force, which is what this article describes. I don't know of any other kind; is there another article we need to deal with? Dicklyon (talk) 15:58, 13 November 2008 (UTC)

A quick search gave this reference, Introducing motion in a circle, John Roche, Phys. Educ. 36 No 5 (September 2001) 399-405 [3]

In it Roche states (on page 402):

"I have identified at least three interpretations of centrifugal force in the literature: a valid meaning in physics, an entirely different but equally valid meaning in engineering, and a cluster of false meanings."

Paraphrasing the rest of the section - the first is the fictitious force that arises in rotating reference frames, the second is the reactive centrifugal force (though he calls it an inertial centrifugal force). I'm pretty sure I've also seen this kind of distinction made in a physics dictionary/encyclopedia. --FyzixFighter (talk) 17:03, 13 November 2008 (UTC)

Three citations are provided in the article Reactive centrifugal force. Brews ohare (talk) 17:50, 13 November 2008 (UTC)

Good, we have articles on the physics and engineering centrifugal forces, and the other interpretations are "false" according to all reliable sources we have seen. Goldstein uses a rotating frame when he writes an equation in r, so his comes under the physics interpretation, even if he wasn't explicit enough about saying so that he confused David. Dicklyon (talk) 17:53, 13 November 2008 (UTC)

Dick, you have just said that Goldstein uses a rotating frame, when he doesn't. You have then said that he wasn't explicit. Too right he wasn't explicit. Your statement that Goldstein uses rotating frames is absolutely untrue. Then you have concluded that Goldstein's lack of explicitness has confused me. Dick, you are the one that is totally confused. You haven't adopted a consistent position since you entered this debate. And FyzixFighter has kindly provided us with a physics journal article as evidence of 'reactive centrifugal force' even though it calls it 'inertial centrifugal force'. I suppose he wasn't being explicit either. And because that author claims that there were a cluster of false interpretations in the literature, Dick Lyon takes that author's opinion to trump all those other authors and further concludes that all reliable sources therefore claim that all other sources are false. No wonder there will never be a reasonable article written here. On explicitness, not one of the three citations on the Reactive centrifugal force article are in any way explicit that there is any such distinct concept. David Tombe (talk) 18:53, 13 November 2008 (UTC)

David, please don't twist what I said. I didn't suggest that that author trumps others; rather, that he is consistent with all others that we have examined here, including Goldstein. In Goldstein, the direction along which he measures r is rotating with the planet; I don't know how else it can be interpreted, but if you have other sources that mention an interpretation, we could examine them. Dicklyon (talk) 19:07, 13 November 2008 (UTC)

David: You say "not one of the three citations on the Reactive centrifugal force article are in any way explicit that there is any such distinct concept."

1. Mook says: "What is sometimes called a "centrifugal force" is a reflection of the force you are exerting on the ball to keep it in a circular path. Similarly the Sun will feel such a reactive, centrifugal force from each of the planets that it holds in orbit.."
2. Signell says: "centrifugal force: the reactive force to the centripetal force, to which it is equal and opposite."
3. Mohanty says: "The centripetal acceleration of magnitude V²/r acts radially inward. Note that the reactive centrifugal force on the [control volume] acts outward."

The underlines are mine. I'd say these authors clearly refer to a reactive centrifugal force. Did you actually look at these references, or simply make an off-the-cuff denial? Brews ohare (talk) 19:23, 13 November 2008 (UTC)

David, if you read the journal article, you would see that what Roche calls the inertial centrifugal force is the same thing that others call the reactive centrifugal force. Also, as for the 1D problem, there are several sources that are explicit that this equation describes the motion in a corotating frame - see Professor Tatum's Celestial Mechanics course notes at the University of Victoria (bottom of the 1st page, top of the 2nd page), also Motion in a central-force field, J.S.S. Whiting, Phys. Educ. 18 No 6 (November 1983) 256-257. So while Goldstein might not be explicit in the 1950 edition, others are explicit. --FyzixFighter (talk) 19:57, 13 November 2008 (UTC)

FyzixFighter, I'm going out of this debate now, but you are welcome to e-mail me through the mechanism. It's best discussed behind the scenes. I'll just respond quickly to your comments. In my way of looking at it, the effect in question would simply be centrifugal force. Anyway, in case you don't want to bother with e-mail, you remember the argument about the force between bar magnets? You said it was F =qvXB. The dilemma was, how do we get a potential energy from a Coriolis type term that clearly has no potential energy? Goldstein, as you can see does it using Lagrangian and obtains grad(A.v). But if you check out Maxwell, that is actually centrifugal force and it exists in addition to F =qvXB. You were trying to explain an effect using the wrong term because the right term has been lost. For attraction it is of course Gauss's law which is still one of the Lorentz force terms. Finally, try looking at that radial equation without getting bogged down with terminologies. Then ask yourself 'can a circular motion exist on centripetal force alone?'. I'll not be replying again on this page again for the foreseeable future. David Tombe (talk) 20:11, 13 November 2008 (UTC)

FyzixFighter: I added a link to Tatum's notes in the article. The other two references are not available to the general reader without a license (or $50) so I didn't include them. Brews ohare (talk) 20:27, 13 November 2008 (UTC)

Why does Wikipedia insist on using Ω for angular velocity?

Several articles in Wikipedia use the uppercase omega symbol (Ω) for angular velocity. Every physics book I have ever read uses lowercase omega (ω). Kinematic variables are always written in lowercase; for example: displacement (s not S), velocity (v not V), and acceleration (a not A). The same is true for rotational kinematics: angle (θ not Θ), angular velocity (ω not Ω), and angular acceleration (α not Α).

What is the origin of this anomalous use of Ω? —Preceding unsigned comment added by 74.64.99.52 (talk) 17:30, 16 November 2008 (UTC)

I raised the same question recently. It seems that the lower case is used for angular velocity of particles whereas the upper case is used for the reference frames contained within rotating rigid bodies. From what I can make out, the topic of 'rotating frames of reference' evolved from rigid body rotation, but I couldn't be absolutely sure about that. At any rate, the use of the higher case omega seems to represent an emphasis on the idea that the angular velocity belongs to the frame of reference and not to the particle. This was the issue in question in the recent debate. Personally, I would prefer to use the lower case as I see the angular velocity as belonging to the object. David Tombe (talk) 19:51, 16 November 2008 (UTC)

David's description of use of upper case Ω is accurate. Take a look at the cited references, for example, Taylor, Landau & Lifshitz, and Arnol'd. However, this choice is not universal. Brews ohare (talk) 20:22, 16 November 2008 (UTC)

In my experience the upper case is used for vector angular rotations, where the vector points along the rotation axis and the length is the rotation speed omega. It's done like that to distinguish it from the scalar.- (User) Wolfkeeper (Talk) 20:26, 16 November 2008 (UTC)