Talk:Kepler's laws of planetary motion/Archive 2

Discussion sections
I organized the discussion on the mathematics content in the article and the discussion on the derivations under separate headings and corrected a few level one headings in the Talk page to level two, to keep it easy on the eyes and add some logical structure to the Talk page. It was going too haphazardly IMHO.

So a request to editors, please keep to the relevant sections rather than adding new ones. This keeps the discussion organized and easier to follow. Thanks. Fatka (talk · contribs) 00:24, 10 November 2008 (UTC)

Math content of the article
Please keep the discussion on the mathematics content in the article under this heading. Maybe the structure will help us come to some consensus. Fatka (talk · contribs) 00:24, 10 November 2008 (UTC)

Very long proofs
Do any of these calculations really belong in this article? AFAIK, Wikipedia isn’t supposed to be a math or physics reference. Maybe it goes into just a little too much detail. —Frungi 22:51, 25 February 2006 (UTC)


 * What does AFAIK stand for? -- Metacomet 23:25, 25 February 2006 (UTC)


 * As Far As I Know -- Roachmeister 14:48, 9 March 2006 (UTC)
 * Maybe you should skip the maths if they don't interest you and read on just assuming a correct proof has been given but to me the mathematical proof makes the article much more convincing and I hope it's going to stay. Wikipedia is supposed to be a good source of information but this doesn't mean that as soon as someone isn't interested in some parts of an article, or if there are readers who cannot understand all of it, these bits should be left out. One way to work around this would be to include the maths in a footnote, so they don't distract from the general text but I would certainly understand if contributors to this page consider the maths an integral part of the text and leave them just where they are. Wikiklaas 19:13, 11 March 2006 (UTC)


 * Today I reworked the proof for the second law, to make it much simpler. I agree that the article needs some strong revision, either as a whole, or in its parts. We also need better source citation -- my little edit tripled the number of inline sources in the article, which is No Good. -- Drostie 07:01, 2 July 2007 (UTC)


 * I agree, there does seem to be an excessive amount of math in this article. I'm sure a lot of people just stop reading as soon as they hit equations, which means they won't get past the first law. Perhaps some reorganization would help. -- Beland (talk) 15:15, 11 August 2008 (UTC)

Too much rambling on mathematics
There is way too much unsourced mathematics in this article. Wikipedia is not a text book. Please read Manual of Style for mathematics, especially the section on proofs: "include proofs when they expose or illuminate the concept or idea; don't include them when they serve only to establish the correctness of result.". The latter seems to be the case for most of the math in this article. I have the impression that various derivations have been added more or less as primary research, at least it is not clear which references were used for which derivation. Han-Kwang (t) 20:53, 19 October 2008 (UTC)


 * Can you be specific? Which parts do you have in mind?  I haven't checked through the mathematical details.  My initial impression is that mathematical derivations of Kepler's laws from Newton's would "expose or illuminate the concept".  Are those the parts you had in mind, or some other?  (It would be astonishing if derivations of Kepler's laws from Newton's were "primary research", since one would expect that Isaac Newton did all that.) Michael Hardy (talk) 21:37, 19 October 2008 (UTC)


 * The proofs are rather elegant, not at all long, and placed in the end, where they do not harm... Do they illustrate an important idea? In some sense, they do. Namely, - the epoch-making idea of Newton, that Kepler's laws (as nearly everything) follow from dynamics (from Newton's laws). Looking at the proofs one can get some impression, what does it mean to derive them, what kind of math is used, and how hard it is. (I do not mean: look, it is terribly hard. Rather, it takes not so many lines. But the lines are ingenious.)


 * In a sense, it is maybe the most historic mathematical proof, at all! Could it deserve a number of lines here?


 * About original research... surely the credit is due to Newton, and a number of others, not to the editor. Maybe the editor did some renaming and simple algebraic rearrangements; such activity is acceptable to Wikipedia, isn't it? Boris Tsirelson (talk) 21:50, 19 October 2008 (UTC)


 * I see, Michael Hardy thinks the same, but was the first to express it; I got an edit conflict... Boris Tsirelson (talk) 21:53, 19 October 2008 (UTC)


 * One example where I emphatically agree with Han-Kwang is that in the first sections, there's a lot more detail than there needs to be. In the first law, the "mathematics of the ellipse" section should be wholesale moved to later in the article. In the second law, the procedure to "calculate the position, (r,ν), of the planet" should be wholesale moved to later in the article. (The third law is probably OK as is.) You don't need to know anything about perihelions, semi-latus rectums, mean anomolies, etc. to understand all three of Kepler's laws at a basic level. --Steve (talk) 22:17, 19 October 2008 (UTC)
 * Re Michael Hardy:
 * (1) I'm referring to the sections "Derivation of the second/first/third law", and to a section that I just removed.
 * (2) Yes, of course Newton did those derivations, but I doubt that Newton used the notation as is presented here. There are many ways to derive the laws, in Cartesian or vector notation, and I feel that the particular approaches in this article should at the very least be attributed to specific sources rather than being presented as derivations from first principles.
 * (3) Our opinions on whether the derivations "expose or illuminate" the concept seem to differ. I am more a physicist than a mathematician and usually I prefer to understand qualitatively why things work the way they do, which is not the case when there is a elaborate mathematical derivation that I can see is correct but does not improve my understanding. For some reason, this article is classified under mathematics rather than physics, which I guess means the standards of mathematicians are applied. But at least WP:MOSMATH recommends against extensive use of "we see that" and so on. I would think a proof here should guide through the main steps of the proof without going into the algebraic details about what to divide by what and so on. For example in the case of the 2nd law, I would reduce the proof to something like "In the approximation of a heavy sun where the acceleration of the sun can be neglected, the equations of motion in terms of the distance vector r and the velocity vector v is (equation). Substituting into Newton's law F=m(r..) and F=GMm/r^2 leads to the differential equation (...) with general solution (...)."
 * Han-Kwang (t) 22:18, 19 October 2008 (UTC)

By the way, a nice compromise might be to use show/hide boxes. See for example here or here for examples of other articles where I've put these. --Steve (talk) 22:23, 19 October 2008 (UTC)


 * In general I don't think most proof deserve to be in wikipedia. The typical wikipedia criteia of verifiability and notability need to be applied. Its often the theorem rather than the proof whic is important. Some proof are however notable and important in their own right. See for example some of those in Category:Proofs, Wikipedia talk:WikiProject Mathematics/Proofs has some discussion on the issue.
 * For these proof my feeling is that they fall into the set of notable proofs. Indeed it is one of the most famous proof about. Of course the proofs need some clean up work and they should really be referenced, ideally with a modern source, rather than newtons original.
 * I'm not a fan of show/hide boxes in articles, it goes against the dominant style on the internet where clicking on a link takes you to a new page, so goes against the Principle of least astonishment guideline. --Salix (talk): 23:19, 19 October 2008 (UTC)

I don't see why Wikipedia should not be a textbook as well. The WP:NOTTEXTBOOK page mentions wikibooks etc. but wikibooks has completely failed. Now, I do see that the algebra on this page is a bit messy, there are too many simple algebraic steps in the derivations. But I don't think we should dumb down wikipedia by omitting derivations from first principles.

Wikipedia is actually a more suitable medium to provide for derivations from first principles than textbooks. A textbook assumes that readers are at a certain level. A textbook in which Kepler's law are derived will be typically written for first year university students studying classical mechanics. But you can derive Kepler's law using just elementary high school algebra. So, while the textbook may be too difficult to work through for some readers, the information on this wiki page could be far easier to understand.

Also, many readers do not have access to textbooks at all, so they would only read about the derivation if it can be found on some webpage. Count Iblis (talk) 00:13, 20 October 2008 (UTC)

If wikipedia is to provide a complete encyclopedia, giving out mathematical proofs is very important. Bunch of people who go through this article would be students who want everything explained well, rather than just referring to the standard laws which you can find in any textbooks. If someone think that this is making wikipedia horribly mathematical and scientific, it would be good if you can try keeping them in hide boxes rather than deleting it. I am not gonna paste the derivation back from history, but if someone thinks that it is a neccessity to make wikipedia complete, they should do it themselves. —Preceding unsigned comment added by 122.167.50.106 (talk) 02:39, 20 October 2008 (UTC)
 * Thanks to Steve (Sbyrnes321) for making the intro of the article more accessible. Actually, I did more or less the same thing a year ago, but there are forces here that seem to prefer rigorous mathematical descriptions above accessibility, which is why it's good that we are having this discussion. (By the way, for some reason this particular talk page doesn't show up on my watchlist, anyone else having the same problem?) Han-Kwang (t) 06:12, 20 October 2008 (UTC)


 * No, I have no such problem. Boris Tsirelson (talk) 07:06, 20 October 2008 (UTC)

I do not understand why the use of show/hide boxes is not being considered seriously. We should realise that Kepler's Laws were proposed as empirical laws which was later found to be consistent with Newtonian mechanics. A qualitative analysis in terms of potential plots, with the mathematical detail provided at the end or using show/hide boxes will make the article more accessible. Although many of the readers consult wikipedia as a reference, we should take into account readers who read just out of curiosity. Fatka (talk · contribs) 19:21, 5 November 2008 (UTC)

Template message math2english added
Hello, I have added a template message to this article tagging it as containing too many technical formulae. As per the talk section above Talk:Kepler's_laws_of_planetary_motion the article needs to be made more accessible to the layman. See also a discussion at Wikipedia_talk:WikiProject_Physics which suggests the use of hidden would help bring some clarity, the conversion of principles derived to prose and the removal of some of the unnecessary mathematical derivations. Jdrewitt (talk) 08:56, 8 November 2008 (UTC)
 * Thank you! I think that the laws themselves are accessible as it is. Do you agree? The technical derivation of Kepler's laws from Newton's laws are inaccessible to the mathematically illiterate, but nevertheless not unnecessary, as they are highlights in the history of thought. I do appreciate your interest in the article. Improvements can be made. Bo Jacoby (talk) 09:24, 8 November 2008 (UTC).
 * The laws are not as accessible as they could be. See this article http://csep10.phys.utk.edu/astr161/lect/history/kepler.html for an example of an explanation that is accessible to the layman.  Beyond this though, the article contains far too much derivation - it is not necessary.  It would be much clearer and accessible to everybody to simply state the key principles required for derivation and then show the final result.  Wikipedia isn't an exercise/text book and as such does not need exhaustive derivation.  Where editors feel this really is unavoidable, then use of hidden would be prudent such that the lay reader can actually follow the derivation by seeing only the main points.Jdrewitt (talk) 12:00, 8 November 2008 (UTC)

The article can be substantially reduced in size, and centred around a few key points, mainly written in prose.

(1) Kepler's laws were deduced by Kepler based on the observations of Tycho Brahe. Then state the laws.

(2) State the equivalence between Kepler's first law and Newton's law of gravitation in that the radial equation solves to an ellipse, parabola or hyperbola.

(3) State that Kepler's second law can be used to prove conservation of angular momentum. As a corollary, the fact that Newton's law of gravity is a radial only 'central force', can be used to prove Kepler's second law.

The only maths that needs to enter the article would be the radial equation and the tangential equation along with a proof of Kepler's second law, asuming conservation of angular momentum. David Tombe (talk) 11:44, 8 November 2008 (UTC)


 * The laws are clearly presented. If the reader does not wish to go through Kepler's derivation, they can skip it without loss of content. However, it is of interest to see how Kepler did it, and for those with an historical bent this part of the article is both interesting and hard to find elsewhere.
 * The derivation from Newton's laws also is historically useful and helpful in understanding the connection to Kepler's laws. Again, the reader that wishes to simply skip the math and focus on the results is not impeded by this math.
 * I really don't share the antipathy of many editors for a little math. If they want to get worked up, they should look at some (arguably most) of the math articles that are totally unintelligible to anyone but an expert in math. Brews ohare (talk) 12:44, 8 November 2008 (UTC)


 * Having articles that are totally unintelligible to anyone but a mathematician is not a good thing and not something that a good article should ever even vaguely resemble. Readers shouldn't have to skip content, they should be able to read and understand the article as it stands in full.  The article should therefore be dedicated to prose and not mathematical derivation.  As has been stated a few times before now but which is not being discussed is the possiblity of using hidden to hide the exhaustive derivations to make the entire more reader friendly for everyone.  Jdrewitt (talk) 13:30, 8 November 2008 (UTC)


 * Roger on unintelligible to anyone but a mathematician. However, I don't see any objection to a reader skipping content. I do it myself all the time. It isn't even a matter of intelligibility, just one of interest. It's no hardship, and after all, the article caters to a diverse audience, possibly including K-12, historians, and, yes, math buffs. One may skip because one doesn't understand the math, or because one just doesn't care about the math. Or, oddly enough, one may read only the math, figuring the rest is just blah-blah. (Math was invented as succinct and precise expression.) No need for the article to cater only to those with a penchant for verbiage. Brews ohare (talk) 18:27, 8 November 2008 (UTC)


 * This is an English encyclopedia, therefore the article should be written in English, not math. You still haven't addressed the issue of the hidden suggestion which is actually a compromise allowing die hard editors to keep their derivations whilst making the article readable to anybody who isn't up to the mathematics.  The formulae need to be summarised by prose to bring it up to standard.  The issues with this article have been raised numerous times by numerous different editors.  Its about time they were actually dealt with. Jdrewitt (talk) 20:12, 8 November 2008 (UTC)


 * "This is an English encyclopedia, therefore the article should be written in English" Can we use ergo or à la or magnum opus? How about ej π or π r 2? Care to write a verbal description of π, of j of exp? How about 1 + 1 or $$\sqrt [3]2$$? Maybe English of one syllable and sentences without subordinate clauses? The point is there is no obvious arbitrary line; we two cannot agree where the line is drawn; the reader has to draw their own line. It suffices that the article has reasonable content at a variety of levels of sophistication. Presently it is organized so the demands on the reader tend to increase the further you read, like a newspaper article. Brews ohare (talk) 20:22, 8 November 2008 (UTC)


 * This isn't about we two, it is about the opinion of numerous other editors who have expressed the same view that article contains too many derivations that are innaccessible to the layman. We have offered a compromise- in the spirit of collaboration are you even willing to consider it? Jdrewitt (talk) 20:50, 8 November 2008 (UTC)

You might respond to this suggestion: It suffices that the article has reasonable content at a variety of levels of sophistication. Presently it is organized so the demands on the reader tend to increase the further you read, like a newspaper article. Is this not a valid objective for the article and description of the article?


 * Articles in Wikipedia should be accessible to the widest possible audience. It is possible to make the entirety of this article more accessible, the reason why you are unwilling to do so, or to compromise or collaborate is beyond my comprehension. Jdrewitt (talk) 23:05, 8 November 2008 (UTC)

All I have to add, as has been proposed many times, is the use of hidden for the derivations. Even published research articles have the detailed calculations listed as appendix. Wikipedia being an online resource can do better, hence the proposal for the use of hidden. Please look at stress to see a very effective implementation of the template. Fatka (talk · contribs) 05:26, 9 November 2008 (UTC)

Split up article
Maybe you'd like to split the article into a disambiguation page Kepler's laws of planetary motion referring to Kepler's laws of planetary motion (Introduction)  and   Kepler's laws of planetary motion (Derivations)? Take a look. Brews ohare (talk) 21:07, 8 November 2008 (UTC)


 * I don't think we need a new article. I think that between Kepler problem and Kepler orbit, there's already a great place to move the derivations from both section 2 and section 3. The derivations are pretty much already in those articles anyway. :-) --Steve (talk) 22:20, 8 November 2008 (UTC)


 * Hmm. Maybe Kepler problem and Kepler orbit should be moved instead? They are pretty short. Brews ohare (talk) 22:36, 8 November 2008 (UTC)


 * I have placed a speedy deletion on each of these articles since they are nothing more than test edits and constitute content forking. Jdrewitt (talk) 22:36, 8 November 2008 (UTC)


 * No contest on speedy deletion, which would happen automatically anyhow. However, it is not content forking, as Wiki policy:
 * Summary style articles, with sub-articles giving greater detail, are not content forking, provided that all the sub-articles, and the summary conform to Neutral Point of View.
 * allows articles on the same subject of different depth if properly linked to other articles. An example is Special relativity, Introduction to special relativity, History of special relativity. Brews ohare (talk) 22:41, 8 November 2008 (UTC)
 * You have already contested the speedy deletion using hangon. Why did you create the articles in the first place? Your reckless actions are a waste of wikipedia resources and editors time.  This is not the way to go about doing things and certainly not good for collaboration.  But then its evident you don't give a toss about collaboration anyway. This is your article after all. Jdrewitt (talk) 23:01, 8 November 2008 (UTC)
 * FYI: Forking can be unintentional or intentional. POV forks usually arise when contributors disagree about the content of an article or other page. Instead of resolving that disagreement by consensus, another version of the article (or another article on the same subject) is created to be developed according to a particular point of view. Jdrewitt (talk) 23:23, 8 November 2008 (UTC)


 * I understand your comments on possible origins for new pages, but here we aren't really arguing POV but depth of treatment. I'd say the analogy with the articles Special relativity, Introduction to special relativity, History of special relativity is pretty close. It's just a way to cater to different levels of interest in the topic. Brews ohare (talk) 03:37, 9 November 2008 (UTC)

It may be frustrating or inspiring to read something that one does not understand, but the cure is not to ban or to hide incomprehensible texts. The only judge for deciding if a text is incomprehensible is the reader. (Hopefully the text is comprehensible to the author). If you don't want to read the article, then simply don't read it. I'd like to ask Jdrewitt: what do you want from this article? Is there a particular formula that you'd like to have explained? The first section, introduction to the three laws may be sufficient for you. Should we put a warning before section 2: STOP READING unless you are interested in the mathematics of Kepler's laws? There is already a warning: So the derivations below involve the art of solving differential equation? Bo Jacoby (talk) 22:43, 8 November 2008 (UTC).


 * Exactly my opinion too. Splitting the article up is just a means of letting reader decide what fits their interest in a more formal manner, like deciding which title to take off a bookshelf. However, I prefer not to split unless stampeded by Jdrewitt. Brews ohare (talk) 22:46, 8 November 2008 (UTC)


 * I have made it blatantly clear. There is too much emphasis in this article on mathematical derivation and not enough on prose. Jdrewitt (talk) 22:57, 8 November 2008 (UTC)


 * Expanding on this, no I don't think there should be a warning as you suggest. I am simply asking if you would consider making the article in its entirety more accessible to the layman.  The whole thing.  Not just the beginning.  I made a suggestion (which incidently has been suggested a number of times both here and at a discussion about this article at the Physics wikiproject discussion page) about using hidden.  I am simply responding to a request for help with this article but didn't expect to start world war 3 simply by suggesting that this article can be improved and that the derivations contained within this article are inaccessible to many readers.  Jdrewitt (talk) 23:20, 8 November 2008 (UTC)


 * All that is blatantly clear is your opinion. That is not a discussion, but an edict. If you wish to contribute here, propose some eloquent language to replace or supplement the math you find so distasteful. Anything less is too vague to matter. Brews ohare (talk) 23:02, 8 November 2008 (UTC)


 * So what you're saying is my opinion doesn't count? Thanks. Jdrewitt (talk) 23:39, 8 November 2008 (UTC)


 * No, just that it isn't only your opinion that counts, and your opinion would count more if it led to specific suggestions. So far, I'd have to guess (based upon limited information) that you would delete the material placed in the trial page Kepler's laws of planetary motion (Derivations). If that is so, or nearly so, perhaps the two pages aren't such a bad idea. They could be tweaked, but leave the simpler one simple and perhaps add some wording you would like to the other? Brews ohare (talk) 03:40, 9 November 2008 (UTC)


 * There have been numerous suggestions on this talk page that have been systematically ignored in favour of a minority (but determined) POV. I have suggested a number of times that the long derivations should be summarised by prose, and using hidden to make the article more accessible to all. And this is not only my opinion.  It is expressed by a number of other editors above.  Jdrewitt (talk) 08:57, 9 November 2008 (UTC)

Jdrewitt, I would agree with you that it would be sufficient to merely state the equivalence between Kepler's first law and $$\ddot r - r\dot\theta^2 = -GMr^{-2}$$ on the one hand, and Kepler's second law and $$r\ddot\theta + 2\dot r\dot\theta = 0$$ on the other hand, and how these two equations together constitute Newton's law of gravitation. Having said that, is there any reason why the full mathematical proofs can't follow in special sections further down the page? The former proof is quite complicated no matter in which direction we do it, and I'd agree with you that it should not be allowed to interfer with the coherence of the article. Hiding is off course another option. David Tombe (talk) 15:33, 9 November 2008 (UTC)

Sbyrnes's split-up / clean-up proposal

 * All the math in Section 2 ("Position as a function of time") is taken out of this page and merged with the presentation which is already written out on either of the pages Kepler problem or Kepler orbit (I'm not sure which would be better). Section 2 would just say something like "Kepler's laws allow one to calculate the position of a planet as a function of time. It involves a few parameters (Mean anomaly, True anomaly, etc.), and the numerical solution of an equation called "Kepler's equation". See the article [article name]."
 * All the math in Section 3.1 (Newtonian equations of motion) and Section 3.2 (Newton's laws imply Kepler's first law) is likewise merged onto Kepler problem or Kepler orbit, and replaced by a few qualitative sentences and a link to the appropriate page
 * Sections 3.3 (Newton's laws imply Kepler's second law) is rewritten to say "this law follows from the fact that all central forces conserve momentum", with the mathematical details put in a show/hide box.
 * Section 3.4 (Newton's laws imply Kepler's third law) is split into two parts. The more important part is that two orbits of the same shape (eccentricity) but different sizes circular orbits would have their periods in the correct proportion. This follows directly from a simple and intuitive scaling argument which we could put in the article. (This is in Landau and Lifshitz, for example.) It probably wouldn't even need a show/hide box. The other half is that orbits with different eccentricity but the same major axis have the same period. For this part, which is more of a detail, we'd send readers to Kepler problem or Kepler orbit.

OK, there's my proposal. Comments? I'm hoping this can start a productive conversation, and maybe people here can stop bickering. :-) --Steve (talk) 07:04, 9 November 2008 (UTC)


 * Thank you for constructive proposals. Regrettably I fail to understand that a split-up of the article solves the problem. Those who dislike formulas in this article also dislike formulas in other articles.


 * 1) I don't mind proposal 4 on explaining a simple special case of the third law, although I don't know why this special case is 'more important'. We already have circular orbits as a very special case.
 * 2) "the fact that all central forces conserve angular momentum" must of course be documented by a link. Do the readers really need to know about angular momentum in order to understand Kepler's laws? Historically it was the other way round, that Kepler's second law was used in order to understand angular momentum.
 * 3) In the article Kepler's laws are derived by integration of Newton's differential equations. Alternatively it could be shown by differentiation that Newton's differential equations are satisfied by the Kepler motions. Kepler's second law proves that the acceleration is towards the sun. Combined with Kepler's first law it shows that the acceleration is inversely proportional to the square of the distance.
 * 4) The formula $$ \left({\frac{P}{2\pi}}\right)^2 = {a^3 \over G (M+m)} $$ might be moved or removed because it is at variance with Kepler's third law which it was supposed to illustrate.
 * 5) The article already has elaborate prose for explaining Newton's and Kepler's laws.
 * Bo Jacoby (talk) 08:14, 9 November 2008 (UTC).


 * Steve, I'd tend to go along with that proposal as a starting point. It would provide a nucleus which could be worked on on an ongoing basis and gradually made more coherent, with an emphasis on prose, but with some key mathematical terms retained.
 * The main problem with the existing article is not one of content dispute. It would seem that everybody is satisfied that the facts are correct. The problem is one of incoherence and too much mathematics.
 * I will once again state the key points,
 * (1) Kepler's laws were obtained from observations made by Tycho Brahe. They should be stated very early on in a listed fashion.
 * (2) Kepler's first law is an elliptical solution to the radial equation (The equation that contains Newton's gravity + centrifugal force, but which also has parabolic and hyperbolic solutions).
 * (3) Kepler's second law is equivalent to the tangential equation in which the Coriolis force and the Euler force sum to zero, which is the equivalent to conservation of angular momentum.
 * (4) Kepler's 3rd law could also be elaborated on in relation to Newton's law of gravitation.
 * That's all that is required. David Tombe (talk) 08:27, 9 November 2008 (UTC)


 * Bo, you say:"The formula $$ \left({\frac{P}{2\pi}}\right)^2 = {a^3 \over G (M+m)} $$ might be moved or removed because it is at variance with Kepler's third law which it was supposed to illustrate." How does this formula contradict the Third Law which states only the proportionality: $$ {P^2} \propto {a^3} $$? The context is to provide the constant of proportionality, which it does. Brews ohare (talk) 17:08, 9 November 2008 (UTC)
 * I see, the problem is the planetary mass: I added a few remarks about this after the formula, and modified the statement of the third law to point out this feature. Brews ohare (talk) 17:24, 9 November 2008 (UTC)


 * Bo, I agree, the circular case is better. I changed that above. It's "more important" in the sense that planets in the solar system have reasonably circular orbits, so you can get pretty far in your solar-system-understanding by just establishing the orbital time relationship for the circular case.
 * It's not unusual that detailed mathematical derivations are unhelpful in Article A but helpful in Article B, in particular when Article A, like this one, has a large readership who isn't interested in those derivations, and Article B is specifically written for more technically-knowledgeable readership, and both Article A and Article B make it very clear to the reader that the other one has thus-and-such content if that's what the reader is looking for.
 * Obviously the angular momentum thing would be explained in the article--how and why a Kepler-second-law relationship is exactly equivalent to conservation of angular momentum, and why Newton's laws imply conservation of angular momentum for a central force, and that gravity is a central force. :-) --Steve (talk) 18:20, 9 November 2008 (UTC)

Further comments on Steve's proposal
Bo says: "Regrettably I fail to understand that a split-up of the article solves the problem. Those who dislike formulas in this article also dislike formulas in other articles." This statement is irrefutable. All that can be done is to remove more technical portions from the introductory article and put the rest on its own. Already such a proposal has failed to interest the editors that hate math. My attitude is: chacun ses goûts. A split is the best compromise, and the "superficial" among us will not have to look at any math - they can settle for the kindergarten version.

*All the math in Section 2 ("Position as a function of time") is taken out of this page and merged with the presentation which is already written out on either of the pages Kepler problem or Kepler orbit  In my view the article Kepler problem at present is an nonentity. Moving this material there is tantamount to installing a new page, and the identity of this page will become that of the new material. The merit of this move is that (i) no more pages are created, and (ii) existing links to Kepler problem can be retained.

*All the math in Section 3.1 (Newtonian equations of motion) and Section 3.2 (Newton's laws imply Kepler's first law) is likewise merged  Same holds true here.

*Section 3.4 (Newton's laws imply Kepler's third law) is split into two parts. I have no problem with this.

I believe that Steve's proposal has merit, and that the material should be moved to Kepler's problem with suitable cross-referencing in the present article. Brews ohare (talk) 17:58, 9 November 2008 (UTC)

Changes made
I created this sub-section so that editors can document the changes being made as per the discussions above (assuming some conclusion is reached ).

I included these to the introduction to the three laws.


 * 1) Introduced the ideas more descriptively with examples
 * 2) Some minor restructuring of the wording and formatting
 * 3) Inclusion of some internal links to ideas like Copernican principle

Fatka (talk · contribs) 00:24, 10 November 2008 (UTC)

Discussion on derivations
Please keep the discussion on the mathematical intricacies of the derivations in the article to this section. This will facilitate a more structured and constructive discussion. Fatka (talk · contribs) 00:24, 10 November 2008 (UTC)

Third law proof
Can it be simplified? -- Beland (talk) 15:15, 11 August 2008 (UTC)

Alternate derivation was from fundamental physics laws
And derivations already included in the article is more comlicated than the alternate derivation. The alternate derivation was derived from fundamentals of physics, and basic coordinate geometry rather than using differntial equation. This should also be taken into consideration. —Preceding unsigned comment added by Nradam (talk • contribs) 02:43, 20 October 2008 (UTC)

Better second law proof
Here's a much quicker proof of the second law. I didn't make it up, but don't recall where I saw it first.

Let $$\mathbf{r}(t)$$ be the vector from the sun to the planet at time t. The rate that the area is traced out is the magnitude of the vector

$$\mathbf{w}(t) \equiv \frac{1}{2}\mathbf{r}(t)\times\dot{\mathbf{r}}(t)$$

(This fact could be explained in detail in a line or two, plus a link to cross product.)

The derivative of this quantity, by a vector calculus identity that we can link to, is

$$\frac{d}{dt}(\frac{1}{2}\mathbf{r}(t)\times\dot{\mathbf{r}}(t)) = \frac{1}{2}(\dot{\mathbf{r}}(t)\times\dot{\mathbf{r}}(t)+\mathbf{r}(t)\times\ddot{\mathbf{r}}(t))$$

The first term vanishes. The second term vanishes because Newton's laws tell us the two vectors are parallel. Therefore, w(t) is constant in time, which proves Kepler's law. By the way, this is essentially the same proof as the proof that central forces conserve angular momentum (w is half the angular momentum).

Obviously in the article I would explain each step in more detail, but I still think it would come out substantially shorter than the derivation in the article right now. Any objections? --Steve (talk) 06:19, 20 October 2008 (UTC)
 * That's what I would call an elegant and illuminating proof. Too bad that there isnt't a source, but it is certainly an improvement above what's already there, which also lacks references. Han-Kwang (t) 06:27, 20 October 2008 (UTC)

Steve, I'm concerned about how you got the second term to vanish. Newton's second law doesn't come into it. If you expand the r(double dot) term, will the expression always be zero? I don't think so. David Tombe (talk) 18:26, 30 October 2008 (UTC)


 * This topic links Kepler's law of areal velocity to the two tangential components of force in polar coordinates, and eliminates them from central force orbital problems. The two tangential components in question are the Coriolis force and the angular acceleration (that has been referred to as the Euler Force in some wiki articles). This has great significance for those that have been erroneously trying to involve a radial Coriolis force in circular motion problems.
 * The approach that I am familiar with is that r^2.theta(dot) is the areal velocity and it is a constant based on observation. Hence, if we differentiate that expression with respect to time, we get zero. Hence 2rr(dot)theta(dot) + r^2theta(double dot) is equal to zero. Divide by r and we can conclude that the sum of the Coriolis force and the angular acceleration is equal to zero. Hence conservation of angular momentum. David Tombe (talk) 17:56, 30 October 2008 (UTC)

Tautology in the main article regarding proving Kepler's second law
The so-called proof of Kepler's second law in the main article is a tautology. It begins by equating the tangential components of acceleration to zero. This will only be so if Kepler's second law already holds in the first place. In fact, in planetary orbital theory, the ability to eliminate the tangential terms is a result that follows from Kepler's second law. Kepler's second law was deduced experimentally from the observations of Tycho Brahe. The law cannot be proved theoretically in isolation. We need to first assume conservation of angular momentum. David Tombe (talk) 17:51, 31 October 2008 (UTC)


 * The laws of Kepler were found experimentally. The laws of Newton were formulated later. The justifications for the laws of Newton included the fact that they reproduce the laws of Kepler. Newton's law says that the force between sun and planet is in the direction of the line between sun and planet. The consequence of that is Kepler's second law. It is not stated that Kepler's law is proved, only that it is derived from Newton's laws. Bo Jacoby (talk) 23:09, 31 October 2008 (UTC).

Bo Jacoby, Yes indeed. If we assume Newton's law of gravitation, then with it being a central force law, we will have conservation of angular momentum, and Kepler's law of areal velocity will follow on.

I think that somebody needs to make a clear distinction between Newton's law of gravitation on the one hand and 'Newton's laws' on the other hand, in the introduction to that section. The latter are normally associated with Newton's three laws of motion. They are not relevant to Kepler's laws.

If the introduction were to make a clear statment on what Newton's law of gravitation is, and emphasize that it is a central force in which angular momentum is conserved and in which tangential components of acceleration vanish, then the proof of Kepler's second law below would fall nicely into place in the context. David Tombe (talk) 11:05, 1 November 2008 (UTC)


 * David Tombe, I think you have got what you asked for in the subsection Kepler's laws of planetary motion: Newton's law of gravitation says that "every object in the universe attracts every other object along a line of the centers of the objects, proportional to each object's mass, and inversely proportional to the square of the distance between the objects," and his second law of motion says that "the mass times the acceleration is equal to the force." The concept of angular momentum was unknown to Kepler and to Newton as far as I know. The tangential component of acceleration does not vanish as the speed of the planet varies from minimum at aphelion to maximum at perihelion. Bo Jacoby (talk) 09:04, 2 November 2008 (UTC).

Bo Jacoby, There are two parts to the tangential component of acceleration in a Keplerian orbit. In general, neither of them vanishes individually. But they both always sum to zero. Hence, the tangential component of acceleration vanishes in a Keplerian orbit. There will be a Coriolis acceleration and an angular acceleration (mutually cancelling) at any moment in time in a Keplerian orbit. (The latter has been referred to as the Euler force in some wiki articles)

On the other issue, yes the facts are all there. But it needs to be written more clearly. At the moment, it is all clouded up with Newton's laws of motion. All you need to say is that Kepler's second law follows from the fact that Newton's law of gravitation is a central force. Then do your mathematical proof in which you equate the tangential component to zero. Newton's three laws of motion don't come into this at all. David Tombe (talk) 15:01, 3 November 2008 (UTC)
 * Hi again David. The first section of your entry above is to me confusing and incorrect. The acceleration of a planet in elliptical orbit does have a tangential component, changing the numerical value of the velocity. As to the second section, please try to write the facts more clearly here in the talk page, but note that the first and third laws also have to be demonstrated. Bo Jacoby (talk) 21:51, 3 November 2008 (UTC).

Bo Jacoby, Yes there is a tangential acceleration that changes the magnitude of the velocity in an elliptical orbit, just as you say. But there is always an equal and opposite Coriolis force acting at the same time. The net tangential acceleration is zero. That is what Kepler's law of areal velocity is all about. On the other point, try to remember that this is an encyclopaedia designed to explain things to people that don't understand. Hence we keep it simple and to the point. The only thing of relevance about Newton's law of gravitation, as regards Kepler's law of areal velocity, is the fact that it is a 'radial only' central force. We do not need to discuss any of Newton's three laws of motion in the context. David Tombe (talk) 02:14, 4 November 2008 (UTC)


 * David, the Coriolis force is not relevant here since it is a force that is only observed for an observer sitting on the planet in an accelerating frame of reference. It is like saying that the centrifugal and centripetal forces cancel out in a circular motion. Han-Kwang (t) 07:44, 4 November 2008 (UTC)

Hankwang, The Coriolis force is relevant. I am surprised that you say otherwise. The entire proof in question totally depends on it being relevant. It's relevance is in the fact that it always cancels with the angular acceleration leaving a net zero tangential acceleration. This then leaves us with a 'radial only' central force equation for planetary orbits. The two radial forces are the inward gravitational force which is the centripetal force, and the outward centrifugal force. In situations where these two radial forces are always balanced, we have the special case of circular motion. In fact, I have just looked and I see that Bo Jacoby has written the very radial equation in question below. David Tombe (talk) 12:56, 4 November 2008 (UTC)


 * The word 'tangential' can mean either with respect to the orbit, or with respect to the polar coordinate system, so perhaps the text can be improved in order to avoid this confusion. Your point is already made in the article: In order to derive Kepler's second law only the tangential acceleration equation is needed. In the equation for tangential acceleration, $$r\ddot\nu + 2\dot r\dot\nu = 0,$$ the first term involves the angular acceleration $$\ddot\nu$$ and the second term is the coriolis acceleration, as is said in the article, but the sum, zero, is not called the 'net tangential acceleration'. Kepler's first law and third law also depend on the differential equation for radial acceleration $$\ddot r - r\dot\nu^2 = -GMr^{-2}.$$ Bo Jacoby (talk) 08:01, 4 November 2008 (UTC).

Bo Jacoby, Yes, the maths was already in the article and I wasn't criticizing the maths. The maths was correct and the statements that went with it confirmed what I have been trying to tell you. There is no net tangential acceleration in a Keplerian orbit. The angular acceleration (Euler acceleration) and the Coriolis force are always mutually cancelling. All I was saying was that you need to simplify the wording of the lead in to this proof. You need to get to the point, which is that Newton's law of gravitation is a central force, and therefore we can equate the tangential acceleration to zero. David Tombe (talk) 12:53, 4 November 2008 (UTC)
 * Hi David. I tried to clarify. I hope it is an improvement. However I am not happy about using the words tangential acceleration because the meaning in the article tangential acceleration is different. Bo Jacoby (talk) 21:23, 4 November 2008 (UTC).

Bo, We have got alot of useful information in that section which relates to an ongoing argument on the talk page at centrifugal force. And thanks goes to Brews ohare for bringing it back into a more readable format with the θ (theta) symbol for angular displacement. Unfortunately, across the two articles I am detecting alot of confusion amongst the editors. This article clearly lays out the two important differential equations that apply in planetary orbital theory. The tangential equation links Kepler's second law to conservation of angular momentum. This conveniently gets Coriolis force out of the analysis, but that is not to say that the Coriolis force is not involved. This seems to have caused alot of confusion, with Hankwang stating that he believes Coriolis force not to be involved. It is not involved in the radial equation which is the bread and butter of the analyis of planetary orbits, but it is nevetheless involved in a discrete kind of way. Regarding the radial equation, this seems to be unknown over on the centrifugal force article, and I have been trying in vain to get it installed centre place in that article. It is the most important of all examples of the centrifugal force. Centrifugal force is a radial force that is involved in the radial equation. Coriolis force is a tangential force that is involved in the tangential equation. Yet we have Hankwang denying the Coriolis force where it really does exist, and over on the centrifugal force article, we have Dick Lyon promoting the Coriolis force in the radial direction where it definitely doesn't exist. This all needs to be sorted out. At the moment, your article here is factually correct but it can still be simplified. There is no need to involve Newton's laws of motion while discussing Kepler's laws of planetary motion. And I don't know what your problem has been with my use of the term tangential. It is exactly as in the article. The two tangential terms are the Coriolis force and the Angular force (Euler Force). David Tombe (talk) 01:21, 5 November 2008 (UTC)
 * To David: you are writing here on the talk page that there is a coriolis force acting on the planet. Quoting Coriolis effect: "The Coriolis force is an example of a fictitious force (or pseudo force), because it does not appear when the motion is expressed in an inertial frame of reference, in which the motion of an object is explained by the real impressed forces, together with inertia." Han-Kwang (t) 09:04, 5 November 2008 (UTC)

Hankwang, You can see for yourself, from the maths in the main article, that Kepler's second law means that the sum of the Coriolis force and the angular force is always zero in a planetary orbit. In the general case, ellipse, parabola, or hyperbola, you can see that the planet will have an angular acceleration. And as the radial distance is changing, the radial direction is also changing. This latter effect is a manifestation of the Coriolis force, which is always mutually cancelling with the angular acceleration, and hence leading to a net zero tangential acceleration, and hence to conservation of angular momentum. The wikipedia article on Coriolis force, which you have quoted, is very wrong and I have stated my views on the matter on the talk page of that article. A rotating frame of reference cannot produce a Coriolis force either real or imaginary. A Coriolis force is a tangential force which causes the direction of a radial motion to change. It occurs as above, or when externally applied, or hydrodynamically as a result of motion through a vortex. David Tombe (talk) 09:34, 5 November 2008 (UTC)

History
(refactoring earlier requests)


 * Where were these laws first promulgated? That is to say, in which of his works did Kepler publish these laws? --Iustinus 08:32, 11 Mar 2005 (UTC)
 * How are these laws "based on the foundation left by Copernicus"?
 * This topic is of great interest to historians of science; an account of how these laws were discovered should be added. The math presented in the article needs to be put in the intellectual context in which it was developed. -- Beland (talk) 15:15, 11 August 2008 (UTC)

Animate?
Idea--someone should create an animated gif showing Keplerian motion for say eccentricity 3/4. I could do it but my hands are full with other matters.---CH (talk) 05:28, 1 September 2005 (UTC)

Undefined
In the paragraph entitled "Third Law", undefined "s" and "m" appear. Small "m" is defined later in the same paragraph. —Preceding unsigned comment added by 86.160.202.155 (talk) 10:38, 2 September 2008 (UTC)
 * "s" and "m" means second and meter. It should not be in italic. Bo Jacoby (talk) 12:11, 2 September 2008 (UTC).

Limitations to Kepler's laws
It may be useful to have a subsection mentioning special cases and limiting cases of Kepler's laws, and differences between Kepler's and later theories. Kepler's first law puts the sun in one focus, while Newton's laws put the centre of mass of sun and planet in that focus. Kepler's third law claims that the period depend only on the great semiaxis, while Newton's solution includes a tiny dependence on the mass of the planet. Kepler's first law says that the orbit is elliptical while Newton's solutions include parabolic, hyperbolic, and straight line orbits. Kepler's laws are not obeyed when perturbations between planets are taken into account. Even in a two-body situation where the planet mass is neglectable compared to that of the sun, there is the relativistic effect of perihelion drift which Kepler's laws do not account for. The animation of two equal bodies does not illustrate Kepler's laws, so it belongs elsewhere. Bo Jacoby (talk) 21:02, 12 October 2008 (UTC).

new subsections
Two new edits concerning alternate derivations have appeared. I don't find them neither necessary nor sufficient, but confusing and superfluous. 'Kepler's third law assuming circular orbit' is not Kepler's third law. The period is called P in the article rather than T. They need to be discussed here. Bo Jacoby (talk) 16:47, 19 October 2008 (UTC).

I dunno what you meant by P. The period is T. But i didnt know how to write pi (3.14) in wikipedia, and maybe thats what you saw as P. The derivation is done with an assumption of circular orbit, which is very easy to derive and still gives the required proportionality. And if the new derivation of first law looked wrong, i have added up the article with more proof. And that derivation is perfect. I am a student learning IIT, and had gone through gravitaion from numerous books and had taken part in many lectures and i know what i am doing.

It would be good if someone can upload a picture of ellipse for my proof. Else the viewers wont have much clarity in the derivation.

Peace. —Preceding unsigned comment added by Nradam (talk • contribs) 19:14, 19 October 2008 (UTC)

Average distance from the sun.
An editor has used the words average distance from the sun meaning the semi major axis. The semi major axis is the average between the maximum and the minimum distances from the sun, but it is not the average distance from the sun, as the planet spends more time far away from the sun than close to the sun. Bo Jacoby (talk) 20:38, 23 October 2008 (UTC).

Symbol for Angular Displacement
Somebody has used the Greek symbol ν for angular displacement. This might easily be confused for v and velocity. It is more normal to use the Greek symbol θ for angular displacement. David Tombe (talk) 20:59, 29 October 2008 (UTC)
 * The symbol for true anomaly was changed from theta to nu in July on request from some other editor. You are welcome to change it back to theta, but please, do it all the way. Bo Jacoby (talk) 00:14, 30 October 2008 (UTC).

That would be a nightmare task. Maybe some day if I have the time. David Tombe (talk) 10:56, 30 October 2008 (UTC)


 * Done. Brews ohare (talk) 17:38, 4 November 2008 (UTC)

Fine! Are you also able to modify the figure ?
 * [[Image:Diagram Anomalies Kepler orbit.svg]]

I don't know how to do it. First the label &nu; must be changed to &theta;. Then it would be nice to have a label, say d, on the point of intersection between line cs and line xp, because this point is part of the missing explanation that
 * $$ a\cos E=a\varepsilon+r\cos \theta,$$

namely that
 * $$\overrightarrow{cd}=\overrightarrow{cs}+\overrightarrow{sd}. $$

Bo Jacoby (talk) 20:48, 4 November 2008 (UTC).

Connection with Coriolis effect article
In fact, Coriolis force as it is treated in this article is correct, and the fact that it is in total contradiction with Coriolis force as per the Coriolis force article is sufficient grounds upon which to investigate the existing Coriolis force article. Because if it is to be held that both articles are correct despite this gross contradiction, then clearly the right hand doesn't know what the left hand is doing. David Tombe (talk) 09:34, 5 November 2008 (UTC)


 * As described in the Wiki article Centrifugal force (rotating reference frame), 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 \ .$$
 * If the first term is called the "Euler force" (per unit mass) and the second the "Coriolis force", this terminology is in common use in discussion of polar coordinates. For example, see Shankar. The angular rate $$\dot {\theta}$$ is the angular rate of the particle in the inertial frame of reference. Obviously, in general this angular rate is non-zero in the inertial frame.


 * 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 &Omega; of a frame of observation. This approach leads to the result for the acceleration a in an inertial frame:
 * $$\boldsymbol{a} = \ \frac{d^2}{dt^2}\boldsymbol{r}$$
 * $$ = \  \left[ \frac{d^2 \boldsymbol{r}}{dt^2} \right] \ + \  \frac{d \boldsymbol{\Omega}}{dt}\boldsymbol{ \times r} \  + \  2 \boldsymbol{\Omega \times} \left[ \frac{d \boldsymbol{r}}{dt} \right] \  + \   \boldsymbol{\Omega \times} ( \boldsymbol{\Omega \times  r})\, $$
 * where square brackets […] denote evaluation in the frame rotating at angular rate &Omega;. This formula is well documented, and is derived in Arnol'd, in Taylor, in Gregory, and in Landau & Lifshitz. Evidently, all the added terms are zero when the angular rate &Omega; = 0, that is, in an inertial frame.
 * where square brackets […] denote evaluation in the frame rotating at angular rate &Omega;. This formula is well documented, and is derived in Arnol'd, in Taylor, in Gregory, and in Landau & Lifshitz. Evidently, all the added terms are zero when the angular rate &Omega; = 0, that is, in an inertial frame.


 * One argues at cross-purposes here by refusing to realize that multiple terminologies are in use: David and this article on Kepler's laws are using the "polar-coordinate" terminology, while the Coriolis effect 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 &Omega; of a non-inertial frame. Brews ohare (talk) 18:53, 5 November 2008 (UTC)

The coordinate system basis vectors $$(\hat{\mathbf{r}}, \hat{\boldsymbol\theta})$$ are not constant, so the coordinate system is not an inertial frame. That's why the coriolis acceleration appears. Bo Jacoby (talk) 08:03, 6 November 2008 (UTC).


 * Bo, the polar coordinate system is referenced to an inertial frame of reference. That much has been agreed between myself and Brews.
 * The argument is whether the Coriolis force and centrifugal force in polar coordinates are the same Coriolis force and centrifugal force that arise in the rotating frame transformation equations.
 * I say that they are the same for co-rotating situations, but that in non-co-rotating situations, we must use the actual angular velocity of the objects in question.
 * Brews says that they are different concepts. He bases this on the fact that in the rotating frame transformation equations, we can even have a centrifugal force acting on objects that are stationary in the inertial frame.
 * Basically the argument comes down to the fact that I have been saying that the correct presentation of centrifugal force and Coriolis force is as it occurs in planetary orbital theory, just as described in this article. And as a corollary, I am disputing the validity of applying the rotating frame transformation equations to objects that are not co-rotating.
 * The edit war on the centrifugal force talk page has been characterized by a total denial of all textbook sources which treat centrifugal force as per this article on Kepler's laws, and an insistence that the article be dominated by centrifugal force is it occurs in some textbooks in relation to rotating frames of reference, with a particular emphasis on non-rotating objects as viewed from the rotating frame. The introduction and title on the centrifugal force page imply that centrifugal force is only ever an illusion in connection with rotating frames of reference. The planetary orbital 'radial' equation which you have supplied us with in this article, and which appears in Goldstein's 'Classical Mechanics' proves this idea to be false. Centrifugal force is not something which is restricted to rotating frames of reference. The edit war is over who owns the angular velocity. I say that the object owns it. Brews says that the frame owns it. I say that it doesn't matter in cases of co-rotation. But it does matter in non-co-rotating situations. I say that the object always owns ω. Brews says that the frame always owns ω. David Tombe (talk) 12:43, 6 November 2008 (UTC)

Rotating frames
Now, take care. The coriolis acceleration and the centrifugal acceleration are related to rotating frames of reference. It is not related to whether cartesian or polar coordinates are used. But a particle moving in a non-rotating frame and having polar coordinates $$(r,\theta)\,$$ defines a rotating cartesian frame $$(\hat{\mathbf{r}}, \hat{\boldsymbol\theta}).$$ This may cause confusion. The equations of motion of a particle obeying Newton's first law in a cartesian inertial frame are $$\ddot x=\ddot y=0. $$ The corresponding equations of motion in a polar inertial frame are not $$\ddot r=r\ddot \theta=0 $$ but $$ \ddot r - r\dot\theta^2=r\ddot\theta + 2\dot r \dot\theta = 0 .$$ This should be no surprise, but it is, and the question arises: What are the forces causing the nonzero 'accelerations' $$\ddot r $$ and $$r\ddot \theta $$ ? The stupid answer to this stupid question is, the centrifugal force $$ mr\dot\theta^2$$ and the coriolis force $$-2m\dot r \dot\theta .$$ These 'forces' compensate for your using wrong formulas for the acceleration. The acceleration of a particle on earth is not $$(\ddot x,\ddot y) $$, when $$(x,y) \,$$ is measured in a frame fixed to this spinning planet. The 'explanation' for the deviation is: the fictituous forces. I hope this helps. Good luck. Bo Jacoby (talk) 14:31, 6 November 2008 (UTC).


 * Bo: I do not agree entirely with your perspective upon the the equations of motion in polar coordinates. You suggest that use of polar coordinates implies an associated rotating frame of reference. I'd like to explore that idea further.
 * In particular, the (purely kinematic; i.e. no reference to impressed forces) equation for acceleration in an inertial frame in polar coordinates is (see this section in Planar motion):


 * $$\boldsymbol{a} = \frac{d\boldsymbol{v}}{dt} =\frac{d^2\mathbf{r}}{dt^2} = (\ddot r - r\dot\theta^2)\hat{\boldsymbol{r}} + (r\ddot\theta + 2\dot r \dot\theta)\hat{\boldsymbol\theta} \, $$


 * This equation is just math, applying the chain rule of differentiation. In this equation the angular rate $$\dot \theta$$ is that of the observed particle as seen from the inertial frame, and is not related to frame-rotation.
 * The corresponding equation in a rotating frame rotating at constant angular rate &Omega; referring to a particle with coordinates (r, θ') is (see this section in Planar motion):



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

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


 * If &Omega;=0 (no rotation), this equation is the same as the first one for the inertial frame. If &Omega; ≠ 0, the last set of terms in Ω are the Newtonian fictitious forces, called centrifugal and Coriolis forces, all of which vanish in an inertial frame.
 * That is, the "extra" terms in the inertial frame are not impressed forces, and are not due to rotation of some "implied, associated" frame, but are artifacts of mathematics in polar coordinates. Needless to say, if you choose spherical-polar coordinates, these "extra" terms are different, even though the same physical problem is observed, and the same actual forces are at play. I doubt that you would advise introducing "cylindrical-coordinate centrifugal force", "spherical-coordinate centrifugal force", "oblate-spheroidal centrifugal force" and so forth.
 * One could choose to call the extra terms "fictitious forces", what is in a name? They do have similar form to the Newtonian fictitious forces in this particular coordinate system. In fact, in discussions of polar coordinates, this is exactly what some authors do. But that choice of terminology does not agree with the Newtonian picture of fictitious forces, which by definition, always are zero in an inertial frame of reference.
 * Unfortunately, we have two conflicting uses for the same terms. Brews ohare (talk) 17:31, 6 November 2008 (UTC)

Your distinction between "State-of-motion" versus "coordinate" fictitious forces in your article on Mechanics of planar particle motion seems to be original research. There is no reference and it does not look like mainstream mechanics. Right? Fictituous forces appear in curvilinear and/or accelerated frames as deviations from the accelerations as computed in rectilinear non-accelerated frames. I am not familiar with your distinction. Bo Jacoby (talk) 23:19, 6 November 2008 (UTC).


 * Needless to say, the designation "Original Research" raises hackles. However, I don't think it is applicable here. The terminology I have used is mine, but the facts described are not my invention. Although there is a tendency to confound terms in the acceleration that appear in curvilinear coordinates with terms that arise because of changes in the observer's state of motion, the two are distinct and have different properties.


 * As an entry to this subject, Hildebrand is a clear exemplar of the approach using $$\ddot r$$ as an acceleration. That approach shows up directly in a Lagrangian formulation of the problem of a particle in a central force field when the "generalized coordinates" are selected as (r, &theta;). If instead that problem is attacked via Newtonian vector mechanics, however, the acceleration is, of course, $$\boldsymbol{\ddot r}$$, which is a different matter entirely than $$\ddot r$$.


 * Naturally, the formulation in Newtonian vector mechanics can be translated into any coordinate system. For example, we might find in vector notation in an inertial frame:


 * $$\frac{d^2}{dt^2} \boldsymbol{r} = -GM\frac{1}{r^2} \hat{\boldsymbol{r}} \ . $$


 * We could solve this problem in any coordinate system using this same inertial frame. Whatever system we choose, we would find one solution is circular motion with the centripetal force provided by the gravitational field. If we choose in particular to use a polar coordinate system, with the particle at position (r, &theta;), we introduce (as you know) unit vectors $$ \boldsymbol{\hat r} \, \boldsymbol{\hat{\theta}} $$, and set $$\boldsymbol{r} = r\boldsymbol{\hat r}$$. Using the chain rule of differentiation and the result that applies for either unit vector:


 * $$\frac{d}{dt}\boldsymbol{\hat u } = \boldsymbol{\omega \times \boldsymbol{\hat u }}\ ,$$
 * with


 * $$\boldsymbol{\omega} = \boldsymbol{ \hat k }\ \frac {d}{dt} \theta \ ,$$


 * we recoup the standard results for the radial and azimuthal directions. In particular:


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


 * where the right side is clearly the radial component of the specific force in Newton's law (as we started out in vector notation above), so the left must be the radial acceleration. (Actually, we know that the left side is the acceleration anyway, because it was derived from $$\ddot {\boldsymbol{r}}$$ by simple substitution and the chain rule.) We now can play games, follow Hildebrand, and take the the "extra" term to the right and call it a "centrifugal force". However, that wording clearly is a departure from a Newtonian terminology, which in the inertial frame says that the only forces are real forces. In different words, by choosing to say this transfer of terms leads to a centrifugal force in an inertial frame, we fly in the face of the Newtonian definition of an inertial frame.


 * A possible escape is to say the Lagrangian approach leads to an "effective fictional One-D problem" (à la Goldstein), and this formulation is not in an inertial frame. Rather, it could be viewed as a solution in the co-rotating frame. (In the co-rotating frame, the particle is immobile, in a condition of equilibrium in which the centrifugal force balances the centripetal force provided by gravity. Reference to a centrifugal force is OK in the co-rotating frame because it is not an inertial frame.)


 * Do you agree with this presentation so far? If not, where do you balk? Brews ohare (talk) 00:42, 7 November 2008 (UTC)

Hi Brews ohare. There is no disagreement on the physics, as far as I can see, but bad terminology leads to difficulties in understanding. I am puzzled by your use of the word Newtonian. Alas I am not an expert on Isaac Newton's writings myself, and few people are, so referring to Newtonian terminology is not helpful to me nor to other readers. What is a Newtonian fictitious force? I don't know the meaning of Newtonian vector mechanics, as Newton knew not about vectors. So, some of your writing I do not understand. Your writing indicate to me that you are eager to explain, but fail to communicate. The reader who knows not about Coriolis force, probably knows not about Lagrangian equations either. I consider 'fictitious forces' to be bad terminology because a fictitious force is not a force. The simplest example is a one-dimensional motion. The coordinate is $$x\,$$, the velocity is $$\dot x\,$$ and the acceleration is  $$\ddot x\,$$. Now introduce a non-linear coordinate transformation such as $$x={y^2\over 2}\,$$. (Assume that x and y are positive). The new coordinate is $$y\,$$, the velocity is $$y\dot y\,$$, and the acceleration is  $$y\ddot y+\dot y^2\,$$. If the velocity is zero then $$\dot y\,$$ is zero too. But even if the acceleration is zero, $$\ddot y\,$$ is not zero. So if you erroneously want the acceleration to be proportional to $$\ddot y\,$$ you are tempted to introduce a fictitious acceleration $$-\dot y^2\,$$ caused by a fictitious force $$-m\dot y^2\,$$ where $$m\,$$ is the mass of the particle. But this is bad physics, and so it is pointless to classify fictitious forces further into "State-of-motion" and "coordinate" fictitious forces. Bo Jacoby (talk) 10:26, 7 November 2008 (UTC).


 * Bo, You said to Brews that "Fictituous forces appear in curvilinear and/or accelerated frames as deviations from the accelerations as computed in rectilinear non-accelerated frames. I am not familiar with your distinction".
 * You are now echoing the username 'Fugal' who has unfortunately dropped out of the debate.
 * There is nevertheless a distinction, but Brews and I are in disagreement about how to rationalize with the distinction. The distinction is that in curvilinear coordinates, the centrifugal force and Coriolis force always apply to objects which own the angular velocity ω in question. However, in rotating frames of reference, the transformation equations are applied to particles whether they own the angular velocity ω or not, with the angular velocity ω being firmly linked to the frame. The latter liberal application of the transformation equations to objects that are stationary in the inertial frame unfortunately does appear in examples in some more modern textbooks, and that is the cause of the entire dispute.
 * Brews rationalizes with this dilemma by deciding that there are two different kinds of centrifugal force, and two different kinds of Coriolis force. Hence, the centrifugal force article has been forked into multi-articles.
 * I rationalize with the distinction on the basis that the rotating frame transformation equations are not supposed to be applied to objects that don't co-rotate with the rotating frame. In other words, as far as I am concerned, centrifugal force is one single topic.
 * The entire body of knowledge surrounding both this article and the centrifugal force article, is contained in these two equations,
 * $$\ddot r - r\dot\theta^2 = -GMr^{-2},$$
 * and
 * $$r\ddot\theta + 2\dot r\dot\theta = 0.$$
 * We have a radial equation and a tangential equation.
 * The radial equation solves to give ellipses, parabolae, and hyperbolae, This is a generalization of Kepler's first law.
 * The tangential equation gives us Kepler's second law (areal velocity).
 * This is the only basis within which the centrifugal force and the Coriolis force should be understood.
 * These two equations generalize to,
 * $$-\frac{\mbox{d} \mathbf{A}} {\mbox{d} t} = - \nabla \phi - \frac { \partial \mathbf{A} } { \partial t } + \mathbf{v} \times \mathbf{B} + \nabla(\mathbf{A} \cdot \mathbf{v})\,$$
 * where,
 * $$\mathbf{B} = \nabla \times \mathbf{A}.$$
 * is the vorticity.
 * The $$\nabla(\mathbf{A} \cdot \mathbf{v})\,$$ term above appears as the centrifugal force term in equation (5) in Maxwell's 1861 paper. It was used to account for magnetic repulsion which is not catered for in the modern Lorentz force. Centrifugal force is the missing fourth term of the Lorentz Force.
 * The Coriolis force and the so-called Euler force both overtly and actively arise in the magnetic field, but never overtly in the gravitational field. They do however both arise in the gravitational field as a subtle mutually cancelling pair. That is the significance of Kepler's second law.David Tombe (talk) 11:17, 7 November 2008 (UTC)
 * Hi David: Your summary of my position is pretty accurate, and I am glad you understand it. You then say:
 * I rationalize with the distinction on the basis that the rotating frame transformation equations are not supposed to be applied to objects that don't co-rotate with the rotating frame. In other words, as far as I am concerned, centrifugal force is one single topic.
 * I interpret your statement as simply indicating adoption of one of the two points of view, and discounting the other. Your choice stems from your personal interest in planetary motion, but there is a much wider field of problems in which the planetary motion viewpoint is misleading because it is incompatible with the concept of fictitious forces in non-inertial frames. A widely used set of these examples is the analysis of various problems on a rotating carousel, such as: attaching a ball to a carousel, tossing a ball on a carousel, rolling a ball on a carousel and so forth, where it is not possible for the two definitions of Coriolis and centrifugal forces to agree. The Newtonian approach makes these forces depend upon the rotation of the carousel, and they vanish when the carousel stops. The other makes the forces depend upon the motion of the ball, and the forces persist when the carousel stops. We have seen the difficulties already with the rotating bucket and rotating spheres. Brews ohare (talk) 19:39, 7 November 2008 (UTC)

Hi Bo: You say:
 * I am puzzled by your use of the word Newtonian.

I adopted this term following Lanczos The Variational Principles of Mechanics who uses the term to distinguish the methods of Newton, based upon vectors like velocity and acceleration, from the methods of Lagrange and Hamilton, based upon the scalars kinetic and potential energy.


 * I consider 'fictitious forces' to be bad terminology because a fictitious force is not a force.

Well one can take issue with terminology, but once it is established in the literature it is quixotic to battle against it. However, it looks like you use the term fictitious in a broader than technical sense to include anything you might treat as a force even though it isn't one. The traditional classical mechanics usage of "fictitious force" is more limited, referring to forces introduced in non-inertial frames of reference by virtue of their motion, as distinct from real forces. That is the historical usage going back to the early discussion of "absolute" rotation, and the sequence of events leading to gravity as a fictitious force, stemming from Mach's principle, general relativity and cosmology models, like the Brans-Dicke theory. This use of "fictitious" based upon state-of-motion, has physical meaning, inasmuch as state-of-motion is real and observable, and these fictitious forces actually can be (and must be) used by observers in the moving frame if they want to apply Newton's laws of motion. The fictitious forces "are treated as real and included on the force-side of the equation with the real forces". (Paraphrase of textbook advice, e.g. Rao.)


 * This is bad physics, and so it is pointless to classify fictitious forces further into "State-of-motion" and "coordinate" fictitious forces

Your example is very close to what is done by Hildebrand, Shankar and by the people in robotic design. I'd agree that it is bad physics, created as you point out, by the temptation to treat $$\ddot \xi $$ as an acceleration, whatever ξ means, and all other terms as "fictitious" forces, using the word "fictitious" to avoid discussion of their reality. (In fact, the situation is worse than this, as conflicting definitions of "centrifugal" and "Coriolis" forces are introduced. See Corliss et al..) This usage contradicts the customary notion of an inertial frame of reference, because these "fictitious forces" can be non-zero in inertial frames. However, again, it is quixotic to battle against established practice, even though the practice is limited to a more specialized arena than all of classical mechanics.

Thus, to sum up and repeat myself, there are two terminologies at work here. Both have their adherents, although the Newtonian version is pervasive in Classical Mechanics, has a longer history, and has valid physical meaning. If you agree, perhaps you can suggest how to handle the matter, inasmuch as my efforts appear to lack transparency. Brews ohare (talk) 16:18, 7 November 2008 (UTC)

Image:Anomalies.PNG
The new drawing is an improvement. Thank you. But there is a misunderstanding. The areas |zcy| and |zsx| are equal, but they are not triangles. yz and xz are arcs along the circle. Bo Jacoby (talk) 23:47, 8 November 2008 (UTC).

Hi Bo: Another attempt; how's it look this time? Brews ohare (talk) 03:27, 9 November 2008 (UTC)
 * Perfect! Thanks a lot. Bo Jacoby (talk) 06:59, 9 November 2008 (UTC).

mathematics of the ellipse
The new formula
 * $$   \varepsilon = \sqrt{ 1 - \left( \frac {b}{a} \right) ^2 } \, $$

is correct, but that is not sufficient reason to include it here. The formula is not used later in the article, and it follows from the given formulas for a and b


 * $$   a=\frac p{1-\varepsilon^2}$$

and
 * $$   b=\frac p{\sqrt{1-\varepsilon^2}}. $$

Any pair of parameters can be chosen as independent. In this article $$(p,\varepsilon)$$ was chosen, and the formulas for $$a,b,r_{max},r_{min}\,$$ were provided. This is necessary and sufficient. None of the other $$(\tbinom 6 2-1)\cdot 4=56$$ correct and potentially useful formulas should be included. Bo Jacoby (talk) 08:34, 10 November 2008 (UTC).


 * Bo : Of course, you are right that this formula is not independent and can be derived from those appearing later for a and b. However, my reasons for including this formula is simply that (i) a and b are shown in the figure, not the eccentricity (ii) they are commonly used in all discussion of the ellipse, and the reader is likely to be on easy terms with these parameters, while the eccentricity is more arcane, and (iii) this formula makes clear why the range of values for the eccentricity is 0 -> 1.


 * So, although there is a certain aesthetic of parsimony, I believe that making things more direct for the reader is useful enough that this formula should appear here. Brews ohare (talk) 15:35, 10 November 2008 (UTC)


 * As another issue of presentation, the equation for the ellipse in terms of the semi latus rectum p is not provided in the article on ellipse, and I'd consider it a rather arcane expression as well. In addition, the presentation in this Kepler article does not specify just where the ellipse from this expression is located relative to the origin (e.g. the center of the ellipse at the origin, or one focus at the origin, or what?) So one might take issue with the starting point in this section. Brews ohare (talk) 15:43, 10 November 2008 (UTC)

Why not illustrate rmin and p and rmax in addition to the nice drawing showing a and b ? Why not illustrate the right angled triangle FCB where F is focus, C is center, |FC|=aε, |FB|=a, |CB|=b. That would explain ε better that a formula involving the square root. Why not choose a formula which is actually used in the article such as $$b=a\sqrt{1-\varepsilon^2}\, $$ if you want more formulas ? The ellipse article contains the formula $$r=\frac{a\cdot(1-\varepsilon^{2})}{1+\varepsilon\cdot\cos\theta}\,.$$ The presentation here says that the coordinate system is heliocentric, and that substituting &theta; = 0 degrees, &theta; = 90 degrees and &theta; = 180 degrees gives respectively  r = rmin, r = p and r = rmax. Bo Jacoby (talk) 16:46, 10 November 2008 (UTC).


 * Hi Bo: I'm confused as to which (if any) of these suggestions are being proposed seriously and which are rhetorical flourishes :-) I added the heliocentric coordinate system to the figure. Brews ohare (talk) 16:59, 10 November 2008 (UTC)

I am serious. Your new drawing shows the variables all right. But I had three simple drawings in mind : Bo Jacoby (talk) 23:09, 10 November 2008 (UTC).
 * 1) one ellipse with rmin drawn from the focus to the right to perihelion, and rmax drawn from the focus to the left to aphelion, and p drawn from the focus and upwards to the ellipse, to illustrate these heliocentric distances.
 * 2) one ellipse with a drawn from the centre to the left to the aphelion and b drawn from the center upwards to the ellipse, to illustrate the semimajor and semiminor axes.
 * 3) one ellipse with right triangle with catheti a&epsilon; between center and focus, and b from the center and upwards, and hypotenuse a, to illustrate the eccentricity.


 * Bo: I'll come back to this eventually. But for now I am diagrammed out. Brews ohare (talk) 23:47, 10 November 2008 (UTC)

More Thoughts on having Maths in the Article
The mathematical link between Newton's law of gravitation and Kepler's first law can be done in two directions. In both cases the maths is quite involved. I can see arguments both ways as to whether or not these mathematical proofs should be included in the article. On the one hand, they interfer with the easy reading of the article. But on the other hand, if they are not there, there will be a certain class of readership who will miss them. In older Encyclopaedia Britannicas there used to be alot of detailed mathematics. As long as there is the space, there is no reason why the maths should not be there. It's merely a question of the sequence. Begin with a few easy reading prose sections. Then further down the page have lengthy maths sections entitled for example "The proof that an elliptical orbit about a focus is the result of an inverse square law central force". That in turn will have to begin with the geometry of the ellipse. It will be a very lengthy and detailed section, but if somebody is willing to do it, it should be in the article further down. Likewise the proof that an inverse square law central force leads to elliptical, parabolic or hyperbolic orbits. There will also be the proof that, assuming conservation of angular momentum, as per Newton's law of gravity, then Kepler's second law holds. And there will also be the mathematical link between Newton's law of gravity and Kepler's third law. But the article should begin in prose with an introduction giving a few historical points of interest. Then a statement of what the laws are. Then a statement of their equivalence to Newton's law of gravitation. The article does not need to deal with matters such as Newton's laws of motion, or for that matter anything to do with mass because, it is a purely kinematical topic. On second thoughts, I can see no reason to use hidden boxes, or forks. An appendix might however be appropriate for conic section geometry because that in turn might break the flow of the calculus parts. David Tombe (talk) 23:07, 13 November 2008 (UTC)

Simplification of the article
Following a few discussions on Bo Jacoby's talk page, I can conclude that this whole topic comes down to two equations with two terms in each.

There is a radial equation,


 * $$ \ddot r = -GMr^{-2} + r\dot\theta^2$$

This contains the gravitational attractive acceleration (inverse square law), and the centrifugal repulsive acceleration (inverse cube law). This solves to give a conic section. Kepler's first law of motion is an example of this.

Then there is the tangential equation,


 * $$2\dot r\dot\theta + r\ddot\theta = 0$$

This contains a Coriolis term and an angular acceleration term. These always sum to zero, hence leading to the conservation of angular momentum and the conservation of areal velocity. This is Kepler's second law. Interestingly, the two individual tangential accelerations can still be observed in non-circular orbits.

That is all that needs to be written in the article following the statement and description of Kepler's three laws. David Tombe (talk) 05:45, 5 January 2009 (UTC)


 * I agree that these two equations are true, although I believe your descriptions of some of the terms go against the common physics definitions of certain words. Also, you need both equations, not just the first equation, to derive that the solutions are conic sections...in other words, there are non-conic-section, non-physical "orbits" that nevertheless satisfy the first equation alone. (On the other hand, you're correct that the second equation, by itself, is equivalent to conservation of areal velocity or angular momentum, as proven in the article.)


 * Anyway, one thing I agree with is the sentiment against Section 2 ("Position as a function of time"). I think Sections 2.1 and 2.2 are helpful and worthwhile, but Section 2.3 ("Details and Proof") should be eliminated, and replaced by the statement "For a detailed derivation and proof of this procedure, see the article Kepler orbit." Section 2.3 is almost 20% of the article, it probably scares off many non-technical readers, and interested readers can just click the link. The article Kepler orbit, on the other hand, doesn't have a wide non-technical readership, so having lots of math there is perfectly appropriate. :-) --Steve (talk) 07:13, 5 January 2009 (UTC)

Steve, Yes agreed. We need both the equations. The conic solution to the radial equation depends on the fact that the tangential equation holds true. But I thought I had made that clear from the context.

Moving on now to terminolgies. We are agreed about the gravity term. It is obviously the other three terms that are causing you a problem. But what names would you like to call them? The Coriolis term is the very term which appeared in Coriolis's 1835 paper. I would agree with you that the name Coriolis acceleration in this context is not in line with the modern usage of the term. So what would you suggest that we call it? I suspect that it is modern usage of the term Coriolis which is actually wrong, because Coriolis himself was not dealing with meteorology or a rotating Earth.

On centrifugal acceleration, some textbooks do call it centrifugal acceleration, but others try to play that name down. Whatever, it is unequivocally the centrifugal acceleration.

On the angular acceleration, that is a tricky one. I thought it through carefully and concluded it was the best name. Technically the term angular acceleration only actually refers to the theta double dot bit (ie. the actual second time derivative of the angular displacement). Tangential acceleration would be more accurate, but the problem is that tangential acceleration also refers to the Coriolis term. So I decided to use angular acceleration, because the effect in question actually goes hand in hand with the effect for which the term would be more accurately applied to.

I have no objection to mathematical proofs appearing in this article. They are indeed very useful for technically minded readers. And that goes for both ways. Kepler to Newton, and Newton to Kepler. But they should go at the end of the article. A coherent prose article should be at the top, with those two equations mentioned. David Tombe (talk) 12:15, 5 January 2009 (UTC)

The problem with the terminologies
It seems that there is general agreement that the two equations listed in the section above embody the essence of Kepler's three laws of planetary motion. The problem seems to be with the terminologies used for the individual terms. We have six individual terms in those two equations, and agreement can only be reached on the name for one of those terms. That is the gravitational attraction term in the radial equation.

Let's look at the radial equation,

$$ \ddot r = -GMr^{-2} + r\dot\theta^2$$

The left hand side is clearly the radial acceleration. Yet this fact is stringently denied by many. There are no end of attempts to reorganize this equation in order to both deny that the left hand side is the radial acceleration, and also to hide the centrifugal term altogether.

The most recent attempt that I witnessed was to the extent that the radial acceleration is actually only the gravitational term alone.

$$-GMr^{-2}$$ was written down as being the total radial acceleration, and the centrifugal term had disappeared entirely from view. It seemed as if somebody was deliberately trying to hide the centrifugal term.

In situations in which the presence of the centrifugal term becomes unavoidable, we still witness attempts to deny that it actually is a centrifugal term, despite the fact that older textbooks call it just that. There have even been attempts to call it the centripetal force. But it can't be the centripetal force because gravity is the centripetal force. Hence it would always be equal to gravity and so we could never have elliptical orbits.

Moving on to the tangential equation,

$$2\dot r\dot\theta + r\ddot\theta = 0$$

it is clear that the first term on the left hand side is a term which appeared in the 1835 paper by the French scientist Coriolis. It is also clear that the second term on the left hand side is an angular acceleration term, and that this angular acceleration is clearly observable in an elliptical orbit. Yet we witness attempts to deny that there is any angular acceleration involved. I have heard it said that there could be no cause to produce an angular acceleration. But we can see that angular acceleration in elliptical orbits, and it's not too hard to realize that the cause is gravity, and that gravity has a tangential component, despite the fact that the total tangential component is zero.

Gravity is predominantly a radial effect. But we must not overlook the fact that the total theory involves a tangential equation also, which involves a tangential component to gravity in elliptical, hyperbolic, and parabolic orbits. David Tombe (talk) 05:41, 7 January 2009 (UTC)

Radial Acceleration
David, you say:
 * Let's look at the radial equation,


 * $$ \ddot r = -GMr^{-2} + r\dot\theta^2$$


 * The left hand side is clearly the radial acceleration.

Acceleration is a vector, and $$ \ddot r $$ is not a vector. The combination of terms $$ \ddot r - r\dot\theta^2$$ is the radial component of the acceleration vector, and $$ \ddot r $$ is not. Brews ohare (talk) 16:46, 7 January 2009 (UTC)


 * Brews, we know that acceleration is a vector. And the equation in question relates to the radial direction. Why do you feel the need to add the centrifugal term to the $$ \ddot r $$ term before you are willing to use the word radial acceleration?


 * What you are saying is basically the same as what Bo Jacoby is saying. You are both saying that the radial acceleration is only the gravitational acceleration. You are making the centrifugal term disappear by merging it in with the radial acceleration and then calling the sum the radial acceleration.


 * You are both basically juggling around with a simple radial equation, and in doing so, you are getting the terminologies all confused.


 * It should be obvious to most people that $$ \ddot r $$ on the left hand side is the radial acceleration, and that the two terms on the right hand side are gravity and centrifugal force. Why the need to juggle around with this equation and mix up the terminologies? David Tombe (talk) 06:47, 8 January 2009 (UTC)

The reason to include two terms in the radial component of acceleration is that the second time derivative of the vector  displacement is:
 * Hi David:
 * $$\boldsymbol a (t) = \ddot{\boldsymbol r} = \left( \ddot r - r\dot\theta^2\right ) \hat {\mathbf r} +\left(r\ddot{\theta}+2\dot r\dot{\theta}\right)\hat{\boldsymbol{ \theta}} $$

which is a kinematic result dependent only on the (arbitrary) trajectory r(t), regardless of how caused or by what type of forces present. It should be pointed out that r(t) can be any trajectory at all, without need to specify the engendering forces, which could be electromagnetic, gravitational or whatever. The path can be any path, be it elliptical, helical, or whatever, and can be traversed with any time dependence whatsoever.

This general result that applies to any motion whatsoever has two terms in the radial component, both of which are needed if the same acceleration vector a(t) is to result in any inertial coordinate frame we might choose, in particular if we switch to a coordinate system related to the first by a static fixed displacement, or to one related by a static fixed rotation and displacement. Acceleration a(t) is a vector, and transforms like a vector because both terms are present in both the radial and the tangential components.

For example, switching coordinates to a set of polar coordinates with origin displaced with respect to the first, the unit vectors in the new system will change, as will the coordinates. However, the acceleration will remain the same. That is:
 * $$\boldsymbol r (t) = r \hat{\boldsymbol r} = r' \hat \boldsymbol {r'} + \boldsymbol d \, $$

with d the displacement of the primed coordinates, and


 * $$\boldsymbol v (t) = \dot{\boldsymbol r} = \dot r \hat \boldsymbol r + r \dot{ \hat {\boldsymbol r}} = \dot r' \hat \boldsymbol r' + r' \dot{ \hat {\boldsymbol r'}} \ $$
 * $$= \dot r \hat{ \boldsymbol r }+ r \dot{ \theta} \hat {\boldsymbol {\theta}} = \dot{ r'} \hat {\boldsymbol {r}}' + r' \dot {\theta}' \hat {\boldsymbol{ \theta}}' \ .$$

Using both terms in the two components:


 * $$\boldsymbol a (t) = \ddot{\boldsymbol r} = \left( \ddot r' - r'\dot\theta'^2\right ) \hat {\mathbf r'} +\left(r'\ddot{\theta'}+2\dot r'\dot{\theta'}\right)\hat{\boldsymbol{ \theta'}} $$

where the complete acceleration vector a(t) is the same as in the unprimed coordinate system. This invariance of a(t) can be checked mathematically by substituting the transformation relating coordinates r to coordinates r' . From a physical standpoint, the accelerations are the same because switching to a displaced coordinate system does not change the physical trajectory, nor any of the forces causing the motion. The switch changes only the coordinates used to describe the same physical events. In contrast, the term: $$\ddot r \hat {\boldsymbol r}$$ is not the same as $$\ddot {r}' \hat{\boldsymbol r}' $$, and doesn't transform as a vector acceleration. [User:Brews ohare|Brews ohare]] (talk) 15:03, 8 January 2009 (UTC)


 * Brews, I'm quite familiar with polar coordinates and all the maths that goes with them. And you are correct in saying that the full expression does not relate to any particular motion. In fact, it was me that first pointed that fact out to you on the centrifugal force talk page, and you denied it at the time.


 * But the two Kepler equations above do relate to an actual situation. They relate to the most fundamental of all actual situations. They fall straight out of visual observation of planetary motion. They shed much more light on the physical significance of our six terms, than does the polar coordinate equation.


 * You and Bo (who has discussed this on his own talk page) are both allowing the general polar coordinate equation to take priority for interpretation purposes over the two Kepler equations, despite the fact that the former lacks any particular physical context.


 * In actual fact, the terms in the polar coordinate equation should be interpreted in the light of the Kepler equations and not vice-versa. The radial Kepler equation tells us unequivocally that the radial acceleration is the sum of the attractive gravitational acceleration and the repulsive centrifugal acceleration. That's the basis upon which the differential equation is ultimately solved.


 * You and Bo insist on grouping the centrifugal term on the same side of the equation as the total radial acceleration and then calling the whole thing the radial acceleration. It is totally illogical to do so, and you are doing so for the reasons that I have just said. You are allowing the format of an unspecific polar equation to govern over the top of the more specific Kepler equations.


 * The polar coordinate equation is indeed very revealing. It is quite surprising what vector calculus can actually reveal about space. But we need the experimentally obtained Kepler equations in order to correctly interpret the six individual terms within the polar coordinate equation.


 * I might as well add that we can see Faraday's law of electromagnetic induction in the polar coordinate equation too. All we need to do is take the curl of it. The two radial terms vanish and we are left with Faraday's law in skeleton form. But we need to have experiments to determine Faraday's law in its richer form. Likewise with Kepler's laws. Kepler's laws beef the polar coordinate equation up. It is not vice-versa, as you seem to think.David Tombe (talk) 03:04, 9 January 2009 (UTC)

Let's try this. Which step(s) do you agree with, and which do you disagree with?
 * (1) $$ \ddot{\mathbf{r}}$$ is the acceleration vector
 * (2) In general, given a vector u and a unit vector $$\hat{\mathbf{n}}$$, the "component of u in the direction $$\hat{\mathbf{n}}$$", is given by the formula
 * $$\mathbf{u}\cdot\hat{\mathbf{n}}$$

--Steve (talk) 07:22, 9 January 2009 (UTC)
 * (3) The "radial direction" is the direction $$\hat{\mathbf{r}}$$
 * (4) Therefore, the "radial component of the acceleration vector" is $$\mathbf{\hat{r}}\cdot \ddot{\mathbf{r}}$$,
 * (5) $$\mathbf{\hat{r}}\cdot \ddot{\mathbf{r}} = \ddot r - r\dot\theta^2$$, where $$r,\theta$$ are the two polar coordinates
 * (6) The term "radial acceleration" is universally understood by all physicists to be shorthand for "the radial component of the acceleration vector".


 * Steve, just as with Brews and Bo, you are looking to the polar coordinate equation for the purposes of establishing your terminologies. The polar coordinate equation is indeed correct, and it is very informative. In fact, I was the one who wanted to introduce that equation into the centrifugal force discussion.


 * But the polar coordinate equation doesn't apply to any specific scenario. Hence we have no basis upon which to interpret the individual terms. I could offer my own theories as to what the terms mean in the general polar coordinate equation context. But it is better if we look to the much more informative Kepler equations in order to establish the meaning of the terms.


 * The radial Kepler equation tells us that the total radial acceleration is the sum of the inward radial gravitational acceleration and the outward radial centrifugal acceleration.


 * But the polar coordinate equation has got all these terms juggled around. In your books, the radial acceleration is therefore the term which I would call the radial acceleration, minus the centrifugal acceleration.


 * It would seem that the polar coordinate equation is exposing the fact that acceleration as we measure it in an inertial frame of reference does not include the inbuilt radial centrifugal component, which is masked under the concept of inertia. It is a very revealing equation. And it is correct. But it is wrong to deduce from it that the actual radial acceleration minus the centrifugal acceleration is the true radial acceleration. That is a very cumbersome and blinkered way of looking at the situation.


 * Stand back and look at the radial Kepler equation.


 * $$ \ddot r = -GMr^{-2} + r\dot\theta^2$$


 * What do you think would be an appropriate name for the $$ \ddot r$$ term?


 * In answer to your specific question above, I have reservations about your very first assertion.


 * $$ \ddot{\mathbf{r}}$$ is only the acceleration vector minus the centrifugal component which is disguised under inertia within an inertial frame of reference. The polar coordinate analysis exposes this additional component. But we need Kepler's laws in order to create a context within which we can understand the meaning of the individual terms.David Tombe (talk) 08:12, 9 January 2009 (UTC)

OK, so now we know that you have "reservations" about saying that the second time derivative of the position vector is the "acceleration vector". Wow, I don't even know where to start. Every physicist and mathematician in the world (besides you, I guess) would say that statement (1) is absolutely straightforward and completely uncontroversial in any context whatsoever. It's the one and only definition of the term "acceleration vector". It's the starting point for Newtonian physics. Most people learn it on the very first day of their college physics class.

(BTW, I would call $$ \ddot r$$ "the second derivative of the radial coordinate", or more likely I would just call it "$$ \ddot r$$". There isn't a well-known standard name, so it's best to just be explicit.) --Steve (talk) 16:02, 9 January 2009 (UTC)

You take the stance that a general result:
 * Hi David:


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

defies interpretation until it is applied to a particular situation. You choose the case of a Kepler motion to make the interpretation. You mean by interpretation the identification of the "centrifugal term" $$r\dot\theta^2$$ as a "centrifugal force" and suggest that the Kepler problem forces one to take this term to the force side of the equation.

Here are a few objections to this statement of your position:


 * 1) A particular application consists simply of specifying the true vector forces generating the acceleration (and the assumed initial conditions) and substituting this F into F = ma using the above expression for a. No particular example forces the "centrifugal term" to the force side of the equation, as this term is present regardless of the application and regardless of the real forces present.
 * 2) The "centrifugal term" cannot be traced to a real generating body of origin, like a nucleon or a charged particle. Rather, it originates from the kinematic side of the equation. It is ubiquitous.
 * 3) The isolated "centrifugal term" has no standing as a force, not only because it does not originate in any identifiable source, but because this term in isolation does not transform like a force on change of coordinates. It differs from one coordinate frame to another, unlike a real force that is the same in every coordinate frame. This point has steadfastly resisted comment by you.
 * 4) If Cartesian coordinates are used, the "centrifugal term" goes away. This disappearance is another indication that the centrifugal term is only a kinematic artifact, as a true force is present in every coordinate system, Cartesian or curvilinear.
 * 5) If a different type of curvilinear coordinate system is used, not polar coordinates, the acceleration vector has still another form, and terms beside double-dot terms appear. Again there is no "centrifugal term", but other terms. Are they too to be considered as forces and taken to the force side of the equation? In Lagrangian mechanics, they are, and the $$\ddot q_k$$ terms alone are equated to the "generalized forces". However, there is no pretense in this approach that the "generalized forces" are the same as the real vector forces entering Newton's law of motion. That is why the word "generalized" is inserted.
 * 6) Although a particular problem may lead to a particular interpretation, it is likely that this interpretation will prove specific to the example, and not turn out to be an interpretation widely applicable. Thus, even given the premise, further argument is required to support generality of the interpretation; for example, a host of other (non-Kepler) examples. Brews ohare (talk) 15:57, 9 January 2009 (UTC)


 * Steve, I'll reply to Brews separately below this reply.


 * What I am saying is, that acceleration in an inertial frame of reference, as it is commonly understood, does not include centrifugal acceleration. Centrifugal acceleration is masked out under the concept of inertia. I think that we all agree with that. When we expand 'your' acceleration term in polar coordinates, we reveal the fact that the radial component of 'your' acceleration derives from a larger radial component minus the centrifugal acceleration.


 * Polar coordinates effectively quantify inertia in terms of an outward radial centrifugal acceleration. Once we realize this, and once we begin to solve the Kepler problem, then it becomes obvious that the full radial acceleration is actually the $$ \ddot r $$ term of the equation,


 * $$ \ddot r = -GMr^{-2} + r\dot\theta^2$$


 * We are not arguing here over the facts. We are arguing over the terminologies. And this argument has been caused because of sensitivities over the concept of centrifugal force. But where you say that the $$ \ddot r $$ doesn't actually have a name, I would say that it is in actual fact the total radial acceleration.


 * Anyhow, the main article can be written without having to use names for the three terms above. I just thought that it would have been helpful to have had three clearly identifiable terms for the radial equation. David Tombe (talk) 03:10, 10 January 2009 (UTC)

Brews, it doesn't actually matter what side of the radial equation that we put the centrifugal term on. But the standard format when solving a differential equation is to isolate the differentiated variable to one side.

I said to Steve above, that you, Bo, and Steve are all looking to a term that lacks the centrifugal effect as being the official radial acceleration. I am saying that :'' the the full radial acceleration is actually the $$ \ddot r $$ term.

We can even then take the areal constant from the tangential equation and make the centrifugal term into an inverse cube law term. An inverse square law and an inverse cube law operating in tandem will circumvent Earnshaw's Theorem and allow for stability nodes.

Yet you have been trying to say above that the centrifugal term is somehow not a real force. It is that belief which I suspect makes you and Bo favour isolating the gravity term on one side of the radial equation, and viewing it to be the total and singular radial acceleration, and making centrifugal force become relegated to something that merely forms part of a general expression for radial acceleration.

On your specific points above, you say that other scenarios can be fit around the polar coordinate equation. Yes. And some of them are so dangerously simple that they have led to absolute total confusion. Consider a weight being swung around on the end of a string. Tension causes the centripetal force (That tension is of course a reaction to the centrifugal force, although that fact is never recognized). The motion is circular and so the centripetal force equals the centrifugal force. Before we know it, the physics books will have caused centrifugal force to have disappeared altogether. They will equate the tension to the centripetal force and that will be it.

The Kepler problem is the most general and fundamental problem for the purposes of interpreting the terms in the polar coordinate equation. On the centrifugal force talk page, it took me nearly two years to actually get any recognition of the two Kepler equations above. Thankfully they were already in this article before I arrived here. They demonstrate the existence of centrifugal force as a real outward force, and their generality exposes the falsity of the explanations for alot of the more simple circular motion problems, which in turn have led to all that unnecessary confusion and splitting of centrifugal force into reactive, and polar, and rotating frames etc. David Tombe (talk) 03:33, 10 January 2009 (UTC)


 * Hi David:
 * Brews, it doesn't actually matter what side of the radial equation that we put the centrifugal term on. But the standard format when solving a differential equation is to isolate the differentiated variable to one side.

True from a math viewpoint, but not true if you then imply that the double-dot r term is acceleration, a term reserved in technical discussion for a vector.
 * if we stand back and look at the full radial equation, we can see...

We don't all see the same thing when we stand back.
 * Before we know it, the physics books will have caused centrifugal force to have disappeared altogether. They will equate the tension to the centripetal force and that will be it. 

You got it; these days are here, and I find it to be the standard view for the last two centuries. In the physics books centrifugal force appears only for a rotating observer, and never for a stationary observer. In particular, in observing planetary motion (the Kepler problem) from a stationary reference frame, there is no centrifugal force at all, whether or not polar coordinates are used.
 * The Kepler problem is the most general and fundamental problem for the purposes of interpreting the terms in the polar coordinate equation.

Whatever the merits of the Kepler problem, it does not demonstrate the existence of centrifugal force.

David, above I've presented a numbered six-point critique of your position, which you do not address in specific detail. By instead resorting to generalities, you make further discussion too vague and too unfocused to make progress. Brews ohare (talk) 06:38, 10 January 2009 (UTC)


 * Brews, I'll deal with your points 3 and 4. The centrifugal force doesn't go away in Cartesian coordinates. It gets camouflaged under inertia. The polar coordinate expansion exposes this fact. It shows that the radial component of acceleration as we measure it in the inertial frame is actually a total radial acceleration, minus the centrifugal force.


 * A straight line motion in a cartesian inertial frame still has mutual radial centrifugal acceleration relative to every other particle.


 * And the only difference in principle between gravity and centrifugal force is the fact that centrifugal force is induced by absolute rotation in the inertial frame. Both are equally as real. We do not need a rotating frame of reference in order to perceive what is happening. A rotating frame of reference is in effect a rather childish visual aid, which only serves to confuse the issue. And if we were to view an elliptical orbit from a rotating frame of reference, it would have to have a varying rate of rotation. We don't need any such childish tools. We can observe an elliptical orbit from the chair in the corner of the room, and we can see clearly that there is a radially outward centrifugal force.


 * Having said all that, it is not my intention to continue the centrifugal force argument on this talk page. My objective was to try and ascertain agreed terminologies for the six terms in the two equations.


 * So let's look at the radial equation one more time,


 * $$ \ddot r = -GMr^{-2} + r\dot\theta^2$$


 * It seems as though we are not going to be able to agree on the names for two of the terms in the equation above. I would have automatically gone for, (reading from left to right), (1) Radial acceleration, (2) Gravity, and (3) centrifugal acceleration.


 * But it seems that apart from the gravity term, we are going to have to simply refer directly to the mathematical expressions instead.


 * So let's end this discussion at that, and move on now to the tangential equation. David Tombe (talk) 02:22, 11 January 2009 (UTC)

You say:
 * Hi David:
 * A straight line motion in a Cartesian inertial frame still has mutual radial centrifugal acceleration relative to every other particle.

This statement contradicts all available sources, and implies a world view that is solely your own. No source I know of suggests centrifugal acceleration appears in straight-line motion. And no source suggests "mutual centrifugal acceleration" between particles, except possibly some hypothetical remark somewhere by Mach, unknown to me, as an illustration of his controversial view that a particle's mass is an effect of the rest of the universe upon a single particle. If you have reputable support for this statement, where is it? Brews ohare (talk) 14:52, 12 January 2009 (UTC)
 * By the way, what is your response to Point 5 above? A Lagrangian approach leads directly to your double-dot r equated to the generalized force, and removes all objections to your approach, except the objection to your interpretation of the "generalized force" as a true Newtonian vector force. Brews ohare (talk) 15:28, 12 January 2009 (UTC)

Tangential acceleration
Here is the tangential equation prior to the application of Kepler's second law,

$$\ddot s = 2\dot r\dot\theta + r\ddot\theta$$

What would you like to call the three terms here?

Reading from left to right, I would strongly suggest (1) tangential acceleration, (2) Coriolis acceleration, and (3) angular acceleration.

Any objections? David Tombe (talk) 02:30, 11 January 2009 (UTC)
 * I'd assume that the name "Coriolis acceleration" is suggested because some connection is implied to the Coriolis force in a rotating frame, to which it bears some formal resemblance but actually no physical connection, inasmuch as there is zero Coriolis force when motion is observed from a stationary (non-rotating) frame, regardless of whether the motion is circular or some other form, while the $$2\dot r\dot\theta$$ term is ubiquitous in polar coordinates, regardless of the form of the motion or the frame of observation. If such a connection is implied, the use of this name for this term is a source of confusion and should be avoided. Brews ohare (talk) 06:17, 11 January 2009 (UTC)

The tangential acceleration is the acceleration component in the direction of the motion. The angular acceleration is $$\scriptstyle \ddot\theta.$$ Bo Jacoby (talk) 08:52, 11 January 2009 (UTC).


 * Bo, OK, so Brews doesn't want to call the term $$2\dot r\dot\theta$$ the Coriolis acceleration even though it refers the the tangential deflection of the radial motion in an elliptical orbit.


 * You have pointed out correctly that angular acceleration accurately refers only to the $$\scriptstyle \ddot\theta$$ term. But what can we then call $$r\ddot\theta$$? I had suggested that we also call it angular acceleration since it always goes hand in hand with angular acceleration. Tangential acceleration would actually be more accurate, but that is already being used for the total tangential acceleration which includes the Coriolis term.


 * By the way, way did you choose for your definition of tangential acceleration an alternative definition which you know has got nothing to do with this article? On your talk page, you said that there are two definitions for tangential acceleration. You agreed that one of those corresponded to $$\ddot s$$ above. So why did you suddenly now choose to switch to the other definition which you know has got nothing to do with this issue?


 * I'm trying to get names for the six terms in the two equations which are central to this topic, and all I am seeing is ducking and dodging. That tells me that these two equations contain truths which are uncomfortable for some, and that the proper names for these terms seem to highlight these truths. (Perhaps you are right that there are no generally accepted names for all the terms. That does not necessarily mean that the truth is uncomfortable, but rather that the translation from algebra to English is not standard? I don't know. Bo Jacoby (talk) 08:15, 12 January 2009 (UTC))


 * And furthermore, your alternative definition of tangential acceleration above is wrong, because we can have a radial acceleration that is in the direction of motion of the particle too. I don't know where you got that definition from, but it has no place in this article. David Tombe (talk) 04:45, 12 January 2009 (UTC)


 * Um, how can tangential acceleration mean something other than acceleration in the direction tangent to the trajectory?? Radial acceleration could also be tangential to the path when the path is a radial path, but that is a very special case that does not impinge the general definition. Brews ohare (talk) 06:51, 12 January 2009 (UTC)

I just clicked on the links tangential acceleration and radial acceleration. There are perhaps no generally accepted names for all the terms. That does not necessarily mean that the truth is uncomfortable, but rather that the English explanation of the algebra is not standard. Bo Jacoby (talk) 08:15, 12 January 2009 (UTC).

It's a new year so let's do something
There was a previous proposal that really should be implemented: all the math in the section titled "Position as a function of time" ought to be taken out of this page and merged with the presentation which is already written out on either of the pages Kepler problem or Kepler orbit.

This would be a very good start to improve this article. I do not support removing the derivation of Kepler's Laws from Newton's Laws, because that derivation was perhaps the greatest achievement of humanity during the seventeenth century. In contrast, the section on "position as a function of time" is bordering on trivial (i.e. if you know Kepler's three laws then anyone with minimal math skills could figure out how to predict position as a function of time).Ferrylodge (talk) 17:57, 11 January 2009 (UTC)


 * I'd suggest that the section Position as a function of time be viewed as an historical more than a mathematical section. It shows how Kepler did it, not necessarily the way it would be done today. That interests me, and possibly some others not too dismayed by a few equations. As an historical note, it fits into this article as well as it fits into the Kepler problem article; it can be skipped by those not interested, and doesn't detract from the present article.Brews ohare (talk) 18:55, 11 January 2009 (UTC)


 * Placing this section before the derivation of Kepler's laws from Newton's laws does several bad things: (1) it gives the readers an impression that this section is more important; (2) it gives the reader the impression that he must read and understand this section in order to understand the derivation from Newton's laws; and (3) it prevents the reader from even becoming aware of the more important derivation from Newton's laws because the reader will have given up reading this article already. If people would like to keep the section titled "position as a function of time" in this article then I suggest moving it down in the article.  However, I still urge that it be split off to another article.  Undergraduate physics textbooks and courses do not typically cover the material in "position as a function of time."Ferrylodge (talk) 19:09, 11 January 2009 (UTC)

The computation of the position of a planet as a function of time is exactly what Kepler's laws are about. I don't know how "anyone with minimal math skills could figure out how to predict position as a function of time" knowing Kepler's three laws. You are welcome to show me. Don't you need to solve Kepler's equation? On the other hand, the derivation from Newton's laws may well be "the greatest achievement of humanity during the seventeenth century", but it does not really belong in an article on Kepler's laws. The Newton orbit is not the same as the Kepler orbit unless the masses of all the planets are neglegted, so the derivation is only approximate. I would prefer a derivation of the acceleration of a planet in a Kepler orbit, as a preparation to an understanding of gravitation. Bo Jacoby (talk) 20:46, 11 January 2009 (UTC).


 * I agree that it would be nice to have a derivation of the acceleration of a planet in a Kepler orbit, as a preparation to an understanding of gravitation. However, the section titled "position as a function of time" does not contain such a derivation.  As far as predicting position as a function of time, using Kepler's laws, one merely observes motion to determine the constants of motion (e.g. eccentricity and rate at which area is swept); it is trivial to forecast future motion once the constants of motion are determined.Ferrylodge (talk) 21:22, 11 January 2009 (UTC)

Ferrylodge, you are jumping ahead a bit. I was trying to slowly work up to the connection between Kepler and Newton. But since you have gone there already, I ought to point out that the link between Kepler and Newton is only with the inverse square law denominator in the gravitational acceleration term. It is only a kinematical link, in that elliptical orbits imply an inverse square law attractive acceleration (and an inverse cube law centrifugal repulsive acceleration). Newton did extra stuff with the numerator regarding reduced mass and the product Mm. That doesn't really belong in this article.

I was trying to keep the link to Newton until the mathematical appendices at the end of the article. And some of the maths proofs are anything but trivial.

Meanwhile, as you can see in the section above, we can't even seem to get agreement on trivial issues such as names for the terms in all the equations which are ultimately going to be involved. David Tombe (talk) 04:31, 12 January 2009 (UTC)


 * I agree that stuff about reduced mass and the product Mm does not belong in this article. There are two kinematical proofs that absolutely do belong in this article: proof that Kepler's laws imply an r^-2 central attactive force, and the converse proof that an r^-2 central force implies that all closed orbits obey Kepler's laws.  The section entitled "position as a function of time" covers neither of those kinematical proofs, and really detracts from the arricle, IMHO.Ferrylodge (talk) 04:43, 12 January 2009 (UTC)

Ferrylodge, Yes I agree with you that the first two proofs that you mention here are indeed essential. In fact, ultimately I had envisaged an appendix section with six mathematical proofs. Two for each of Kepler's laws. One for each direction, Kepler to Newton, and Newton to Kepler.

I'll have a look at the section which you are objecting to. Meanwhile I'd like to hear your views on what we should call the six terms in the core maths.

I should further add that some of these proofs are anything but trivial and as you know, involve the need to be familiar with conic section geometry. That is a maths appendix headache in its own right.

Also, there is the issue of the inverse cube law centrifugal acceleration which is being camouflaged by some as being merely a part of the radial acceleration term. That's why we need to sort out the terminologies. David Tombe (talk) 04:49, 12 January 2009 (UTC)


 * Please see footnote 5. If it were up to me, this article would contain no trigonometry whatsoever.Ferrylodge (talk) 04:58, 12 January 2009 (UTC)

Ferrylodge, I've looked at that paper. I can't see the benefits of a cartesian treatment of the Kepler problem. Polar coordinates suit both the astronomical scale and the microscopic scale. Cartesian coordinates only suit terrestial situations. So what is the point? And besides, despite the title, the proof was anything but simple.

You are not seriously suggesting that a cartesian proof should be put in the main article are you?

As regards the section that you want eliminated, I read it. It is heavy going. But I think it is directly relevant to what Kepler did. I haven't studied the details, but from what I could read, assuming it to be true, it is how Kepler calculated the ellipse from his observations.

Just like yourself, I am much more interested in the link between the inverse square law and the ellipse, but for this particular article, that section that you want eliminated may actually be more relevant to Kepler.

I'd be inclined to keep it, and to put the stuff that we are more interested in into the appendices. Having said that, I am very much in favour of the use of polar coordinates for any proofs.David Tombe (talk) 05:33, 12 January 2009 (UTC)


 * Newton's Principia is full of complicated arguments and proofs, but hardly any of it belongs in Wikipedia. Likewise, I have a hard time understanding why the section titled "position as a function of time" says anything that belongs on Wikipedia.  Anyway, I've got a bunch of other things going on, and so probably won't be able to participate much at this article.  Sincere thanks for considering my opinions.  Cheers.Ferrylodge (talk) 05:42, 12 January 2009 (UTC)

''it is trivial to forecast future motion once the constants of motion are determined. Ferrylodge'' Please show us how you do it trivially. Then your explanation can replace the complicated deviation in "position as a function of time". Bo Jacoby (talk) 08:21, 12 January 2009 (UTC).


 * Ferrylodge, it was your specific interest in the cartesian proof which worried me, and it was anything but trivial, contrary to what the title of the paper claimed. David Tombe (talk) 12:34, 12 January 2009 (UTC)

Conclusion on Terminologies
Well it would certainly seem that the obvious English names for the terms are not standard. We have Brews objecting to the use of the terms centrifugal and Coriolis, and we have Brews, Bo, and Steve objecting to the term radial acceleration as applied to what I would consider to be the total radial acceleration. We also have issues over the terms tangential acceleration and angular acceleration. I'm sure that if no underlying controversy over centrifugal force were to exist, that these six terms would all be swiftly named without any problems whatsoever.

Radial equation

$$ \ddot r = -GMr^{-2} + r\dot\theta^2$$

Tangential equation

$$\ddot s = 2\dot r\dot\theta + r\ddot\theta$$

I would have thought that (1) radial acceleration, (2) gravity, (3) centrifugal acceleration, (4) tangential acceleration, (5) Coriolis acceleration, and (6) angular acceleration, would have precisely matched the six terms above.

The irony is that the very term, which I would call angular acceleration, and which is very obviously real in an elliptical orbit, is the only term which can ever actually appear fictitiously. Rotating frames of reference merely superimpose a circular motion artifact on top of existing motion. They do not cause centrifugal or Coriolis effects, real or apparent. The centrifugal or Coriolis effects are either already there or they are not, based on actual rotation.

But if we angularly accelerate a rotating frame of reference, we can indeed get an angular acceleration artifact which has been referred to in other articles as the Euler force. The Euler force is the only true fictitious force. But in the Kepler elliptical orbit we are actually witnessing a real version of the Euler force. The centrifugal force and the Coriolis force on the other hand are always real and can never be produced as artifacts from a rotating frame of reference. This is where modern physics has messed up so badly.

OK. So we'll leave the two equations with no names and we'll discuss them where necessary in the main article by referring directly to the mathematical symbols. David Tombe (talk) 12:26, 12 January 2009 (UTC)


 * There are two unresolved issues: one is the use of the word "acceleration", which technically refers to a vector used in Newton's second law of motion. Whatever else the various terms you wish to discuss may be, they are not accelerations, do not originate in identifiable sources as do true forces, and do not transform as vectors.
 * The remaining issue is: just what has to be discussed? David, your views of this matter are way off the beam, and certainly not supported by the literature, for example, Taylor, Arnol'd, or Landau & Lifshitz.
 * The refutation of your notions by reference to verifiable sources is already well documented at the talk pages on Talk: Coriolis effect and Talk:Centrifugal force (rotating reference frame). It seems unnecessary to revive all that discussion and regurgitate it here. Brews ohare (talk) 14:02, 12 January 2009 (UTC)

Brews, I appreciate that in wikipedia we need to keep to the textbooks. But the textbooks are not always clear, and they do not always whistle exactly the same tune. There is some leeway for slanting wikipedia articles.

If you look at those two equations above, I think that you will find that they contain the larger body of total human knowledge in physics, and that that knowledge will be enhanced if the six terms are named as I have suggested.

There is something seriously wrong with all your reputable sources if they are trying to push the idea that radial acceleration includes a negative centrifugal term as an inbuilt constituent. Centrifugal force is a rotationally induced effect which has repercussions in electromagnetism. Maxwell clearly exposed this fact, and I would give much greater credibility to Maxwell's writings than I ever would to the likes of Taylor, Arnol'd, or Landau & Lifshitz.

The Kepler problem illustrates that centrifugal force and Coriolis force are never fictitious. The only one of the six terms above that can be fictitious, but doesn't have to be fictitious, is the one that I would call the angular acceleration. You have called it the Euler force in the fictitious forces article.

Obstructing plain language is a form of propaganda. And that is what we are witnessing here. We are witnessing a point blank refusal to allow obvious names for mathematical terms.

The $$ \ddot r$$ term is obviously the total radial acceleration. But that term is disallowed. The $$r\dot\theta^2$$ term refers to a real outward radial acceleration which is obviously the centrifugal acceleration, but that term is also disallowed.

The $$\ddot s$$ term is obviously the total tangential acceleration, but that term has been quibbled over and obfuscated with irrelevencies. The $$2\dot r\dot\theta$$ is obviously a very real Coriolis acceleration, but that has been ruled out because of the religiously held belief that Coriolis accelerations are never real. And the $$r\ddot\theta$$ is a very real tangential component of gravity, clearly visible in elliptical orbits, but swept under the carpet because of the current teaching that gravity is only ever a central force. David Tombe (talk) 05:47, 13 January 2009 (UTC)


 * Hi David:
 * A Lagrangian approach leads directly to your $$\ddot r $$ equated to the generalized radial force, and removes all objections to your approach, except the objection to your interpretation of the "generalized force" as a true Newtonian vector force. That would solve many of your problems. It amounts to using the word "term" in place of "acceleration", as in "Coriolis term", "centrifugal term". Also refer to $$\ddot r $$, $$\ddot \theta $$ in place of radial acceleration and angular acceleration and avoid suggestion of a vector behavior for any of these terms. See, for example, . This author is sloppy in his usage and often forgets to put in the adjective "generalized": I'd suggest a bit more care. (On p. 157, Hildebrand is unusually careful (for him) to distinguish between generalized and actual forces.)


 * The way forward is not to battle well-known and widely accepted authorities like Arnol'd and Landau & Lifshitz. Brews ohare (talk) 14:22, 13 January 2009 (UTC)

Brews, it just goes to show that if you search through enough of the literature, you will find that other authors have seen it the same way as I have. The real problem is with those textbooks, of which there aren't that many, which extrapolate the Coriolis and centrifugal terms to objects that are at rest in the inertial frame, as viewed from the rotating frame. That viewpoint has become a cult which has destroyed the correct understanding of the Kepler problem. David Tombe (talk) 15:00, 13 January 2009 (UTC)


 * Perhaps you could explicitly switch to a Lagrangian approach and use the generalized force terminology to align yourself with many reputable discussions that do exactly this. It would modify your wording in places, but probably you could make your points without appearing as a radical?


 * The standard texts using a Newtonian vector mechanics viewpoint are not incorrect; but a Lagrangian approach uses different terminology that cannot be interpreted as if it were the Newtonian vector mechanics standpoint. Brews ohare (talk) 15:05, 13 January 2009 (UTC)

State Laws in Intro
I was surprised that the laws were not stated in the introduction. It seems to me that they should be stated right at the begining. They are the essence of this article. This is an old mature article so I was reluctant to edit it, but on reflection I decided to give it a try.

Nick Beeson (talk) 19:52, 6 March 2009 (UTC)

Kepler's Second Law
FyzixFighter, There was no need to remove that edit. It is a standard textbook fact that Kepler's law of areal velocity means that the two tangential terms sum to zero mathematically. I hardly need a source for that. But if you insist, then I'll use Herbert Goldstein 'Classical Mechanics'. Anybody who has done planetary orbital theory will be familiar with that relationship. And anybody who has examined an elliptical orbit can see that one of the two terms causes an angular acceleration or deceleration, and that the other causes a tangential deflection of the radial motion. I intend to put that edit back again but I will wait until you have thought about it for a while. David Tombe (talk) 01:02, 23 March 2009 (UTC)


 * Only the first sentence and half of the second sentence of your edit are standard textbook facts. Kepler's second law can be expressed as
 * $$ 2\dot r\dot\theta + r\ddot\theta = 0$$
 * but it can also be expressed as
 * $$r^2 \dot\theta = const$$
 * or
 * $$x \ddot y - \ddot x y = 0$$
 * In the first representation the terms do sum to zero. However they do not exist individually except in a mathematical sense since writing them using other coordinate systems would yield different separate terms. All of Kepler's laws are kinematic equations, not dynamic equations so you can't say that terms cause angular acceleration or deceleration. You need a reliable source for the statements (1) that the two terms "exist" individually, (2) that they don't cancel physically, and (3) that this requires an analysis in terms of two different phenomena. All standard textbooks I've seen show that all of Kepler's law is the result of Newton's laws of motion and a single phenomenon - a central attractive force that goes as 1/r^2. I will continue to remove any edits that fail WP:OR and WP:RS. --FyzixFighter (talk) 03:16, 23 March 2009 (UTC)

FyzixFighter, Let's then take one of the two terms at a time. Let's look at the $$r\ddot\theta$$ term. Imagine an elliptical orbit. Can you see that term in operation or not? I can. It refers to the fact that the angular velocity is constantly changing. As a comet moves towards the perihelion, it's angular speed will be increasing. David Tombe (talk) 10:55, 23 March 2009 (UTC)


 * I agree with you that $$2\dot r\dot\theta + r\ddot\theta = 0$$ is a valid symbolic representation of Kepler's second law and that the angular velocity is constantly changing for elliptical orbits. But this does not mean that there is a nonzero torque/non-radial force. Kepler's second law, the conservation of areal velocity, is a general property of central force motion and is not restricted to an inverse-square law of force (see Goldstein, pg 60-61 in the 1st edition, pg 72-73 in the 3rd edition). Goldstein shows this using Lagrangian mechanics (where V=V(r) for a central force) but it can also be shown using Newton's laws of motion. Kepler's second law does not indicate that there are other forces in the theta-hat direction, in fact it indicates that the theta-hat component of the net force must be zero. What I have problems with are the three statements which I indicated above as needing reliable sources. --FyzixFighter (talk) 23:56, 23 March 2009 (UTC)

FyzixFighter, I've done a reworded version and I have left out references to the analogy with Faraday's law. As regards the inverse square law, I would absolutely agree with you that the tangential equation has got nothing to do with the inverse square law. But you were the one that said yesterday that the entire phenomenon came down to a single inverse square law force.

In fact there are four acceleration terms over two equations and only one of those terms is the inverse square law force, and that is in the radial equation. So at least we are now agreed that Kepler's second law has got nothing to do with the inverse square law.

As regards the radial equation, I have been the one that has been trying to point out that it contains both an inverse square law term and an inverse cube law centrifugal term. If gravity were to be an inverse cube law, we could not have a stable orbit because there could be no stability nodes (Earnshaw's Theorem). We could still theoretically have a circular orbit, but it would be unstable and if it were disturbed, it would spiral in. I saw that analysis many years ago. Other power laws are an interesting concept but unfortunately few of them are analytical. The direct force law is however analytical and it does not permit open orbits. It only allows closed and bound orbits.

As regards Faraday's law, the two terms in Kepler's second law do correspond mathematically to the two terms in Faraday's law. My original point was in relation to Feynman's assertion that it is only in Faraday's law that we see two separate aspects to one phenomenon. We can see this very same division in Kepler's second law too. David Tombe (talk) 11:55, 24 March 2009 (UTC)


 * I apologize for the confusion - looking back I see I missed a few typos in my previous edit. What I was meaning to say was that all of Kepler's laws are the result of a central 1/r^2 net force.


 * As to your edit, David, most of it duplicates information that is already in the first paragraph. The first paragraph already states that the speed of a planet increases as it moves inward. Also, since the only force in operation is a purely radial force (gravity), the last statement is redundant with the statement that there is no net tangential force (tangential here meaning in the $$\hat\theta$$ direction, not tangent to the orbit). Your statement is also somewhat misleading as it implies that there are non radial components to the net force. Other parts are meaningless and border on incorrect. The statement that the two terms can be seen individually is meaningless; like any kinematics equation, it shows the relationship between physical quantities, all of which can be observed individually regardless of whether they occur as separate terms (we can observe the $$r$$ and $$\ddot\theta$$ individually as the radius and angular acceleration). Are you trying to imply that the two terms correspond to separate forces by saying they can be observed individually? If so, this is not supported by reliable sources. Lastly, the statement that the radial motion will be continually deflected into the tangential direction is incorrect as the only force is in the radial direction - deflection of motion implies a force in the direction of the deflection. The statement is also ambiguous due to the two meanings of "tangential direction" - (1) tangent to the path and (2) in the azimuthal direction (the meaning used when saying the net tangential force is zero). --FyzixFighter (talk) 17:39, 24 March 2009 (UTC)

FyzixFighter, You have built up quite a record of reverting my edits. It would be interesting to calculate what percentage of your physics edits in the last year and a half have been exclusively to revert edits of mine.

What you have said above is quite wrong. There are two equations involved in the Kepler problem. There is the radial equation and the tangential equation. Each of these two equations contains two acceleration terms. The inverse square law gravity term is only one of four terms. Here are the two equations which cover the entire topic,

There is a radial equation,


 * $$ \ddot r = -GMr^{-2} + r\dot\theta^2$$

This contains the gravitational attractive acceleration (inverse square law), and the centrifugal repulsive acceleration (inverse cube law). This solves to give a conic section. Kepler's first law of motion is an example of this.

Then there is the tangential equation,


 * $$2\dot r\dot\theta + r\ddot\theta = 0$$

You clearly do not understand this topic if you think that the inverse square law term in the radial equation takes some kind of primacy over the other three terms. When you said that the only force involved is in the radial direction, you were basing that on the fact that the net tangential force is zero. But as I pointed out, each of the two tangential terms can be observed individually. You have even acknowledged one of these yourself. You have acknowledged that the planet speeds up in the tangential direction as it moves towards its point of closest approach. The other term is clearly observed by virtue of the inward radial motion continually changing its direction.

It would be preferable if you would discuss issues on the talk page before storming in and making reverts because you have proved on previous occasions that you have made reverts before fully studying the merits of the edits in question. This is another clear cut case of that. You are now only realizing for the first time that there are actually two tangential terms which although they cancel mathematically, they are still individually observable. It may be a hard concept to grasp, but it is an undeniable fact.David Tombe (talk) 18:55, 24 March 2009 (UTC)


 * David, if you feel that I've dealt unjustly with you then I invite you to pursue WP:RFC or WP:3O or any of the other dispute resolution avenues. I'm not going to argue physics with you. I've already done it on numerous occasions, as have many others, and that's not how wikipedia works. I will discuss reliable sources with you, how to include what they say and how their content relates to the topic at hand. You want me to stop reverting your edits here, then provide reliable sources that support your statements. The last time I reverted you I didn't do so until I had a chance to look at Goldstein (since you appealed broadly to that reference). Unfortunately, I couldn't find anything Goldstein that could possibly support the fringe science and original opinion that you're trying to push into this article. Goldstein derives Kepler's laws using only a single, central, radial, inward gravitational force - no centrifugal force, no coriolis force, no euler force. He says nothing about the two terms being individual and distinct physical "things". He makes no comparison between Faraday's induction law and Kepler's second law. The same can be said for every other reliable source I can find. Like I said before, if you want me to stop reverting your edits here, then provide reliable sources, not your OR arguments about why you're right and I'm wrong. --FyzixFighter (talk) 04:19, 25 March 2009 (UTC)

FyzixFighter, First of all, it is quite clear from what you say above, that you don't have the slightest comprehension of the subject matter. And secondly, I have made a careful check and discovered that there has never been a time since I opened my account, that you have done a physics edit other than to revert an edit which I have made. There is no exception to the rule. I will list the chronology here,

25th April 2008, You came to centrifugal force to revert an edit of mine. On the 28th April, you went to the adminsitrator's noticeboard and accused me of disruptive editing. You then argued with me on the centrifugal force pages into May and maybe June. But every single physics edit which you made in that period was exclusively in antagonism to my edits.

Then on 23rd July 2008, you once again reverted an edit of mine on centrifugal force.

And on 23rd October 2008 you entered the arena again exclusively to undermine a citation which I had given at the centrifugal force talk page, and you did so on totally specious premises.

On 31st January 2009, you reverted an edit of mine at Faraday's law of electromagnetic induction.

On 16th February 2009, you reverted an edit of mine at Kepler's laws of planetary motion.

On 22nd March 2009,you reverted an edit of mine at Kepler's laws of planatery motion.

On 23rd March 2009, you reverted a new edit on the same article.

And on 24th March 2009 you reverted one of my edits at Faraday's law of induction.

In the entire period since I opened my account in April 2008, your only physics edits have been to undo my edits. It is clear that you don't comprehend the subject matter in these subjects, and so it is clear that your motivation has got nothing whatsoever to do with what you say it is. This has got nothing to do with original research. You are trying to undermine my attempts to explain the Kepler problem, which is a problem that I have studied in depth over many years. David Tombe (talk) 18:18, 25 March 2009 (UTC)

The connection with Newton's law of gravitation
There was a discussion a few months ago relating to the importance of the connection between Kepler's laws and Newton's law of gravitation. This connection is certainly very important and it involves interesting mathematical proofs in both directions, for both the radial equation and the tangential equation.

But I now think that these mathematical proofs rightfully belong in the 'Newton's law of gravitation' article. Newton's law of gravitation came later than Kepler's law in the historical chronology.

This article could be drastically simplified by limiting the after discussion to the fact that the three Kepler laws can be summarized by two equations with four acceleration terms, and that the inverse square law of Newton's law of gravitation is one of these four terms.

The tangential equation which arises from Kepler's second law, leads to an areal constant $$r^2\dot\theta$$, normally denoted by the symbol $$ {l} $$. Combining this constant with the centrifugal term $$r\dot\theta^2$$ leads to a radial equation of the form,

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

The presence of two different power laws accounts for the stability of the Keplerian orbits. I intend to add a discussion of these points after the exposition of the three laws. I will move all the remaining mathematical stuff to the bottom and suggest that it be transferred to the 'Newton's law of gravity page'. David Tombe (talk) 00:37, 27 March 2009 (UTC)


 * I can agree with removing most of the Newton's law derivation. And actually you don't need to move it to the Newton's law of gravitation, since the derivation is over at Kepler orbit. A link to that page for the Newton's law derivation should be sufficient in my opinion. We should, however, have a paragraph or two discussing Newton's influence on the modern understanding of Kepler's laws. That is, Newton gave us the dynamic theory that explains why Kepler's kinematic laws must be obeyed. In addition, Newton generalized Kepler's laws in a number of ways - adding unbound orbits, center of mass (not the sun) at the focus, and a few generalizations to the the third law.


 * As for the equations, I disagree with you on some of it. First, barring a reliable source, I would say that it is incorrect to call any of these individual terms "acceleration terms" as they do not correspond to components of the acceleration vector, i.e. $$\ddot r$$ is not the radial acceleration and $$r\ddot\theta$$ is neither the tangential nor angular acceleration. We should use terminology and descriptions that are established in reliable sources. While I might agree with throwing in your tangential equation, I don't agree with including your radial equation. The tangential equation is a trivial outcome of Kepler's second law, but the radial equation is usually derived not from Kepler's laws, but from Newton's laws or Lagrangian mechanics and then shown to be consistent with Kepler's laws. The equations that really sum up Kepler's laws would be (with polar coordinates):


 * $$r=\frac{a (1-\varepsilon^{2})}{1+\varepsilon \cos\theta}$$
 * $$\frac{d}{dt}\left ( \frac{1}{2}r^2\dot\theta \right )=r\dot r\dot\theta+\frac{1}{2}r^2\ddot\theta=0$$
 * $$ {P^2} \propto {a^3} $$


 * Getting your radial equation from these is not trivial. If you can provide a reliable source that does this, it would be whole lot easier to assess how much space and how simply the derivation could be shown. --FyzixFighter (talk) 17:03, 28 March 2009 (UTC)

FyzixFighter, First of all, I will remind you that the only physics edits which you have done since I opened my account last April, have been to revert my edits. And we must not lose sight of that fact when analyzing what you have just written above. You have just written out Kepler's three laws. You have written out the second law in both its integral form and its differential form, but you have chosen to write the first law only in its integral form. You have then suggested that the differential form of the first law, which is the radial equation,

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

shouldn't be included in the article. Your argument above is totally specious. You are the one that went running to the adminstrator's notice board last April to complain when I was trying to clarify centrifugal force. And now you are intervening again because the radial equation which you don't want to be seen in this article contains the inverse cube law centrifugal force.

At least you are not denying the actual radial equation itself, but you have put forward some specious argument that it shouldn't be included because its conversion to the integral form and vice-versa is non-trivial. The conversion does actually appear in the Kepler problem page which you drew my attention too. That page shows us the differential to integral conversion, which is actually much more complicated than the integral to differential conversion, because the latter only involves differentiation.

Basically you want to hide any suggestion that centrifugal force is a real force. And you also want to hide any suggestion that the Coriolis force or the Euler force are real, which is why you don't like having names for the terms in the tangential equation. You haven't been able to suggest yet what the correct names should be in your opinion.

This argument has never been about sources. Sources are no problem. The radial equation above appears at 3-12 on page 74 of the second edition of Herbert Goldstein's 'Classical Mechanics', and you are not disputing it anyway. And below the equation at 3-22, Goldstein reminds us all, not that we need to be reminded, that the inverse cube law term in 3-12 is the centrifugal force.

So long as you continue to follow me around and revert my physics edits, there is not really much point in me continuing. I am trying to make these articles both easier to read and easier to understand, but you are clearly out to obstruct my efforts. David Tombe (talk) 20:26, 28 March 2009 (UTC)


 * David, like I said before, if you think I have dealt unjustly with you, take it up through the proper dispute resolution channels. I am willing to discuss these issues, in the proper place and through the proper channels. I refuse to answer or discuss your accusations on an article talk page. The talk page is for discussing the article, not for editors to hurl accusations back and forth against each other.


 * The argument between you and others may not have always been about sources, but it should be and I'm trying to establish that by refusing to argue physics with you but instead requiring reliable sources. Reliable sources are a basic principle of wikipedia. Rather than trying to teach each other why the other one doesn't understand physics, we should be bringing reliable sources to the table that support our edits. The radial equation 3-12 in Goldstein is actually really good. For those who might not have access to the book, the equation states
 * $$m\ddot r-\frac{l^2}{mr^3}=f(r)$$
 * where f(r) is the central conservative force (and the only force that Goldstein introduces when setting up the problem). Writing the radial equation this way is good because all the terms that come from $$\ddot\vec r$$ (the acceleration vector) are on the left-hand side, and the terms that come from forces are on right-hand side. However, Goldstein does not derive it from Kepler's laws, but from Lagrangian mechanics, so I fail to see why the reference supports inclusion of the equation in an article on Kepler's laws. By the time Goldstein gets to 3-22, he is discussing the equivalent 1D problem and he always prefaces statements about the centrifugal force as applying to the fictitious one-dimensional problem, which returns the same mathematical equation of motion in r as the real two-dimensional problem. Only in the fictitious 1D problem is the inverse cube term the centrifugal force; in the real 2D problem, as shown in 3-12, it appears on the other side of the equation (with a negative sign) as a contributing term that, along with $$\ddot r$$, gives the radial acceleration.


 * As for my opinion of what the correct names for those terms should be, it doesn't matter. My opinion is irrelevant without proper reliable sources. Do you have a reliable source that gives names to these terms? Do you have a reliable source that shows the radial equation derived from Kepler's laws? --FyzixFighter (talk) 22:25, 28 March 2009 (UTC)

FyzixFighter, Yes. Shankar 1994. And also Goldstein. The same reference which you have just tried to wriggle out of. Goldstein refers to that term in general as the centrifugal force. It took quite a bit of imagination on your part to believe that he only meant the name 'centrifugal force' to apply to the fictitious one-dimensional problem. What Goldstein was saying was that the radial equation as per 3-12 is the same as the fictitious one-dimensional problem. How could there possibly be centrifugal force in the one dimensional fictitious problem? That wouldn't make any sense. He was drawing attention to the fact that the one dimensional fictitious problem and the radial orbital problem are mathematically identical. They both lead to a stability node due to the two different power laws. In the actual orbital analysis, we have eliminated the tangential equation on the grounds of Kepler's second law, and then substituted the areal constant into the centrifugal term in the radial equation to get an inverse cube law term. The resulting radial equation is then the same as in the fictitious one dimensional problem.

It seems to me that you have been learning alot about this subject while reverting my edits. Would it not be more polite to ask questions on the talk pages first and to discuss possible amendments, rather than to go storming in and reverting edits wholesale when it is obvious that you knew very little about the subject at the moments when you did the reverts? David Tombe (talk) 17:55, 29 March 2009 (UTC)

Notation
I understand that this proof was originally given by Newton using his own notation, but is it really advisable for us to be using fluxions as well? Or would Newton turn over in his grave if we used his nemesis' notation instead?

Tebello TheWHAT!!?? 06:56, 8 April 2009 (UTC)

sorry
Ok sorry but i keep ending up here. I'm looking for the wiki page on a special counter/ratio/formula that explains the distances of the planets in relation to the plaet before but i cannot remember the name. Google keeps directing me here. If anybody could help, that would be much appreciated. 91.104.179.109 (talk) 01:10, 27 April 2009 (UTC)
 * See Bode's_Law. Bo Jacoby (talk) 18:03, 17 May 2009 (UTC).

Selective
Stephen Hawking only re-printed the fifth book of Kepler's Harmonice Mundi, because the first four books put Kepler in a bad light, being full of weather prophecies, attempts to link astronomy with music and the like. —Preceding unsigned comment added by 86.137.170.8 (talk) 10:12, 15 May 2009 (UTC)

File:Anomalies.PNG
The proportions of the drawing are misleading: The semimajor axis is |CZ| = a = 33 mm and the linear eccentricity is |CS| = 21 mm, corresponding to an eccentricity of 21/33=0.64. So the semiminor axis should be 33*(1-0.642)1/2 = 31.7 mm. Actually the semiminor axis is only b = 27 mm. Bo Jacoby (talk) 14:48, 28 May 2009 (UTC).