Talk:Laplace-Runge-Lenz vector

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Contents 1 SO(4) 2 Explanation of edits (14 June 2006) 2.1 A few technical points 3 So what is it? 4 Kepler's problem 5 Symmetry 6 Improvements to the writing 7 Turning a new leaf 8 Symmetry 9 Accessibility 10 The O(4) symmetry 11 Ummm... 12 few quick comments 12.1 New stuff 13 More comments 14 Include reflection symmetry?

[edit] SO(4)

Hi, I reverted the "burial" of SO(4). Its not that the Runge-Lenz is "related to SO(4)", its rather that the symmetry of planetary motion is SO(4). Its easy to understand the SO(3) part of planetary motion, and Runge-Lenz gives up the harder-to-see part of it. linas 17:08, 13 June 2006 (UTC)

In fact, the article already disucssses this, in the mis-titled section called "applications to quanum mechanics" In fact, the theory applies to the classical motion as well, not just the quantum motion. linas 17:30, 13 June 2006 (UTC)

I whacked on that section so as to give a name for the actual group that the commutators generate, and to make it clear that it applies to classical mechanics, not just to quantum mechanics. Perhaps you'll find these edits acceptable. linas 18:03, 13 June 2006 (UTC)

[edit] Explanation of edits (14 June 2006)

Hi Linas, I agree that the SO(4) pertains classically as well as quantum mechanically, and deserves mention in the introduction. However, I reduced it to a "teaser" , for the following reasons.

My general approach to writing scientific articles is to "lay out a honey trail to enlightenment", that is, to begin very simply and gradually draw the reader onwards to ever greater complexity, all the while mixing the new with the familiar. This approach may be frustratingly slow for experts, but I believe that our target audience for this article consists mostly of people who are

• interested in physics and/or astronomy
• have a basic knowledge of algebra and calculus (~1 semester course)

Relatively few will have had any group theory, even fewer will understand the full Pioncare group and, I daresay, only experts will understand the SO(4) symmetry of the Lagrangian and how the LRL vector follows from it, especially given our (presently) terse explanation. I'm not saying that we shouldn't write for those few, on the contrary; however, I believe that we need to "ramp up" to that level gradually, to keep our non-expert readers engaged for as long as possible.

Does that seem sensible to you?

[edit] A few technical points

Strictly speaking, the LRL vector is not conserved exactly for real planetary motion. The orbit of every planet around a star precesses inevitably due to several factors, e.g., the corrections of general relativity, the imperfectly spherical distribution of mass in the star and the presence of mass (such as cosmic dust) between the star and the planet. Thus, the LRL vector is conserved exactly only in the two-body problem of classical gravity or electrostatics.

Your edit note suggests that SO(4) symmetry is the only reason for the importance of the LRL vector. I can't really agree with that, since the LRL was being used for planetary motion long before SO(4) was ever formulated. Perhaps that's not what you meant, though. Out of curiosity, when was the relationship between the LRL and SO(4) first pointed out explicitly? It'd be nice to find the original reference for the article. WillowW 09:59, 14 June 2006 (UTC)

[edit] So what is it?

So, what fundamentally does the Laplace-Runge-Lenz vector represent? Are there intuitive ways of thinking about it that the average person would understand? If so, they should be given in the first few sentences of the article. Mike Peel 19:08, 5 November 2006 (UTC)

Hi, Mike, thanks to your and Lambiam's suggestions, I tried to make the lead less math-y and more intelligible to the average reader. Does it read better now, do you think? Do you have any other suggestions before the article ventures out onto the dark, dangerous waters of peer review? Thanks for your help! Willow 19:13, 9 November 2006 (UTC)

Much better, thanks. I've just moved some content from the start section to an Introduction section, as it was getting a bit long and complicated in places. I guess that the first couple of sections ideally want to be orientated at the lay person, providing a gradual buildup to the more complex stuff later on in the article. Mike Peel 08:55, 18 November 2006 (UTC)

[edit] Kepler's problem

What is "Kepler's problem"? Is it related to Kepler's laws of planetary motion, or the Kepler conjecture? Mike Peel 08:52, 18 November 2006 (UTC)

[edit] Symmetry

The vector A is constant, because the Kepler problem has an unusually high symmetry; it has SO(4) symmetry, whereas most central force problems of classical mechanics have only SO(3) symmetry.

I doubt this is true; there should be such vectors, differently defined, for most if not all central force problems. In any case, it does not belong in the introduction. Septentrionalis 23:06, 20 November 2006 (UTC)

Hi, Septentrionalis Nice name :) — "a person characterized by 73", did I guess right?

It is indeed true that the Kepler problem has a higher symmetry than most central-force problems, as I tried to describe later in the article. That's why, for instance, the energy levels of the hydrogen atom depend only on n and not on the other quantum numbers, l and m. That is not true, for instance, if other potentials are added, such as those that cause level splitting in hydrogen.

I have to mention it in the introduction, because it is discussed at great length in the article itself (please see Wikipedia:Lead section). I agree with the method of gradually working up to ever more complex concepts in a WP article, and I realize that the symmetry sentence might be daunting to beginners. But it might also lure them onwards with curiosity and, besides, it's only one sentence. I hope that you'll understand my reasons and forgive my reversion. Willow 23:22, 20 November 2006 (UTC)

P.S. That reason also holds for including in the intro the precession of A under a perturbing potential, which is granted a whole section of the article, as requested in the scientific peer review. Secondly, the answer to the question "why should I care that A is constant?" — namely, that it lets one solve the Kepler problem by elementary geometry, rather than solving a differential or integral equation — also seems pertinent to the introduction. Sincere apologies on our difference of opinion, Willow 23:44, 20 November 2006 (UTC)

Let me be clearer; it's the word because I don't believe. Septentrionalis 23:28, 20 November 2006 (UTC)

OK, I'm beginning to understand your point now; you're referring to the Fradkin (1967) paper referenced in the article and discussed in the scientific peer review? Namely, that other, possibly multi-valued functions can be defined for other central forces? Would you please describe your objection in more detail, so that we can find a good way to resolve it? Thanks! :) Willow 23:44, 20 November 2006 (UTC)

I didn't read Fradkin; I did the mathematics. As this article proves,

.

If the right-hand side is integrable to g(r), and in general it will be, consider the vector p x L - g(r). This will not be A; call it B. B will be a constant of the motion. Septentrionalis 23:58, 20 November 2006 (UTC)

Please allow me to encourage you to read the Fradkin reference, which treats this problem at length. The approach is valid, but g(r) is generally not a simple function of position. Have to run now and stalk some hapless mountaineers, but I'll check in tomorrow. Thanks for your comments and insights! The Abominable Snowgirl 00:09, 21 November 2006 (UTC)

[edit] Improvements to the writing

Introducing bad writing because you think FA will require it is abominable. In fact, what WillowW cites is a suggestion from a guideline, not a requirement. FA requires that articles be comprehensible. Introducing SO(3) as though it were more than symmetry about a central point is jargon, justly deprecated. Septentrionalis 23:39, 20 November 2006 (UTC)

I'm speaking from my experience on Photon, in which I was required to add a paragraph to the lead to describe a section on the technological applications of photons. I believe that was right and a desirable thing to do, wouldn't you agree? Here, we have more than one section on the SO(4) symmetry, which you'll find is very important to some mathematicians (see above).

In Photon, the added sentence is unobjectionable, and therefore a benefit to the article. Guidelines (and FA when it's working) impose things that should be done, unless there is a good reason not to.Septentrionalis 00:23, 21 November 2006 (UTC)

That said, I'm perfectly willing to replace SO(3) and SO(4) with more accessible language. Is that the main problem, or the Fradkin issue, or maybe both? Just wondering, The Abominable Snowgirl ;) 23:50, 20 November 2006 (UTC)

You reverted edits intended to solve three problems.

• The Fradkin issue, where the text was (at best) misleading, and I suspect false; if there is a constant vector B without SO(4), the text is wrong
• "SO(3)" does not belong in the lead of an article, unless Lie group is preliminary reading; especially here, where "rotational symmetry of space" is so easy.
• A should not be used before it is introduced. Septentrionalis 00:11, 21 November 2006 (UTC)

In this edit, I think "any" is what you want; but do think about it. Septentrionalis 00:15, 21 November 2006 (UTC)

[edit] Turning a new leaf

Hi Septentrionalis,

I couldn't resist flipping open my Lewis&Short as soon as my family was asleep, and discovering that your Latin name means "of or pertaining to the North", presumably due to the related word septentriones, the seven stars of the Big Dipper (Great Wain) and I guess Polaris. Fun name! :) I'm a Latin fan as well; you might enjoy some of my translations and original texts at Wikisource and Vicifons, although I know my own limitations and would welcome any suggestions on them. I'll confess, though, that the music of Greek — especially Homer and Sappho — has always appealed to me more than any Latin text.

I probably won't have much time to edit today, but I thought we might turn over a new leaf. I really wasn't aware that I had destroyed your efforts to clarify the above three problems; from my perspective, I was adding back explanatory text that had been deleted from the lead section. I firmly agree with your goal of making the article as well-written, precise and accessible as possible, and I hope that we can work together to unite our differing perspectives.

My sense is that the because issue is semantic. My intended meaning was: "Given such-and-such a symmetry, such-and-such a quantity is conserved." It says nothing about what might be conserved under different symmetries. Perhaps we might eliminate the causal confusion by associative phrasing, e.g., "The conservation of A corresponds to (or is asociated with) a higher symmetry"?

I'm perfectly happy to remove the technical name SO(3) from the lead; it's much more accessible and less daunting without it.

I'm not sure about which A is being introduced before it is defined — do you mean Figure 1? It seems appropriate to mention in the caption, don't you agree?

Serenely, Willow 16:54, 21 November 2006 (UTC)

I have much Latin "and less Greek"; but I consider myself a Hellenist. As for the name, I mean Northerner in the strict American sense.

I have no problem with Figure I; but I hope what is now said about geometry in the lead is sufficient. Anything more definite belongs in Introduction, not the lead IMHO. Septentrionalis 18:14, 21 November 2006 (UTC)

Shakespeare is great, too! :) Happy to hear that Figure 1 is OK. :) Willow 20:51, 22 November 2006 (UTC)

[edit] Symmetry

I read Fradkin, and have good news and bad news.

• You don;t have worry about "multivalued"; what he means is that θ is multivalued as a function of r; i.e. that the period does not divide 2п.
• On the other hand, he proves that all central forces have O(4) and SU(3) symmetry. This means that most of what this article says about symmetry is wrong.

Septentrionalis 06:59, 22 November 2006 (UTC)

I don't believe that what the article said was wrong per se, but I understand how the SO(4) arguments might be misconstrued. Please review the additions/corrections/refinements I made today and let me know what you all think. We might be ready for FAC next week, do you all agree? I'll probably be busy for the next few days with Thanksgiving, though. See you all, Willow 20:51, 22 November 2006 (UTC)

I do think parts of the article are wrong, i.e. the sentence "The higher symmetry results in the conservation of both the angular momentum vector L and the Laplace-Runge-Lenz vector A." I thought that this was the case, and I might have misled you in this regard, but the fact that all central forces have O(4) symmetry means that it is at least more subtle than a simple application of Noether's theorem. I should have realized this immediately, because O(4) is six-dimensional so Noether's theorem would lead to six functionally independent conserved quantities, but there are only five. So, can you explain (or point to a reference explaining exactly how the O(4) symmetry leads to conservation of L and A?

Needless to say, this should be resolved before thinking of FAC. -- Jitse Niesen (talk) 22:39, 22 November 2006 (UTC)

You raise a really good point with the 5 vs. 6 conserved quantities. Of the five, four are defined by the energy and the angular momentum vector; so the LRL vector seems to correspond to only one additional symmetry. What do you all think of the hodograph inversion/rotation mapping? To me, the key is being able to find a mapping that transforms orbits of the same total energy but different angular momentum into one another; but whether that mapping will turn out to be SO(4), I haven't thought through yet.

In most papers, the identification of SO(4) seems to follow from the Lie algebra defined by the Poisson brackets; with the exception of the Rogers reference, no one seems to spell out the mapping explicitly. I'll keep looking, though; the 2nd external link is a promising review paper.

Terribly tired from cooking tonight, but looking forward to understanding this all, Willow 05:59, 23 November 2006 (UTC)

Fradkin derives L, A and the symmetries, both in the special case of (relativistic) inverse square forces and for general central forces. He doesn't expressly write the O(4) symmetry (and it must be O(4), not just SO(4); the reflection of any orbit is an orbit) but he sets up the Lie algebra. I'm not a physicist, and it's not clear to me what the rotation is; but it looks like it should be simple enough to explain if I really understood this.

I suspect, without reading everything (and I can't on Thanksgiving anyway) that five versus six is whether to count the energy as an invariant or take it for granted; so there should be two more after the energy and the angular momentum. Septentrionalis 17:02, 23 November 2006 (UTC)

The other review paper wasn't that helpful in understanding the O(4) rotation, unfortunately. :(

Six dynamically independent constants might be one too many for the six-dimensional phase space, being more than maximally superintegrable. The orbit would presumably be reduced to isolated points, right? That suggests two important considerations:

• Maybe we're dealing with a restricted set of O(4) transformations? Presumably, there are other rotations of the orbits on the 4-sphere that change the energy of the stereographically projected orbits as well.

• The vector A really adds only one dynamical constraint to those of E and L, which in turn suggests that we look for a single-parameter transformation that is independent of the normal O(3) spatial rotations. I like the Möbius transformation of the hodographs, but the Rogers reference has another type of pseudo-4-rotation. I'll also go check out the old Fock and Bargmann papers, they probably have some good insights.

Talk to you all soon, Willow 16:37, 27 November 2006 (UTC)

[edit] Accessibility

I really think this article needs an explanation of the vector with as little technical machinery as possible. This includes such machinery as the angular momentum vector. Everybody writing this article knows what one is; but our readership doesn't. (For those who care about it, this will include the readership which is going to review any FA nomination, btw. See Wikipedia:Featured article candidates/General relativity, where they are complaining of unit vector.)

If you don't think there can be such a thing as excessive precision, wait until we have the pedant who insists (correctly) that angular momentum is really not a vector, but a bivector.

Comments? Septentrionalis 23:42, 27 November 2006 (UTC)

I agree that we'll need to make the article more accessible to our readers, but first we should make it complete and correct, don't you agree? Then we can figure out how to lay out that "honey trail to enlightenment". Along those lines...

We should make it correct; we can hardly make it complete. But the context section is correct now. It is also clear to you, me, Linas, and Jitse. Let me get back to you with a demonstration.... Septentrionalis 22:51, 28 November 2006 (UTC)

[edit] The O(4) symmetry

...I think I might understand the O(4) symmetry of the motion. The momentum hodograph approach was indeed correct, but the true mapping is much simpler and more direct than that cute little Möbius transformation. Rather, the mapping is a straightforward rotation of the 4d sphere that mixes the p_y and p_w components. The key insight is that the 4d orbit is a great circle that intersects the p_x axis; thus, rotation about that axis, followed by stereographic projection, produces all the Apollonian circles directly. The foci p_x=±p₀ are pinned throughout, lying as they do on the rotation axis. As in the original description, the normal O(3) rotational symmetry can be eliminated by choosing the Cartesian coordinates such that p_z is aligned with the angular momentum vector L (and, thus, p_z=0); by considering only the three-dimensional unit sphere defined by (p_w, p_x, p_y), the symmetry also becomes more visualizable to the average reader. I'll try to add some Figures and text to explain this better over the next few days. Willow 19:18, 28 November 2006 (UTC)

Good. What is p_w? Septentrionalis 22:51, 28 November 2006 (UTC)

Oops, sorry, I changed the variable names a few edits ago. I mean η_w; the w component is the fourth dimension in addition to the normal x, y, and z. Rotate a great circle on a 3d unit sphere that intersects the x-axis about the x axis, and project stereographically from the North pole unit vector w (1, 0, 0) onto the x-y plane and you produce a family of Apollonian circles with foci at x=±1. The eccentricity e of the orbit equals the sine of the dihedral angle between the great circle and the x-y plane. Hoping that this helps to clarify the transformation, Willow 23:24, 28 November 2006 (UTC)

[edit] Ummm...

...does anyone have any comments on all the material I've been adding — perhaps you're waiting until I've finished?

Confused by the silence, but still looking forward to FA status for our favorite vector, Willow 19:57, 1 December 2006 (UTC)

Just a short note that I'm still planning to look through your changes; unfortunately, I don't know when I'll find the time. -- Jitse Niesen (talk) 02:53, 6 December 2006 (UTC)

[edit] few quick comments

Sorry this isn't more thorough; I've been kind of pressed for unbroken blocks of time lately, and this article needs a bit of time to digest :)

• These diagrams are exactly what people want converted to SVG. The current PNGs look quite pixelated on my screen; SVG will scale better.

I totally agree, but it's a little hard for me; my old version of Xfig doesn't do SVG (well). I'll try to figure something out.

• Minor wording issue, but: for some reason, I don't like 'overview' and its variants in headings; an 'overview' is what the lead is for.
• "The Laplace-Runge-Lenz vector A is defined below for a single point particle of mass m..." in the overview section - why not just put this in the mathematical definition section with the content it describes?
• Does "Kepler problem" = "two-body problem", as suggested by the piped link in the Kepler orbits section? If so, explicitly say that, perhaps in the lead, where the linking goes to Kepler but not two-body problem.
• In the 'conservation under inverse-square forces' section, the sentence 'none are as conserved as A' may be a bit ambiguous, since it's referring to the A defined immediately above rather than the general A used in the rest of the article. Maybe use a subscript?
• Same section, 'these constant vectors are multivalued functions of the angle θ' - though it's not hard to figure out, θ hasn't been explicitly introduced in the preceding derivation.
• I have a dim memory that the precession of Mercury agreeing with Einstein rather than Newton was a notable event. If what I'm thinking of is in fact what's described here, this section could be expanded to describe that discovery in more detail, which would also provide a bit of a practical touchstone for those who aren't just here for the vectors.

My memory is that the anomalous precession of Mercury was known well before Einstein did his calculations and, if I recall correctly, he cited it in an early paper on general relativity. I think the historical discovery that made the news was that Eddington expedition showing that the bending of starlight in a gravitational field was twice what Newtonian gravity would predict, and agreed with Einstein's predictions.

See Vulcan (hypothetical planet). Septentrionalis PMAnderson 22:55, 11 December 2006 (UTC)

Thank you, Septentrionalis, I do fondly recall Le Verrier's initial hypothesis for the anomalous precession; the planet Vulcan was a popular conceit of the science fiction of my youth. ;) There's also no doubting that Einstein's 1915 paper and the nearly exact agreement between his prediction and experiment did make a splash for general relativity. However, since the magnitude of the precession was known beforehand, an ungenerous scientist might have suspected that Einstein tinkered with his theory to produce the agreement. The starlight bending was a more powerful result, since it was not known beforehand, don't you agree? It seemed to get more press coverage, at any rate. Willow 23:23, 11 December 2006 (UTC)

I think I had these two events conflated in my mind. I'm not familiar with the history, but just a bit more on this - ie, the fact that it was a notable 'post-diction' for general relativity - might be an interesting aside. Opabinia regalis 06:26, 13 December 2006 (UTC)

• On that note, there's a lot of derivations in the middle (up to 'lie transformation') with little connection to their practical uses, ie, what is Noether's theorem used for? I think a bit more on what the practical applications of these vectors to physical problems would be useful, as several sections of the article seems to be spent going through alternative definitions and derivations, which can get rather abstract and cause the reader to forget why he wanted to read this article in the first place.

That is a crucial point; thanks! I'll work at making that part more sensible.

Opabinia regalis 06:06, 7 December 2006 (UTC)

I think I may have fixed some of those up — does it read better now? Thanks super-muchly for taking the time to review it! :D Willow 18:45, 7 December 2006 (UTC)

Looks nice! The main concern I have, I think, is that Noether's theorem and the Lie transformations both read as rather disconnected from the preceding and following sections. The Lie transformation in particular jumps back to Kepler's third law after some discussion of the applications of these vectors in QM. I can see that Noether's theorem comes into the conservation and symmetry section, but it might be better placed as a subsection? Or a reference in the Noether's theorem section to the fact that the implications are discussed further below? Opabinia regalis 06:58, 9 December 2006 (UTC)

[edit] New stuff

Separating this out for clarity (mostly nitpicky stuff, so the article must be good :)

• Much prefer the current positioning of Noether's theorem and Lie transformations.
• I'm not sure about the positioning of the Hamilton-Jacobi section. It comes between two sections discussing quantum, but the section itself only describes another way of deriving the vector's conservation. Seems like it might go more logically before the hydrogen atom section - possibly even before Poisson brackets, since that section follows rather clearly into the next.
• Inconsistent wikilinking. Quantum number seems like an obvious candidate but isn't (perhaps because this text is new, I think?). Yet the rotational symmetry section wikilinks things like Cartesian coordinates (twice) and vector - subjects that no one who makes it that deep into the article will need clarification on.
• I've never thought about this before, but do we have a label system for figures yet? I doubt this article will get too much editing after you guys are done with it, but it worries me to see things like "figure 7" in plain text, easily lost or obsoleted by the addition of a new figure or removal of an old one.
• Picky writing question: was the Schrodinger equation "discovered" or "invented"? Opabinia regalis 06:26, 13 December 2006 (UTC)

[edit] More comments

I have some more comments, but no time now to write them down. Just quickly, it's much better now than two weeks ago. I didn't find any major issues However, I think the connection between the sections "Conservation and symmetry" and "Four-dimensional rotational symmetry" on the one hand and the rest on the other hand can still be tightened. More tomorrow. O yes, Willow, I have a recent version of XFig, so if you mail me the xfig files I can convert them easily to SVG. -- Jitse Niesen (talk) 12:09, 10 December 2006 (UTC)

I made some corrections, which I expect to be uncontroversial, myself. I split it in lots of small edits so that I could justify them individually; hope you don't mind. Here are some more comments:

• In the first sentence, I think the fragment "a special case of the two-body problem" can be removed.

Umm, that one I'd like to keep, since it introduces the concept of "two body problem" and, indeed, clarifies that there are two bodies interacting. Forget that, you are totally right. It's much better this way.

• In the first figure, use the proper symbol for the cross product, i.e., write p×L instead of pxL.

OK; did you fix this already?

• This may be controversial, but in my opinion you should not include inline cites in the lead section. They distract the reader and they do not help with verifiability because the references are mentioned in the body of the article.

Maybe, I suppose; I don't have strong feelings about it. My thought was that the lead is meant to function as a "mini-article", so I thought that it should have the key references as well. Let's both look into this, and find out what's customary!

• The sentence "Similarly, there is not consensus for its symbol, although A is used most commonly." seems to be not important enough to appear in the lead.

Sure, that makes sense.

• Typographical nitpicking: In my opinion, one should use en dashes as in Laplace–Runge–Lenz vector and not hyphens as in Laplace-Runge-Lenz vector (look at the length of the dashes).

My keyboard doesn't have an en-dash key; is there a simple way besides writing "& n d a s h ;"?

It's on the edit screen, above the Greek alphabet; look for Insert. Septentrionalis PMAnderson 22:50, 11 December 2006 (UTC)

• Section "History of rediscovery": what is the word "anschaulich" doing here?

Intuitive clarification without intuitive clarification. ;) (apologies to J. A. Wheeler)

• Section "Mathematical definition": I think it's clearer to write for the definition without using the middle expression. The middle expression is not needed, and the two equal signs in have a slightly different meaning (definition and equality, respectively).

Sure, that's fine; I introduced the longer equation back when I hadn't defined the unit vector, as I recall.

• I'm used to adding a full stop after formulas if they end the sentence, e.g., the formula . However, I have a vague memory that this is an instance where different conventions are followed in maths in physics.

I would really prefer to not punctuate the indented equations, since it might confuse lay-readers, rather than edifying them, which is the purpose of punctuation.

• Section "Alternative scalings ...": Is it necessary to say "However, the choice of scaling and symbol for the Laplace-Runge-Lenz vector do not affect its conservation"? Perhaps somebody who made it this far will realize this themselves?

It is ghastly, I know, but I felt that it had to go in there, since even college students (such as me, once upon a time — how embarassing! ;) don't always "get" this, to say nothing of lay-readers.

• The phrase "dyadic tensor" sounds rather quaint and outdated to me. Is it still being used in physics?

Indeed it is, albeit rarely. I used "dyadic" and the corresponding notation to clarify the relationship between W, A and B.

• In the last formula in this section, you take the inner product of L with both W and A. Do you consider L as a dyadic tensor on the left-hand side and as a vector on the right-hand side? If yes, that's rather confusing, if no, I don't know what L ⋅ W is supposed to mean.

In a dyad, the two vectors stand "next" to each other in a specific order., e.g., X Y. Taking the right dot product with another vector Z results in the vector X scaled by the dot product Y⋅ Z, that is, X (Y⋅ Z). Conversely, the left dot product of the dyadic tensor X Y with Z results in the vector Y scaled by the dot product Z⋅ X, that is, (Z⋅ X) Y. That's why dyadic tensors are still around; they're useful for presenting some tensor calculations (contractions) in a brief way.

• Section "Evolution under perturbing potentials": I'd call them perturbed potentials.

OK — did I fix it correctly?

• Section "Poisson brackets": Are you sure that the sign of [D_i,D_j] depends on the energy? I find that rather odd. It also raises the question what happens what the energy is zero.

Yes, it's odd but I'm sure that it's true; it's because of the absolute value in the denominator or, seen another way, because of the radical in the denominator. The E=0 case is annoying but I know of no other definition. The hodograph argument still works, though; the circles are tangential to the p_x axis.

• Section "Quantum mechanics ...": Is it necessary to use the same symbol for the ladder operator as for the Laplace-Runge-Lenz vector?

No, but it's customary, along with the subscripts. What letter would you prefer? I'm game. ;) (Added later: do you like J?)

• What does "the eigenstates of the first Casimir operator are n^2-1" mean?

Oops, I mean, the "eigenvalues of the"; that's how we derive the Rydberg formula.

• Section "Lie transformation": It's not clear what "the above equation" refers to.

It's that formula for A² in terms of L and E that follows immediately; should I re-word that for clarity? (Added later: did so — is it any better?)

• Section "Rotational symmetry ...": If the problem has an easy formulation in action-angle variables, perhaps you could include it in the article. I'm intrigued by this remark.

I'd be happy to, but I do give the reference and the article is already somewhat long and technical. I'm sensitive to Septentrionalis' criticism that the article is too technically exalted to make it as a FA, and would prefer to keep such details to a minimum.

• How come that the coordinates x, y and z are not symmetric in the action-angle variables?

If I recall correctly, symmetry is broken by aligning z with the angular momentum, as in the above derivation.

• You use β at three places: when defining the rank-2 tensor W, when discussing the Hamilton-Jacobi equations, and in the action-angle variables. Are they the related?

They're totally unrelated, and you raise a very good point. I'll try to replace them with disambiguous symbols.

That brings me to the end. I trust you're not too disheartened by the length of the list ;) -- Jitse Niesen (talk) 12:55, 11 December 2006 (UTC)

Not at all disheartened! I'm very appreciative of the time and care you took with it, and for your fresh insights. If we do ever make it to FA, it will be in good part thanks to your efforts! :) Willow 18:01, 11 December 2006 (UTC)

[edit] Include reflection symmetry?

Am I wrong in thinking the symmetry groups should be O(4), O(3,1), and O(3) throughout? The difference is not much if so, and they have the same Lie algebras; but I don't see why reflection is excluded. Septentrionalis PMAnderson 23:02, 11 December 2006 (UTC)

I have no real objection to including them if you'd like. It's just that the conservation laws are derived from continuous infinitesimal symmetry transformations (check out the Noether and Lie sections), whereas reflection is a discontinuous operation. To me, that suggests that the reflection symmetry is irrelevant for showing that the Laplace-Runge-Lenz vector is conserved, and therefore a distraction. But what do you think? Willow 23:12, 11 December 2006 (UTC)

Yes, the conservation laws derive from the Lie algebras, which are the results of the principal component of the Lie groups; and so the improper component makes no difference there. But it seems to me we will have two classes of readers;

• Those who don't know exactly what SO(3) is, who won't care.
• Those who do, who will see that O(3) is more than necessary for the proof, which depends on o(3); but who may wonder why we exclude improper rotations. I think the article on the groups is less than clear on this point. Septentrionalis PMAnderson 06:12, 12 December 2006 (UTC)

Hi, Septentrionalis, do you like new paragraph? Your point is very well-taken, and I tried to address it there. Willow 20:07, 12 December 2006 (UTC)

Well, I was thinking of just dropping the S's, but flattery will get you everywhere. I think I've simplified things a bit; but feel free to revert. Septentrionalis PMAnderson 04:35, 13 December 2006 (UTC)

Well, let me flatter you some more — they were excellent edits! :)

Compliments aren't flattery if they're sincerely meant, no? You've made some excellent additions and changes to the article over the past month, making it much more accessible, insightful and accurate than it was earlier. Doesn't it seem like we've all blended the best parts of our differing perspectives, in the best tradition of Wikipedia? I tend to be over-enthusiastic and warmly approving, but I suspect that even the worst curmudgeon would concede that we've all done well with this article. We might do better still; but I think we can pause for a moment and be proud of our work. :) Willow 15:45, 13 December 2006 (UTC)