Hi, I'd like to take this article to FA status. I think it's correct and complete, but I would appreciate other people looking it over critically. Thanks! :) Willow 23:06, 8 November 2006 (UTC)[reply]

The article starts much too abruptly; the lead paragraph should not contain formulas but explain in words the essential characteristics and significance. Something starting along the lines of:

In orbital mechanics, the Laplace-Runge-Lenz vector describes a quantity that is useful for studying the orbital motion of a body, for example a planet or a particle. It is a vector, defined for an orbiting body, that depends on its position and momentum. The significance is that this vector is a constant of motion when an inverse-square central force — such as gravity or electrostatics — acts on the body.

 --Lambiam^Talk

Thanks for the insight, Lambiam! Mike Peel said as much, too. Do you think it reads better now? Willow 19:07, 9 November 2006 (UTC)[reply]

• In the first displayed equation of "Quantum mechanics of the hydrogen atom", is there any particular reason to use curly braces instead of just parentheses?
• I'd like to see a description of this SO(4) symmetry in the classical problem. In particular, is there a simple fourth generator of symmetry on the phase space? And even if there isn't a simple one, can we see what a complicated one would look like? Or, instead of an infinitesimal symmetry, how about an explicit function (x', p')(x, p, theta)?
• For that matter... In what sense, precisely, is the two-body problem equivalent to motion on a 4D hypersphere? I get the feeling from skimming the Baez external link that there's a lot of necessary information that isn't included here. Maybe we need a new stub, separate from this article, to take on the responsibility for this deep topic? Or is there already somewhere to look on Wikipedia?
• Would it be worthwhile to include a discussion of perturbations to the vector due to a small perturbing force? Or does that not turn out to useful in applications...? I mean, I know it's useful to consider perturbations to the individual orbital elements in celestial mechanics, but is it useful to cast the idea in vector form?

Melchoir 19:13, 9 November 2006 (UTC)[reply]

Hi, Melchoir, thanks for your review! I think it might be a good idea to discuss the effects of perturbations on the evolution of A here, but I'd actually have to read up on that, or rederive it from scratch. The same is true for the classical SO(4) symmetry, although I think it's just a simple stereographic projection from the four-sphere onto 3D. I'll let you know what I can dig up.

P.S. I fixed the curly braces; it was a holdover from an anticommutator. Willow 19:25, 9 November 2006 (UTC)[reply]

That's what I figured (about the braces)! The perturbations shouldn't be too hard to derive; it probably just involves expanding out the iterated cross products and hoping for a lucky cancellation to crop up. But I'd advise against rederiving it, both because of my general belief in WP:NOR and because if you can't find the derivation or at least a formula or two somewhere, then it probably isn't notable enough to include in the article anyway. Good luck with the rest! Melchoir 19:35, 9 November 2006 (UTC)[reply]

Thanks for your keen-sighted review, Jitse! I hope I've dealt with most of these objections, but if you disagree, please let me know. I'll read up on superintegrability as well. Willow 13:44, 13 November 2006 (UTC)[reply]

• You need to define k before you can define A.

OK, did that.

• For poor mathematicians like myself, mention that r = |r| and that you use Einstein summation.

Eliminated summation convention and specified typeface conventions in lead.

You didn't have to eliminate the summation convention for my sake, but I won't complain.

• "The orbits of the Kepler problem can be shown to be elliptical …" — they can also be hyperbolic or parabolic. Perhaps worth mentioning what the LRL vector is in those contexts? Also, I think that just before that sentence, you should say that the vector is conserved for the Kepler problem, and refer to the proof in Section 4.

Clarified that LRL vector pertains to all conic-section orbits of the Kepler problem. I'd been limiting it to ellipses to help the average reader visualize it.

Please check this edit. As I understand it, the eccentricity is 0 for circles, between 0 and 1 for ellipses, 1 for parabola, and greater than 1 for hyperbola (cf. conic section#Polar coordinates).

• When counting the degrees of freedom and constants of motion, mention that you're in 3d (since it's quite natural to consider the problem in 2d). The one interesting fact that I missed is that 5 independent constants of motion implies immediately that you've closed orbits (in the bounded case). I think it's called superintegrability.

I need to read up on this. I'm a little surprised to hear this conclusion, since it seems as though there are always six constants of motion (the initial conditions in phase space) but I'm willing to learn better.

I might have a look myself as I'm quite interested in how this works out. Just to clarify, I'm not claiming that I understand this stuff myself.

• Are the alternative scalings really important enough to spend a whole section on them?

I think so; an amateur reader is apt to encounter a different scaling or symbol in a textbook and then complain/change the article drastically. I'm trying to accommodate everyone and spare us all future work.

• "Analogous conserved quantities can be defined for other central forces" — this seems to mean that every central force has an analogous conserved quantity, but I don't think that's true (because other central forces do not have closed orbits). Do you mean that some other central forces also admit analogous conserved quantities?

No, indeed, there is a corresponding conserved quantity for all central forces, as shown in the Fradkin (1967) reference.

I saw that you added that they're multivalued functions. That's cheating ;) But that explains it; I was thinking about globally defined constants of motions. Jitse Niesen (talk) 01:08, 14 November 2006 (UTC)[reply]

• The section on "conservation and symmetry" is written in the context of quantum system (e.g., the many mentions of mixing of orbitals), but I think the discussion is also valid and important in a classical setting.

You're completely right. I was trying to avoid juggling both balls (classical symmetry and quantum mechanical symmetry) at once, but it's better this way. Thanks for pointing that out! :)

• I don't understand the second "connection to four-dimensional rotations". It should explain how a four-dimensional rotation acts on a six-dimensional vector (x,y,z,p_x,p_y,p_z) describing the state in the Kepler problem, I presume?

I'm sorry, I didn't explain that very well. The idea is that the orbits of a free particle on a four-dimensional hypersphere are mathematically equivalent to the Keplerian ellipses via a stereographic projection, and that rotations of the hypersphere correspond to continuous mappings of the Keplerian orbits of the same energy onto other Keplerian orbits of the same energy but different angular momentum. In Cartesian spaces, a rotation is always a mixing of two coordinates; hence, in d dimensions, there are _dC₂ rotations, i.e., d(d-1)/2 rotations. Thus, in four dimensions, there are six independent rotations, whereas in six dimensions, there are 15 independent rotations. I'll work on clarifying that later today; the whole section does need work. :( Willow 13:44, 13 November 2006 (UTC)[reply]

Jitse Niesen (talk) 07:07, 13 November 2006 (UTC)[reply]