Talk:Laplace–Runge–Lenz vector

message: 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)
replies: text: 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)
replies: text: 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)
thread_title: SO(4)

message: 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?
===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)
thread_title: Explanation of edits (14 June 2006)

message: 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)
thread_title: A few technical points

message: 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)
replies: text: 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)
replies: text:  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)
thread_title:  So what is it? 

message: 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)
thread_title:  Kepler's problem 

message: 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)
replies: text: 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)

text: 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)
thread_title: Improvements to the writing

message: ...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 py and pw components.  The key insight is that the 4d orbit is a great circle that intersects the px axis; thus, rotation about that axis, followed by stereographic projection, produces all the Apollonian circles directly.  The foci px=±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 pz is aligned with the angular momentum vector L (and, thus, pz=0); by considering only the three-dimensional unit sphere defined by (pw, px, py), 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)
replies: text: Good. What is pw? Septentrionalis 22:51, 28 November 2006 (UTC)
thread_title: The O(4) symmetry

message: ...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)
replies: text: 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)
thread_title: Ummm...

message: 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)
replies: text: 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)
replies: text: 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)

text: That is a crucial point; thanks!  I'll work at making that part more sensible.
Opabinia regalis 06:06, 7 December 2006 (UTC)

text: 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)
replies: text: 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)
thread_title:  few quick comments 

message: 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)
thread_title: New stuff

message: This definition 
:Laplace-Runge-Lenz vector is a constant of motion in a key problem of classical mechanics: Kepler's problem...
does not look complete. May be it is better to write
:Laplace-Runge-Lenz vector is a constant of motion for a particle moving in Kepler or Coulomb potential.
or similar to that. Alexander Mayorov 21:45, 16 December 2006 (UTC)
thread_title: Definition

message: The Derivation of the Kepler orbits, Circular momentum hodographs, and Poisson brackets sections contain no references.  Could you please add references that show the derivations (just to demonstrate that the derivations are not original derivations)?  Thank you, Dr. Submillimeter 12:19, 31 December 2006 (UTC)
thread_title:  Unreferenced sections 

message: This phrase is not correct, because the observed precession of Mercury is about 5600" per 100 years, but the value 43" per 100 years which we are interested is just a general relativity correction unexplained by Newtonian gravitation. Alexander Mayorov 11:59, 13 January 2007 (UTC)
thread_title: "which closely matches the observed orbital precession of Mercury"

message: I found the Poisson bracket formula
 [ D_i, L_j ] = \epsilon_{ijs} D_s 
confusing at first.  I had expected
 [ L_i, D_j ] = \epsilon_{ijs} D_s 
since this says (to those who know how to read it) that the Runge--Lenz vector transforms like a vector under rotations.  A moment's scribbling convinced me that the formulas are equivalent.  But, I think mine would be clearer to people in the know.  
Also, why not use the usual notation for Poisson brackets?  Why Lie brackets?  Maybe 'cause nobody knows how to draw curly brackets on this system?
John Baez (talk) 17:46, 6 March 2008 (UTC)
thread_title: Switching conventions

message: I would not dare to add something to such a distinguished article, but for the fun and the readers interested in geometric algebra (see Hestenes "New Foundations for Classical Mechanics") I would like to show how easy and instructive is the manipulation of the LRL vector with that mathematical tool.
First some self explaining definitions and relations :
:r=\rho r \qquad \qquad \dot r=\dot\rho \hat{r}+\rho \dot \hat{r} \qquad \qquad  \hat{r} \cdot \dot{\hat{r}}=0
:r \wedge \dot r=\rho^2\hat{r} \wedge \dot{\hat{r}}=\rho^2\hat{r} \dot{\hat{r}}
Then we show that the bivector L is constant.
:L=m r\wedge v=m r \wedge \dot r=m \rho^2\hat{r} \dot{\hat{r}}\qquad \qquad \ddot{r}=-k m^{-1}\rho^{-2}\hat{r}\qquad \qquad \dot L=m(\dot r \wedge \dot r+r \wedge\ddot{r})=0
And finally we find the eccentricity vector :
:\dot(Lv)=L \dot{v}=[m \rho^2\hat{r} \dot{\hat{r}}][-k m^{-1}\rho^{-2}\hat{r}]=-k\hat r \dot \hat r \hat r=k\dot \hat r 
:\dot (L v-k\hat r)=0\qquad \qquad Lv=k(\hat{r}+e)\qquad \qquad L=\lambda \hat L\qquad \qquad L^2=-\lambda^2
And now  we can do something which is not possible in classical vector algebra ; we can immediately calculate v :
:v=k L^{-1}(\hat{r}+e)=-k\lambda^{-1}\hat L(\hat{r}+e)
Of course that gives us immediately a coordinate free equation of the hodograph. One interesting feature, seldom cited, is the fact that apart of a dilatation factor the velocity vector v is deduced from the vector (\hat{r}+e) by a 90 degrees rotation, where \hat L acts as the rotation operator. That is a lot more illustrative than trying to visualize the cross product ...
Chessfan (talk) 10:00, 29 May 2010 (UTC)
replies: text: It is plainly unsound to claim that a direct calculation of the momentum is inaccessible to standard vector algebra: you simply take the cross product of the first eqn in the Circular momentum hodographs section with L; and use the textbook identity to obtain an eqn for p L2. That hodographic equation is coordinate independent. I fear you are rewriting the equations of the article in a language whose obscurity prevents you from identifying the underlying transcriptive identity of the two.              Cuzkatzimhut (talk) 15:16, 2 June 2010 (UTC)
thread_title: Why not geometric algebra ?

message: Hi, I'm sincerely confused by this article, and I'd be grateful if you could answer a question or two.  Thanks!
My question is whether this template is working?  Thanks! Willow (talk) 00:58, 11 July 2010 (UTC)
replies: text: I've removed it and said why here. --JohnBlackburnewordsdeeds 01:02, 11 July 2010 (UTC)
thread_title:  Clarification needed on [[Laplace–Runge–Lenz vector]] 

message: Wikipedia has featured many great articles about technical matters, but I'm afraid this isn't one of them.  As I look over this for the first time, I think...
# The lead tells us that "the Kepler problem is mathematically equivalent to a particle moving freely on the boundary of a four-dimensional ball,", which is really kewl, if only I understood it or the section that explains it.  But... the lead does not tell us what the LRL vector is, mathematically speaking.  Might I suggest something like "A vector directed along the major axis of an ellipse, pointing in the direction toward which the central body is displaced relative to the center of the ellipse"?
# The next section tells us that "Whereas for typical conserved quantities, there is a corresponding cyclic coordinate in the three-dimensional Lagrangian of the system, there does not exist such a coordinate for the LRL vector."  WETF that means.  But where's the explanation?  Oh, yeah, and where's the inline citation if I wanted to go hunting for the explanation?
# After that we finally get the mathematical derivation.  It might be a bit simpler in spots - it might be worth saying p x (r x p) - mkr/|r| - oh, yeah, but there's a trouble.  What is k?  Does it say anywhere in the article how to calculate k?
# The figure is a noble effort but falls short - for example, where the legend says that the angular momentum vector L is perpendicular to the orbit, and people look and see that p x L is perpendicular to the orbit at each spot (in 2-D), they're going to get confused.  And the figure shows just one orbit - the question is, how does A vary over all orbits?  I assume that for an elliptical (i.e. inverse square) orbit that A will be a vector related to the displacement of the focus to one side of the center of the ellipse.  (Hmmm, is there some simple way to calculate A as a scalar function of this displacement?)
# There's an effort here to express the generality of this to non inverse square forces, but is there any tangible example to give?  Does someone actually work this out with a deuterium nucleus and the strong force or something?  Can we see an example of how A varies under a situation more radical than a gradually advancing perhelion?
# Question: in political articles people are eager to farm out "details" to additional articles.  Here, why can't we displace the whole part about how the elliptical orbit is a four dimensional symmetry to some other article?
# It would be useful to say when A is zero.  Is it accurate to say that this is true for any perfectly circular orbit?
Sorry - I know I've inevitably made some errors in the above.  But when you want to put an article like this in front of a broad audience you need some ignorant feedback, IMHO. Wnt (talk) 02:45, 31 December 2010 (UTC)
replies: text: Agreed, mostly. I'm literate enough to follow, for example, a proof o Kepler's laws from Newton's laws (at ODE or [US] "calculus II" level), and I kinda intuited that something else is conserved in there. Eventually I found Laplace name-droped in a book as the other thing conserved... and I found this article... but it's anything but clear to me. The writing in it seems to go around in circles. 188.25.77.29 (talk) 14:02, 14 January 2016 (UTC)
thread_title:  Feature failure 

message: The article states "the Kepler problem is mathematically equivalent to a particle moving freely on the boundary of a four-dimensional sphere". But this is nonsense. A sphere, four-dimensional or otherwise, has no boundary. Do you mean four-dimensional ball? -lethe talk [ +] 06:17, 5 December 2012 (UTC)
thread_title:  boundary of a sphere??? 

message: I'm beginning a sort of WP:Expert review process for articles independent of the featured article system which I've realized has problems. As such, I've rated this article a level 'A' which means it is of the quality that would be expected from a professional reference work on the subject. I say this as a person with graduate degrees in astrophysics, but I encourage others who have similar qualifications to make comments if they believe my judgement to be incorrect.
jps (talk) 02:18, 12 September 2013 (UTC)
replies: text: It might contain a collection of correct statements, but it's not an article for someone who doesn't already know the topic to learn much from. 188.25.77.29 (talk) 14:15, 14 January 2016 (UTC)
thread_title: WP:PHYSICS review: A-level article

message: Philosophaies revisions of today are wrong, and have spoilt dimensional consistency. e should be dimensionles. They have been reverted. 
Please discuss misgivings before incorrect edits. Cuzkatzimhut (talk) 14:48, 25 September 2013 (UTC)
thread_title: Bad dimensions

message: I have spotted three Big Scary Red Error Messages on the page, the first being under heading "Rotational symmetry in four dimensions"
Failed to parse(syntax error): \begin{align} \boldsymbol\eta & = \displaystyle \frac{p^2 - p_0^2}{p^2 + p_0^2} \mathbf{\hat{w}} + \frac{2 p_0}{p^2 + p_0^2} \mathbf{p} \\[1em] & = \displaystyle \frac{mk — r p_0^2}{mk} \mathbf{\hat{w}} + \frac{rp_0}{mk} \mathbf{p} \end{align}
I suspect that they are both on the server end, as I found them in both Firefox (26.0) and IE 11.0.9600.16476.   Simon Crase.  — Preceding unsigned comment added by 118.93.231.190 (talk) 23:36, 16 December 2013 (UTC)
thread_title: Big Scary Red Error Messages

message: The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion:
* 
Participate in the deletion discussion at the . —Community Tech bot (talk) 23:37, 9 February 2020 (UTC)
thread_title:  A Commons file used on this page or its Wikidata item has been nominated for deletion 

message: If one ever wanted to kick this into the domain of abstract mathematics, one could tie this into the theory of symmetric spaces. It goes something like this: this article already explains that SO(4)/Z_2 = SO(3) x SO(3)  but one could also write this as saying SO(4)/SO(3)= S^3 the three-sphere and the three-sphere is the Lie group manifold of SU(2) and of course SU(2)/Z_2 = SO(3) so that's just an alternative factorization of "the same stuff". The Z_2 is the involution. So the sphere is a symmetric space and symmetric spaces have an involution, the Cartan involution. The Cartan involution has two eigenvalues, +1 and -1. Now, if I recall correctly (its been a while) the -1 eigenvalue corresponds to the bound orbits and the eigenvalue +1 to the ionized orbits. The involution corresponds to a factorization of \mathfrak{g} = \mathfrak{k}\oplus\mathfrak{p}. Now, if you look at this article, and look at the part that says "Poisson bracket" -- "unscaled functions" you will see commutators that are effectively the same as those for "symmetric space -- algebraic definition" (which uses different notation \mathfrak{g} = \mathfrak{h}\oplus\mathfrak{m} but its the same thing).
Then \mathfrak{p} can be further factorized to pieces corresponding to different orbital angular momentum (for the hydrogen atom).  This is the same factorization as for Riemanninan manifolds, where each distinct factor of \mathfrak{p} has a distinct eigenvalue for the Killing form (the quadratic   Casimir invariant) which for this particular example is just the total orbital angular momentum L^2 (of the hydrogen atom; don't know how this works for the non-quantized version). The spaces here are the spherical harmonics; the spherical harmonics generalize to arbitrary symmetric spaces as eigenvalues of \nabla^2 on whatever symmetric space you're interested in. So "hydrogen atom" in a generic setting. 
But wait, there's more (for the low, low price of $12.99 you also get ...) an extended bilinear form ds^2 that is a sum of the metric on the underlying space, plus the sum of weighted Killing forms on each of the \mathfrak{p} factors... just like in this article, subsection titled "Conservation and symmetry".
I think I've spewed enough key-words here that your favorite search engine will find more detailed expositions of this stuff ... I think it would be cool to articulate this stuff ... somewhere if not here ... (because otherwise its just dry, dusty, pointless and dead abstractions in textbooks on Riemannian geometry) 
p.s. Going through the article edit history, someone else (not me) attempted to wikilink this to potential theory which is the theory where the eigenvalues of the Laplacian for symmetric spaces are studied (even though our wikipedia article never quite comes out and says this). I have no clue if there is a generic LRL vector for these generic cases, but maybe there is one, any time you take a quotient G/H !??? That would also be cool to articulate, if it was true... it seems believable ... 67.198.37.16 (talk) 04:16, 7 November 2020 (UTC)
thread_title:  abstract math geekery 

message: Looking at this as part of the ongoing FA sweeps.  Currently, there is a very large amount of uncited text in here, which is problematic against the FA criteria.  While some of it may not need a citation under WP:WTC as self-proving math stuff, the very high percentage of uncited text is problematic and may require a featured article review if not addressed. Hog Farm  Talk 05:23, 17 April 2021 (UTC)
thread_title:  Featured article concerns 

message: Could you get my message on page User:Pmanderson (talk).What is a reason to remove the article: Fock symmetry in theory of hydrogen.
And, what is essential here, the operator of Laplace-Runge-Lenz vector doesn't exist in the momentum space ! So, Fock's symmetry found in momentum space, has nothing to do with that article.
Sincerely Yours
EfimovSP (talk) 11:27, 6 September 2021 (UTC)
thread_title:  Dear editor Septentrionalls Pmanderson