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On Oct 25, 12:44*am, Martin Brown
wrote: I remember being interested in astronomy too at about that age and the main thing that frustrated me was that the star maps in books did not include the planets! Obvious why when you know that they move about! "Planet" literally means "wandering star". And then came along an English clown called Newton who couldn't comprehend that the 'wandering' nature of planets refers to retrogrades and retrogrades are an illusion caused by the Earth's own orbital motion between Venus and Mars and around the central Sun.A teenager with the benefit of contemporary imaging and time lapse footage can figure out what Isaac and his followers couldn't - http://apod.nasa.gov/apod/ap011220.html There's your wandering motion for you Brown and the same teenager could probably tell you that Isaac's idiosyncratic view of retrogrades is a technical non sequitur as it doesn't involve a hypothetical observer on the Sun - only an intelligent observer who realizes he is standing on a moving Earth ! - "For to the earth planetary motions appear sometimes direct, sometimes stationary, nay, and sometimes retrograde. But from the sun they are always seen direct,..." Newton As a genuine astronomer,I can see what Isaac was trying to do with his absolute/relative time,space and motion using that worthless idea of retrogrades but I wouldn't hold my breath waiting for somebody else to ask what exactly he was up to and why it is catastrophically disruptive for 21st century purposes.Apparently the English like their iconic figures and certain sections of your nation seem terrified of certain individuals then as now judging from the recent celebrity exposure and Newton has such a grip on science that demonstrating what he was actually doing looks like an assault on the English nation. Wlliam Blake got it right even though he didn't know the technical ins and outs of Newton's clockwork solar system approach which borrows from Flamsteed's muddleheaded conclusion which takes a step too far with a rotating celestial sphere of Ra/Dec. "I turn my eyes to the Schools & Universities of Europe And there behold the Loom of Locke whose Woof rages dire Washd by the Water- wheels of Newton. black the cloth In heavy wreathes folds over every Nation; cruel Works Of many Wheels I view, wheel without wheel, with cogs tyrannic Moving by compulsion each other: not as those in Eden: which Wheel within Wheel in freedom revolve in harmony & peace." William Blake,Jerusalem Cruel works indeed !,the cruelty will be returned if the English do not deal with the mess which occurred within their borders and specifically the train wreck involving astronomy and human timekeeping.The Americans have already begun the recovery process- http://www.youtube.com/watch?v=kDWHM00sZJc BTW It would be nice to know if these replies are reaching spacebanter.com and if the OP is still there. I hope he hasn't been frightened off by Oriel36 blather. -- Regards, Martin Brown |
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On 25/10/12 08:44, Martin Brown wrote:
On 24/10/2012 21:14, Dr J R Stockton wrote: Contrary to common opinion, Lagrange did not discover the Lagrange Points - although the final step to the Points from what he did is trivial, he did not take it in the /Essai/, and, AFAICS, nowhere else either. Interesting. It's not just "common" opinion; eg, Kopal's "Close Binary Systems" says explicitly [p546] "The five point- solutions were discovered by J. L. Lagrange in his 'Essai [...] (cf his /Collected Works/, *6*, p.229)," Kopal was a meticulous researcher with access to a huge library and would certainly have read the /Essai/, so I'm surprised he got it wrong. [...] Most physics undergraduates today would struggle to derive the orbital Lagrangian points from first principles. This may well be true, esp if they are simply given the problem with no hints or "signposts". However, the derivation is not particularly difficult, either for the Lagrange problem of finding persistent configurations or for the usual restricted three-body problem, as long as vector algebra is used to keep the equations simple. I see no reason why a student shouldn't be able to follow such a derivation, or to construct it given reasonable pointers as to how to proceed. The Lagrange points are also very easy to derive from the Jacobi integral, by either vectorial or algebraic methods. As this is essentially the potential energy of the system, this derivation is also accessible to anyone who has done Hamiltonian or Lagrangian mechanics -- surely still in the physics syllabus at decent universities, even if not common knowledge among 8yos! -- and gives scope then for discussion of stability. -- Andy Walker, Nottingham. |
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On Oct 25, 5:21*am, Andy Walker wrote:
On 25/10/12 08:44, Martin Brown wrote: On 24/10/2012 21:14, Dr J R Stockton wrote: Contrary to common opinion, Lagrange did not discover the Lagrange Points - although the final step to the Points from what he did is trivial, he did not take it in the /Essai/, and, AFAICS, nowhere else either. * * * * Interesting. *It's not just "common" opinion; *eg, Kopal's "Close Binary Systems" says explicitly [p546] "The five point- solutions were discovered by J. L. Lagrange in his 'Essai [...] (cf his /Collected Works/, *6*, p.229)," *Kopal was a meticulous researcher with access to a huge library and would certainly have read the /Essai/, so I'm surprised he got it wrong. [...] *Most physics undergraduates today would struggle to derive the orbital Lagrangian points from first principles. * * * * This may well be true, esp if they are simply given the problem with no hints or "signposts". , The older English scientists didn't chant empirical voodoo,they could actually present difficulties they had with problems inherited from the past - Rouse Ball being among them - "The demonstrations throughout the book [Principia] are geometrical, but to readers of ordinary ability are rendered unnecessarily difficult by the absence of illustrations and explanations, and by the fact that no clue is given to the method by which Newton arrived at his results." Rouse Ball 1908 Men can actually talk about these things and be understood,Edgar Allan Poe being among the few who was more expansive on the iconic theory that answers everything and says nothing - "To explain: — The Newtonian Gravity — a law of Nature — a law whose existence as such no one out of Bedlam questions — a law whose admission as such enables us to account for nine-tenths of the Universal phænomena — a law which, merely because it does so enable us to account for these phænomena, we are perfectly willing, without reference to any other considerations, to admit, and cannot help admitting, as a law — a law, nevertheless, of which neither the principle nor the modus operandi of the principle, has ever yet been traced by the human analysis — a law, in short, which, neither in its detail nor in its generality, has been found susceptible of explanation at all — is at length seen to be at every point thoroughly explicable, provided we only yield our assent to —— what? To an hypothesis? Why if an hypothesis — if the merest hypothesis — if an hypothesis for whose assumption — as in the case of that pure hypothesis the Newtonian law itself — no shadow of à priori reason could be assigned — if an hypothesis, even so absolute as all this implies, would enable us to perceive a principle for the Newtonian law — would enable us to understand as satisfied, conditions so miraculously — so ineffably complex and seemingly irreconcileable as those involved in the relations of which Gravity tells us, — what rational being could so expose his fatuity as to call even this absolute hypothesis an hypothesis any longer — unless, indeed, he were to persist in so calling it, with the understanding that he did so, simply for the sake of consistency in words?" Allan Poe No offence to the magicians of Oxford and Cambridge who have managed to run a tight ship for the last number of centuries but once the cracks start to appear with the iconic character and his iconic theory the amazing series of events that led to its acceptance and the damage it actually caused is an amazing story and includes some of the most renowned English personalities including the brilliant John Harrison and his equally dismissive comments to welfare empiricists of his time - " But indeed, had I continued under the hands of the rude commissioners, this completion, or great accomplishment, neither would, nor could, ever have been obtained; but however, providence otherwise ordered the matter, and I can now boldly say, that if the provision for the heat and cold could properly be in the balance itself, as it is in the pendulum, the watch [or my longitude time- keeper] would then perform to a few seconds in a year, yea, to such perfection now are imaginary impossibilities conquered; so the priests at Cambridge and Oxford, &c. may cease their pursuit in the longitude affair, and as otherwise then to occupy their time." John Harrison Carry on guys,the indignity is not that Flamsteed made a mistake and empiricists built on that mistake,after all the error is not immediately recognizable and certainly not in older times without imaging power and data we possess today but it sure is now.Chanting voodoo is quaint but the actual nuts and bolts of astronomy and any links between astronomy and terrestrial sciences is perfectly understandable including the awful recklessness that occurred within English borders. However, the derivation is not particularly difficult, either for the Lagrange problem of finding persistent configurations or for the usual restricted three-body problem, as long as vector algebra is used to keep the equations simple. *I see no reason why a student shouldn't be able to follow such a derivation, or to construct it given reasonable pointers as to how to proceed. * * * * The Lagrange points are also very easy to derive from the Jacobi integral, by either vectorial or algebraic methods. As this is essentially the potential energy of the system, this derivation is also accessible to anyone who has done Hamiltonian or Lagrangian mechanics -- surely still in the physics syllabus at decent universities, even if not common knowledge among 8yos! -- and gives scope then for discussion of stability. -- Andy Walker, Nottingham. |
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In uk.sci.astronomy message , Thu, 25
Oct 2012 08:44:35, Martin Brown posted: On 24/10/2012 21:14, Dr J R Stockton wrote: In uk.sci.astronomy message , Tue, 23 Oct 2012 16:42:09, Martin Brown posted: On 23/10/2012 12:48, Lunar wrote: Hi everyone, I'm new to this forum and want to learn more about our solar system and beyond. I'm eight years old and feel too advanced for what my school are teaching me (poems about the order of the planets in our solar system!) I want to learn more. Ask away and we will try to answer at the right level. There is also http://starchild.gsfc.nasa.gov/docs/...StarChild.html H'mmm - all it has relevant to the Lagrange Points (popular nowadays) is a GIF of Lagrange. The deficiency has been pointed out. Contrary to common opinion, Lagrange did not discover the Lagrange Points - although the final step to the Points from what he did is trivial, he did not take it in the /Essai/, and, AFAICS, nowhere else either. Euler discovered L1 & L2, quietly. Details on my site. Be fair John! The intricacies of the Lagrange points are not within easy grasp of an average eight year old. A picture of the guy is more than enough - he was a great mathematician. It is easy enough to state, without proof that (and roughly where) L1 L2 L3 exist, and are unstable as is a ball on the nose of a seal; and that L4 L5 exist, and are stable as is a ball on the inside of a wok. The site has two age-ranges, and refers over-thirteens elsewhere. At least the upper range, bearing in mind that only the more intelligent will be reading the site, should be able to cope with the ideas behind a ** well-informed ** description. See http://www.merlyn.demon.co.uk/gravity4.htm and the associated pages linked to it. Most physics undergraduates today would struggle to derive the orbital Lagrangian points from first principles. Yes, especially if they have been taught by the average general-purpose lecturer. Another interesting site for the OP is Stellarium which provides a realtime simulated view of the sky on a PC which shows where to look for planets and comets. Jupiter is easy in the evening sky now. Here it seems to be generally eclipsed by instances of Pluvial Nimbus. http://www.heavens-above.com/ is also good. http://sourceforge.net/projects/stellarium/ You might be interested in my http://www.merlyn.demon.co.uk/astron-5.htm, though it needs more work on the spherical trig. Especially if you've read Hal Clement's "Mistaken for Granted". -- (c) John Stockton, nr London, UK. Mail via homepage. Turnpike v6.05 MIME. Web http://www.merlyn.demon.co.uk/ - FAQqish topics, acronyms and links; Astro stuff via astron-1.htm, gravity0.htm ; quotings.htm, pascal.htm, etc. |
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In uk.sci.astronomy message , Thu, 25
Oct 2012 13:21:04, Andy Walker posted: On 25/10/12 08:44, Martin Brown wrote: On 24/10/2012 21:14, Dr J R Stockton wrote: Contrary to common opinion, Lagrange did not discover the Lagrange Points - although the final step to the Points from what he did is trivial, he did not take it in the /Essai/, and, AFAICS, nowhere else either. Interesting. It's not just "common" opinion; eg, Kopal's "Close Binary Systems" says explicitly [p546] "The five point- solutions were discovered by J. L. Lagrange in his 'Essai [...] (cf his /Collected Works/, *6*, p.229)," Kopal was a meticulous researcher with access to a huge library and would certainly have read the /Essai/, so I'm surprised he got it wrong. Well, if that's an accurate quote, he did not know how Lagrange's initials are usually written. Lagrange (born 1736 and named Giuseppe Luigi Lagrancia) used at least these : Ludovico de la Grange Tournier, Ludovicum de la Grange, Luigi di La Grange Tournier, Louis de la Grange, De la Grange, De Lagrange, Louis de Lagrange, De Lagrange, Lagrange, L. G., Joseph-Louis Lagrange, J.-L. Lagrange. He died in 1813 as, and is now generally known as Joseph-Louis Lagrange. But, IIRC, his tomb does lack the hyphen. The relevant chapters of the Essay contain no instances of the word or number five. Read it yourself - it's quite an easy read, if the actual maths is disregarded. [...] Most physics undergraduates today would struggle to derive the orbital Lagrangian points from first principles. This may well be true, esp if they are simply given the problem with no hints or "signposts". However, the derivation is not particularly difficult, either for the Lagrange problem of finding persistent configurations or for the usual restricted three-body problem, as long as vector algebra is used to keep the equations simple. Then you don't know the simple way, which needs no vector algebra. See http://www.merlyn.demon.co.uk/gravity6.htm and the associated pages linked to it. It is often said that Lagrange treated the circular restricted three- body problem. He did not. He treated the three-body problem, for which the restricted and circular are merely special cases. Granted, he did initially discuss circular in Chapter II, but later in the Chapter he did the general shape. He did not consider restricted, which is why he did not discover L3 L4 L5. The final step from what he did discover, restricting, is trivial, but was not taken. As for predicting bodies being found in such configurations - he apparently predicted the opposite : "quoique ces cas n'aient pas lieu dans le Systeme du monde", in Chapter II section XXIII. Lagrange's aim was to win the 1772 Paris Prize for the set topic of Theory of the Moon, in which he was half successful, as was Euler. Chapter I attempts the General Three-Body Problem, Chapter II finds the two special constant-pattern configurations, and Chapters III & IV are more directly related to the Moon. Chapter II is definitely a side- line. I see no reason why a student shouldn't be able to follow such a derivation, or to construct it given reasonable pointers as to how to proceed. The Lagrange points are also very easy to derive from the Jacobi integral, by either vectorial or algebraic methods. As this is essentially the potential energy of the system, this derivation is also accessible to anyone who has done Hamiltonian or Lagrangian mechanics -- surely still in the physics syllabus at decent universities, even if not common knowledge among 8yos! -- and gives scope then for discussion of stability. Some of us have been around long enough to have forgotten whether we were ever taught such things! But they are not needed. NOTE : Lagrange did not consider the stability of bodies at the Points; and I've only thought about considering it. -- (c) John Stockton, nr London, UK. E-mail, see Home Page. Turnpike v6.05. Website http://www.merlyn.demon.co.uk/ - w. FAQish topics, links, acronyms PAS EXE etc. : http://www.merlyn.demon.co.uk/programs/ - see in 00index.htm Dates - miscdate.htm estrdate.htm js-dates.htm pas-time.htm critdate.htm etc. |
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On 26/10/12 22:53, Dr J R Stockton wrote:
It is easy enough to state, without proof that (and roughly where) L1 L2 L3 exist, and are unstable as is a ball on the nose of a seal; and that L4 L5 exist, and are stable as is a ball on the inside of a wok. Easy to state, but wrong; eg, in the restricted circular problem, L4 and L5 are stable only if 27PS (P+S)^2, where P and S are the masses of primary and secondary; even then, it's more like rolling around on an upturned wok [ie, "naturally" unstable], but with the wok being twisted around so as to keep the ball up. [Proof beyond the scope of this article, and just about at the limits of undergraduate mechanics, I'd guess.] -- Andy Walker, Nottingham. |
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On Oct 26, 4:59*pm, Andy Walker wrote:
On 26/10/12 22:53, Dr J R Stockton wrote: It is easy enough to state, without proof that (and roughly where) L1 L2 L3 exist, and are unstable as is a ball on the nose of a seal; and that L4 L5 exist, and are stable as is a ball on the inside of a wok. * * * * Easy to state, but wrong; *eg, in the restricted circular problem, L4 and L5 are stable only if 27PS (P+S)^2, where P and S are the masses of primary and secondary; *even then, it's more like rolling around on an upturned wok [ie, "naturally" unstable], but with the wok being twisted around so as to keep the ball up. [Proof beyond the scope of this article, and just about at the limits of undergraduate mechanics, I'd guess.] -- Andy Walker, Nottingham. Empirical voodoo chanting is so diverting,people might even imagine you both are saying something. I look at where the modeling/predictions agenda originally arose and specifically the technical details surrounding its acceptance in the late 17th century with the idea that there is no perceptual boundary between the motion and behavior of objects at a human level on one side with the motion and behavior of objects at a planetary and solar system scale on the other side. What they did in the late 17th century was bundle the separate AM/PM system and the Lat/Long system, which together contain the information that the Earth turns once in 24 hours,into a calendar based clockwork system known as Ra/Dec hence the clockwork solar system beloved of modelers and why today it is close to impossible to find a scientist who can keep one 24 hour day in step with one rotation.Without that basic fact,our era can’t explain why the temperature goes up and down daily in response to one rotation of the Earth but we can,with the Ra/ Dec system,predict when a star or the moon will rise and set,when a lunar or solar eclipse will occur and things like that.The price for being able to predict the locations of celestial objects within a rotating celestial sphere (Ra/Dec) is terrible as we lose cause and effect between planetary dynamics and terrestrial experiences such the day/night cycle,the seasons,climate and many more topics. I have a high regard for John Harrison and the monarchy that eventually supported him as a triumph of mechanical innovation in tandem with the astronomical principles which supply the core facts on which clocks and watches are based and little regard for those who can't follow those principles and especially the 'celestial mechanics' - people who think they are following Newton but are really following John Flamsteed's muddleheaded Ra/Dec conclusion.People make mistakes,even a catastrophically disruptive one like this one,it is how a people and a nation deals with that mistake and the iconic characters that created them that matters and the English have every opportunity to use Harrison as representative of AM/PM system and the Lat/Long system which contains the Earth's core facts and the Ra/Dec system which doesn't. So far,this forum has behaved as the empiricists did with John Harrison but there are signs in the wider community that people can and will deal with iconic figures in an open and honest way.The idea is to get the 'celestial mechanics' who followed the Ra/Dec system in to adjust to its limitations and adopt the stable AM/PM and Lat/Long systems once more otherwise it would be appealing to a Nazi mentality that can't change and that is a distinct and dismal possibility. |
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On 26/10/12 23:38, Dr J R Stockton wrote:
[...] It's not just "common" opinion; eg, Kopal's "Close Binary Systems" says explicitly [p546] "The five point- solutions were discovered by J. L. Lagrange in his 'Essai [...] (cf his /Collected Works/, *6*, p.229)," Kopal was a meticulous researcher with access to a huge library and would certainly have read the /Essai/, so I'm surprised he got it wrong. Well, if that's an accurate quote, he did not know how Lagrange's initials are usually written. [...] But, IIRC, his tomb does lack the hyphen. Back in the '50s, there was less concern about historical consistency. But ZK must have read the /Essai/, and must have checked to find the page number, so it's surprising that he got the discovery wrong. My other usual source on celestial mechanics, Roy's "Orbital Motion" is much more circumspect, and seems to agree with you about the history. You're right, BTW, that the tomb lacks the hyphen. The relevant chapters of the Essay contain no instances of the word or number five. Read it yourself - it's quite an easy read, if the actual maths is disregarded. It's quite an easy read with the maths included! But I'm v happy to take your word for it. [...] However, the derivation is not particularly difficult, either for the Lagrange problem of finding persistent configurations or for the usual restricted three-body problem, as long as vector algebra is used to keep the equations simple. Then you don't know the simple way, which needs no vector algebra. See http://www.merlyn.demon.co.uk/gravity6.htm and the associated pages linked to it. Yes, I've seen those. It's quite easy to show that the equilateral triangle persists, somewhat harder to find it in the first place [the usual maths difference between verifying that something is a solution, and finding/deriving it]. [...] The Lagrange points are also very easy to derive from the Jacobi integral,[...] -- and gives scope then for discussion of stability. Some of us have been around long enough to have forgotten whether we were ever taught such things! But they are not needed. Um. I don't think you can get stability without doing a decent amount of calculus. Note that L4 and L5 are *maxima* of the [modified] energy, so the dynamical stability [essentially brought about by Coriolis forces] if the masses are sufficiently disparate is definitely non-trivial. NOTE : Lagrange did not consider the stability of bodies at the Points; and I've only thought about considering it. As hinted above, it's an interesting problem. There is a derivation, eg, in Roy, op cit, p134. I've taught it in a different, and perhaps simpler, way, but I don't think it can be reduced to [eg] sixth-form level, except perhaps in a very arm-wavy way. -- Andy Walker, Nottingham. |
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On Oct 27, 2:16*am, Andy Walker wrote:
* * * * As hinted above, it's an interesting problem. *There is a derivation, eg, in Roy, op cit, p134. *I've taught it in a different, and perhaps simpler, way, but I don't think it can be reduced to [eg] sixth-form level, except perhaps in a very arm-wavy way. -- Andy Walker, Nottingham. I know,you must work at the empirical Klingon language institute - http://en.wikipedia.org/wiki/Klingon_Language_Institute As a direct result of what occurred within English borders,and especially concentrated around the area of Longitude,the rest of the world is going to slowly recover a stable narrative which retains the 24 AM/PM cycle and the Lat/Long system as containing the information of the Earth's rotation.With no visible signs that the English representation have any intention of shifting away from the flawed Ra/ Dec clockwork solar system it will be external national concerns that will isolate the iconic error for what it is. Enjoy yourselves with empirical language that merely disguises a mistake that when seen and understood,and it doesn't take that much,is breathtaking in the debris area it creates around all other sciences and especially terrestrial sciences.In the decade since I came here I see little stomach for adjusting to a better perspective so this is no longer a matter of intellectual cowardice any longer - it is what it is and UK.sci.astronomy has acted as a reflection of your nation,wish I could report otherwise,but that is it. |
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On 27/10/2012 10:16, Andy Walker wrote:
On 26/10/12 23:38, Dr J R Stockton wrote: [...] It's not just "common" opinion; eg, Kopal's "Close Binary Systems" says explicitly [p546] "The five point- solutions were discovered by J. L. Lagrange in his 'Essai [...] (cf his /Collected Works/, *6*, p.229)," Kopal was a meticulous researcher with access to a huge library and would certainly have read the /Essai/, so I'm surprised he got it wrong. Well, if that's an accurate quote, he did not know how Lagrange's initials are usually written. [...] But, IIRC, his tomb does lack the hyphen. Back in the '50s, there was less concern about historical consistency. But ZK must have read the /Essai/, and must have checked to find the page number, so it's surprising that he got the discovery wrong. My other usual source on celestial mechanics, Roy's "Orbital Motion" is much more circumspect, and seems to agree with you about the history. You're right, BTW, that the tomb lacks the hyphen. I remember Prof Kopal he was one of the people who got me interested in astronomy as a youngster. He was patron of the local astrosoc and did an annual lecture. The relevant chapters of the Essay contain no instances of the word or number five. Read it yourself - it's quite an easy read, if the actual maths is disregarded. It's quite an easy read with the maths included! But I'm v happy to take your word for it. I think that it taking things a little bit too literally. Lagrange derived the always an equilateral triangle stable solution for the three body problem independently as a part of his rediscovery of the Euler solutions of 1767 which he published in 1772. I am paraphrasing from Celestial Encounters - another book on the history of orbital dynamics discoveries. Its referencing is not that hot but based on dates I think the book to be checked is Lagrange, J.L, Oeurves, vol 6, p272-292 Paris 1873 He almost certainly didn't call them L4, L5 (later authors did). I did find a copy from 1873 online at http://gallica.bnf.fr/ark:/12148/bpt...agrange.langFR You want page 292 under the heading XXXIII. He did know about the pure equilateral planar solution at least in his French writings. It would be necessary to work back through the references chain to see whether later authors were rewriting history here or just clarifying things that Lagrange had actually said in earlier Latin papers. My assessment of the book as a whole is that it could use a few more diagrams and a lot less turgid French prose! YMMV [...] However, the derivation is not particularly difficult, either for the Lagrange problem of finding persistent configurations or for the usual restricted three-body problem, as long as vector algebra is used to keep the equations simple. Then you don't know the simple way, which needs no vector algebra. See http://www.merlyn.demon.co.uk/gravity6.htm and the associated pages linked to it. Yes, I've seen those. It's quite easy to show that the equilateral triangle persists, somewhat harder to find it in the first place [the usual maths difference between verifying that something is a solution, and finding/deriving it]. [...] Lagrange actually proved a more general result that with the right initial conditions a three body solution exists where the initial conditions mean it remains always an equilateral triangle. Trivialising this to the planar case would be easy. And it seems that he had doen it according to the contemporaneous French Oeuvres. The Lagrange points are also very easy to derive from the Jacobi integral,[...] -- and gives scope then for discussion of stability. Some of us have been around long enough to have forgotten whether we were ever taught such things! But they are not needed. Um. I don't think you can get stability without doing a decent amount of calculus. Note that L4 and L5 are *maxima* of the [modified] energy, so the dynamical stability [essentially brought about by Coriolis forces] if the masses are sufficiently disparate is definitely non-trivial. NOTE : Lagrange did not consider the stability of bodies at the Points; and I've only thought about considering it. As hinted above, it's an interesting problem. There is a derivation, eg, in Roy, op cit, p134. I've taught it in a different, and perhaps simpler, way, but I don't think it can be reduced to [eg] sixth-form level, except perhaps in a very arm-wavy way. I think it could just about be done with 6th form maths, but I am not convinced that (m)any sixth formers would be able to follow it! -- Regards, Martin Brown |
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