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On Oct 28, 2:29*am, Martin Brown
wrote: 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 athttp://gallica.bnf.fr/ark:/12148/bpt6k229225j/f294.image.r=Oeuvres%20... 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 Regards, Martin Brown I have this wonderful magazine in front of me from 1983,it is the Discover Magazine's 'Year in Science' and it is such a wonderful read - there are adds for whiskey and cigarettes,vhs tapes and vinyl records and dotted between are stories which now look really dated -nuclear winter,acid rain and those trial balloons until they hit the jackpot a decade later with 'climate change' nee,global warming where everyone got implicated in destroying the planet or rather infested with the belief that humans can control the planet's temperature Among those articles is a wonderful essay about Langrangians,now this was written at a time when people could actually reason and there is a sense of dismay,almost inevitability about the whole thing as mathematicians drift further and further away from common sense that most people call reality.I am taking the time out to post the relevant passage in that essay as it is not available online,not so much to expose the 30 year lament but rather its prescience - "A Langrangian is not a physical thing;it is a mathematical thing - a kind of differential equation to be exact.But physics and maths are so closely connected these days that it is hard to separate the numbers from the things they describe.In fact,a month after [Philip] Morrison's remarks,Nobel Prize winner Burton Richter of the Stanford Linear Accelerator Center said something that eerily echoed it: " Mathematics is a language that is used to describe nature" he said "But the theorists are beginning to think it is nature.To them the Langrangians are the reality " Discover 1983 A people enveloped in their own imagination and using a language they understand among themselves are hardly the people who would stop to look at the geometrical language of astronomy and the extraordinary and one-off event where a conclusion was reached using an extension of the 24 hour AM/PM system and the Lat/Long system by inserting an explanation for planetary dynamics using right ascension when it is impossible.In this case mathematics do not substitute for reality but affirm it as a reasonable,if not sane ,person can safely affirm that all their experiences within a 24 hour cycle reflect one rotation of the Earth and they never,ever fall out of step.Long before Langrangians there was Ra/Dec,a system on which Newton built his absolute/relative time,space and motion agenda and the mathematicians in the late 17th and following centuries never stopped to consider whether the clockwork solar system arising from the Ra/Dec system reflected reality - it doesn't. |
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In uk.sci.astronomy message , Sat, 27
Oct 2012 00:59:39, Andy Walker posted: 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.] True as to the stability criterion. Do you know if Gascheau's 1843 article is accessible on the Web? Danby, Astron J, 69, 4, pp.294-6, May 1964 (GIF) looks a good read - for some. Its abstract indicates that the same holds for non-circular orbits, but with a varying numerical constant. A.N.Other implied that moderately elliptical orbits are stable for moderately lighter secondaries, but no further. -- (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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In uk.sci.astronomy message , Sat, 27 Oct
2012 10:16:05, Andy Walker posted: 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. http://en.wikipedia.org/wiki/File:Lagrange's_tomb_at_the_Pantheon.jpg for those who want to check -- IIRC, I saw that just as friends were visiting Paris - too late to ask them to look. Zdenek Kopal may have read the Essai at one time, mis-remembered what it said, and got the page number from somewhere else without a full re- read. One of my papers, Principles and theory of wattmeters operating on the basis of regularly spaced sample pairs, F J J Clarke and J R Stockton J. Phys. E: Sci. Instrum., was correctly printed in the journal. But in the issue's index page, Frank got truncated to a mere Clark - and the paper has often been cited with that. Google shows that it has also been cited as in "J Phis E" .... 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]. Indeed. I think that the section "Two Constant-Pattern Solutions Exist" in http://www.merlyn.demon.co.uk/lagrpapr.htm would, if written before 1772, have amounted to a discovery. The actual algebra can be taken as correct, having been independently checked. As far as I know, the sole assumption (known, but not proved, valid) is that only motion in a fixed plane needs to be considered. That uses a form of Heron's formula for the area of a triangle, which made me realise that the constant-pattern solutions of the general three-body problem put the masses at the corners of a triangle of extremal area (max/min) for the sum of its sides - and then wonder whether there might be a neater proof of the extremal property than chewing through the algebra. [...] 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 know of Roy, but have not yet seen it. In fact, the only books that might help to which I have ready access are Sir Thomas Heath on Euclid! .... ... IoP has published the 4th edition, and Google Books will show some of it, including some of the nearby pages. Looks as if http://www.scribd.com/doc/70409658/A-E-Roy-Orbital-Motion can show/supply it. -- (c) John Stockton, nr London, UK. For Mail, see Home Page. Turnpike, WinXP. Web http://www.merlyn.demon.co.uk/ - FAQ-type topics, acronyms, and links. Command-prompt MiniTrue is useful for viewing/searching/altering files. Free, DOS/Win/UNIX now 2.0.6; see URL:http://www.merlyn.demon.co.uk/pc-links.htm. |
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In uk.sci.astronomy message , Sun, 28
Oct 2012 09:29:53, Martin Brown posted: 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. Though Lagrange did not derive "stable" in the sense in which L1 L2 L3 differ from L4 L5. Giving the date of an Euler paper is not very useful, for two reasons; he wrote so many papers per year, and each tends to have several dates. For example, "De motu vibratorio tympanorum", read at Berlin 1761, presented at Petersburg 1762, published 1766. But the Eneström Index number is unambiguous, and that one is E.302. I've not checked whether Euler's papers all had different titles. Euler's E.304, read 1762, presented 1762, published 1766, reveals L1 & L2 (by considering putting the Moon there), but not L3, nor L4 L5. Translation is on my site. 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 No. OEuvres de Lagrange, Tome 6, pp. 272-292 Paris 1873 by J-A Serrat. The link to Tome 6 is on my site, but you want pp. 229-292, to include Chapter I. It is more easily read from the University of Liege, however. And an older copy can be read also in "Recueil des pieces qui ont remporte les prix de l'Academie Royale des Sciences", Tome 9 (1764-1772), link in my gravity4.htm. Liege and Gallica evidently have different scans of the same 1873 edition; I imagine of different copies of that edition. He almost certainly didn't call them L4, L5 (later authors did). He did not refer to them at all; he only found the constant-pattern solutions of the general three-body problem, from where the five points are a trivial step which he did not take, at least in writing. My guess is that they were named Lagrange Points soon after the time in 1906 when the first two asteroids were discovered to be co-moving with Jupiter, since Lagrange (1772) would have been reasonably well-known among astronomers for other chapters of the /Essai/. I did find a copy from 1873 online at http://gallica.bnf.fr/ark:/12148/bpt...r=Oeuvres%20de %20Lagrange.langFR You want page 292 under the heading XXXIII. He did know about the pure equilateral planar solution at least in his French writings. No, you need to start at page 229 for a proper understanding; the maths can be skipped. He obtained the collinear and equilateral solutions in Chapter II. If your French is rusty, read instead http://www.merlyn.demon.co.uk/essai-3c.htm; otherwise, check that page (it has been checked by a real Frenchman) ! 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. I have read the entire index of the OEuvres de Lagrange, and few other papers seemed at all likely to be relevant to the Points - and I have read those, and they are not. E&OE, of course; observe St Luke: Chapter 10, Verse 37, tail. 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 If you mean the OEuvres, it is not a composed book, but a collection of earlier material. [...] 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 two constant-pattern solutions exist only in a fixed plane. It is obvious that, if the velocities are at any time all in the plane of the bodies, the plane of the bodies cannot itself be changing at that time. But I think that, to satisfy the initial condition that the distances between the bodies change at rates instantaneously proportional to themselves, it must be necessary for the initial velocities to be in the initial plane. That is in barycentre-centred inertial coordinates. The OEuvres are not contemporaneous, although their contents are. They are a reprint in new type, and might have introduced their own errors. They have, for example, been in at least one place re-spelt in comparison with the version in the Recueil; e.g. Avertissement, para 5, first phrase. And the Recueil uses the long s, the OEuvres the modern one. -- (c) John Stockton, Surrey, UK. Turnpike v6.05 MIME. Web http://www.merlyn.demon.co.uk/ - FAQish topics, acronyms, & links. Proper = 4-line sig. separator as above, a line exactly "-- " (SonOfRFC1036) Do not Mail News to me. Before a reply, quote with "" or " " (SonOfRFC1036) |
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On 28/10/12 09:29, Martin Brown wrote:
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. I have a fund of ZK stories, but mostly not suitable for here. His soft Czechoslovakian accent was very easily imitated. so almost all of us did it. Occasionally, a group of postgrads would be holding a conversation in ZK-speak when ZK would turn up and join in; whether he didn't notice or didn't mind, we never discovered, [Stability of L4/5:] 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! Mechanics in rotating co-ordinates is not 6th-form maths! Nor is the general theory of small oscillations. I think it might be the other way round -- that you couldn't do it with 6th-form maths, except by spending a lot of time developing a framework for the proof, but on the other hand that there are some very talented sixth-formers who would be able to read a proof such as that in Roy, and fill in the gaps themselves. But that's just an opinion. -- Andy Walker, Nottingham. |
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On 28/10/12 20:30, Dr J R Stockton wrote:
[...] Do you know if Gascheau's 1843 article is accessible on the Web? I've never pursued any historical research into these areas, just read what I needed to for my own research and other interests. Ie, I don't know, and your googling is likely to be at least as well-directed as mine. Danby, Astron J, 69, 4, pp.294-6, May 1964 (GIF) looks a good read - for some. Its abstract indicates that the same holds for non-circular orbits, but with a varying numerical constant. A.N.Other implied that moderately elliptical orbits are stable for moderately lighter secondaries, but no further. It would be very surprising if it was not so. The usual restricted circular result is not "fragile", but comes merely from the shape of the [modified] equipotentials, esp near the maximum at the Trojan points. If the Trojan asteroids are stable to small perturbations in their own orbits, we can expect them to be stable equally to small perturbations in Jupiter's orbit. Putting some numbers in to that is a whole lot harder, though; not least because co-rotating co-ordinates are no longer rotating uniformly. But a computer simulation can't be hard, and could be Quite Interesting, esp for highly elliptic orbits. On looking at Danby's paper, eg at http://articles.adsabs.harvard.edu/c...J.....69..165D his Figure 1 [p171] is interesting; the shape of the stability region, with a minimum at Danby's point B [where only the circular case is stable] and a cusp at D [where some elliptical cases are stable even when the circular case isn't] is more complicated than might be expected. -- Andy Walker, Nottingham. |
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On 01/11/2012 13:06, Andy Walker wrote:
On 28/10/12 09:29, Martin Brown wrote: 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. I have a fund of ZK stories, but mostly not suitable for here. His soft Czechoslovakian accent was very easily imitated. so almost all of us did it. Occasionally, a group of postgrads would be holding a conversation in ZK-speak when ZK would turn up and join in; whether he didn't notice or didn't mind, we never discovered, Does that mean you were once at Manchester University in the 1970's? I used to know a few folk at Jodrell and in Kopal's group. [Stability of L4/5:] 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! Mechanics in rotating co-ordinates is not 6th-form maths! Centripetal force as they pedantically called it when I did A level was. Full derivation of general motion in rotating frames was second year physics although a few problems that required it cropped up sooner. Nor is the general theory of small oscillations. I think it might I am certain the simple and conical pendulum was. I don't think it is that much of a stretch to the general theory of SHM. And I am pretty sure at least in further maths integrating through the DE to get conservation of energy was taught. It is a long time ago though. be the other way round -- that you couldn't do it with 6th-form maths, except by spending a lot of time developing a framework for the proof, but on the other hand that there are some very talented sixth-formers who would be able to read a proof such as that in Roy, and fill in the gaps themselves. But that's just an opinion. Roy's treatment is certainly the most easily accessible. I came across it in my first year undergraduate about the same time as the first Meuss book/monograph was published in English translation. At the time I was trying to back solve initial observations of new comets into orbital elements - I became very impressed that Gauss had done it by hand in the days before computers. What is a bit of a shame is that the OP has not asked any questions. I guess the complicated thread drift has put him off. Message to OP: Ask away and we will try to answer a *lot* more simply. -- Regards, Martin Brown |
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On 01/11/12 17:56, Martin Brown wrote:
Does that mean you were once at Manchester University in the 1970's? I used to know a few folk at Jodrell and in Kopal's group. 1965-68. Exciting times in astronomy. [...] Mechanics in rotating co-ordinates is not 6th-form maths! Centripetal force as they pedantically called it when I did A level was. Yes, but that is to do with circular motion in non-rotating co-ordinates, not with small oscillations in rotating co-ordinates. Full derivation of general motion in rotating frames was second year physics although a few problems that required it cropped up sooner. It was first-year maths for me. Nor is the general theory of small oscillations. I am certain the simple and conical pendulum was. I don't think it is that much of a stretch to the general theory of SHM. SHM isn't too bad, but here we're looking rather at normal modes and such-like. And I am pretty sure at least in further maths integrating through the DE to get conservation of energy was taught. Yes, but that doesn't really help with the stability of the Trojan points, which are maxima of the [modified] PE. [...] At the time I was trying to back solve initial observations of new comets into orbital elements - I became very impressed that Gauss had done it by hand in the days before computers. The calculating skills of [eg] Newton and Gauss are indeed very impressive. What then surprised me was the primitive state of numerical analysis at the time, despite the existence of Newton's and Gauss's methods all over the place. I don't think there is any corner of NA that has been untouched in the computer era, even in such elementary areas as multiplication. What is a bit of a shame is that the OP has not asked any questions. I guess the complicated thread drift has put him off. Seems likely -- if he read the group at all, that is. At about his age, I was reading books on the solar system, the stars, cosmology, etc. There are surely still some such books aimed at quite young children -- I'd expect to find an Usborne book, eg, tho' I haven't looked. I used to enjoy Eddington, but I think the OP would perhaps need to wait a year or two, and would then find that however interesting the material was, it's now 80-odd years out of date and so completely useless. There must be modern equivalents! Message to OP: Ask away and we will try to answer a *lot* more simply. Right. -- Andy Walker, Nottingham. |
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On Nov 1, 6:21*pm, Andy Walker wrote:
* * * * The calculating skills of [eg] Newton and Gauss are indeed very impressive. *What then surprised me was the primitive state of numerical analysis at the time, despite the existence of Newton's and Gauss's methods all over the place. -- Andy Walker, Nottingham. Newton's method indeed !,they couldn't stand the Ra/Dec generated clockwork solar system on which Isaac built his agenda as it left no room to study anything else but they couldn't work out what method Newton used to make it appear that predictive elements within Ra/Dec transfer to experimental sciences and they didn't really care.Once it was possible to get a mathematician to admit this but not after the early 20th century - "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 Newton's agenda is like modeled global warming,it answers any question you care to ask and precious few people have comprehended the horror when so much is answered by so little,one of the best commentaries was by Edgar Allan Poe which in someway mirrors the comments of Rouse Ball in an expanded way - "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 theprinciple 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?" Edgar Allan Poe We inherit a highly dysfunctional system and mathematicians couldn't care less or at least the ones I have encountered and it is now pretty obvious that people here are trying to discuss mathematical things as though they were astronomical things but recently jargon has overtaken an semblance to real astronomical objects and their effects. |
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On 02/11/2012 01:21, Andy Walker wrote:
On 01/11/12 17:56, Martin Brown wrote: Does that mean you were once at Manchester University in the 1970's? I used to know a few folk at Jodrell and in Kopal's group. 1965-68. Exciting times in astronomy. Well before my time. Trying to remember a few other names from way back when I can only conjure up John Meaburn & Ken Elliott (sp?). One of them had a cute Fabry-Perot device. Didn't Kopal help map the moon for the NASA moon landings or something - ah yes: http://www.amazon.co.uk/Photographic...1870829&sr=1-1 I reckon we now live in a golden observational age where there are enough amateur scopes and survey instruments with CCDs watching the skies to catch most things that happen pretty quickly and record it. [...] Mechanics in rotating co-ordinates is not 6th-form maths! Centripetal force as they pedantically called it when I did A level was. Yes, but that is to do with circular motion in non-rotating co-ordinates, not with small oscillations in rotating co-ordinates. Agreed. I haven't worked it through in detail but I think you can get to the modified potential that way. Full derivation of general motion in rotating frames was second year physics although a few problems that required it cropped up sooner. It was first-year maths for me. Our lot spent most of the first term proving 0 != 1. Nor is the general theory of small oscillations. I am certain the simple and conical pendulum was. I don't think it is that much of a stretch to the general theory of SHM. SHM isn't too bad, but here we're looking rather at normal modes and such-like. And I am pretty sure at least in further maths integrating through the DE to get conservation of energy was taught. Yes, but that doesn't really help with the stability of the Trojan points, which are maxima of the [modified] PE. Hmm. Yes maybe this requires a bit more thought. [...] At the time I was trying to back solve initial observations of new comets into orbital elements - I became very impressed that Gauss had done it by hand in the days before computers. The calculating skills of [eg] Newton and Gauss are indeed very impressive. What then surprised me was the primitive state of numerical analysis at the time, despite the existence of Newton's and Gauss's methods all over the place. I don't think there is any corner of NA that has been untouched in the computer era, even in such elementary areas as multiplication. What is a bit of a shame is that the OP has not asked any questions. I guess the complicated thread drift has put him off. Seems likely -- if he read the group at all, that is. At The thread is there on SpaceBanter, but I suspect the OP isn't. about his age, I was reading books on the solar system, the stars, cosmology, etc. There are surely still some such books aimed at quite young children -- I'd expect to find an Usborne book, eg, tho' I haven't looked. I used to enjoy Eddington, but I think the OP would perhaps need to wait a year or two, and would then find that however interesting the material was, it's now 80-odd years out of date and so completely useless. There must be modern equivalents! Main problem is picking a good one. There seemed to be very mixed reviews of the childrens astronomy books on Amazon so without having seen them I would hesitate to recommend one. Message to OP: Ask away and we will try to answer a *lot* more simply. Right. PS weak aurora last night in northern UK. Seen in Northumberland. -- Regards, Martin Brown |
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