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.
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