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