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.

--
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Web http://www.merlyn.demon.co.uk/ - FAQ-type topics, acronyms, and links.
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