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  #11  
Old October 25th 12, 11:19 AM posted to uk.sci.astronomy
oriel36[_2_]
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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


  #12  
Old October 25th 12, 01:21 PM posted to uk.sci.astronomy
Andy Walker[_2_]
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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.
  #13  
Old October 25th 12, 03:01 PM posted to uk.sci.astronomy
oriel36[_2_]
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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.


  #14  
Old October 26th 12, 10:53 PM posted to uk.sci.astronomy
Dr J R Stockton[_183_]
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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.
  #15  
Old October 26th 12, 11:38 PM posted to uk.sci.astronomy
Dr J R Stockton[_183_]
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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.
  #16  
Old October 27th 12, 12:59 AM posted to uk.sci.astronomy
Andy Walker[_2_]
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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.
  #17  
Old October 27th 12, 09:13 AM posted to uk.sci.astronomy
oriel36[_2_]
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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.


  #18  
Old October 27th 12, 10:16 AM posted to uk.sci.astronomy
Andy Walker[_2_]
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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.
  #19  
Old October 27th 12, 01:10 PM posted to uk.sci.astronomy
oriel36[_2_]
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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.


  #20  
Old October 28th 12, 10:29 AM posted to uk.sci.astronomy
Martin Brown
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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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