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.