Hamilton's principle/History/Lagrange and the Mecanique analytique - Revision history

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	Lagrange's ''Mécanique analytique'' rebuilt mechanics as a branch of analysis. It		Lagrange's ''Mécanique analytique'' rebuilt mechanics as a branch of analysis. It
	supplied the calculus of variations in the form still taught, absorbed		supplied the calculus of variations in the form still taught, absorbed
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Lagrange's ''Mécanique analytique'' rebuilt mechanics as a branch of analysis. It
supplied the calculus of variations in the form still taught, absorbed
d'Alembert's treatment of constraints, and produced a formalism in which the
equations of any mechanical system follow from a single function by a fixed
recipe.

Hamilton, fifty years later, opened his own essay by praising it and then saying
what it could not do.

== Provenance note ==

[[File:OPL-HP-F-58.png|thumb|right|400px|The two long gaps in the subject's history.]]

Two editions exist and this volume treats them as '''two sources'''.

The first, of 1788, is on Gallica as a page-image facsimile. The second, of
1811–15, is on Gallica and at the Internet Archive.

Lagrange described the second in his own preface as, in many respects, a new work,
more extensive than the first. Quoting the later text under an 1788 attribution
would be a provenance error, so the rows are kept apart.

'''Neither has been opened for this volume.''' No equation and no wording is
attributed to Lagrange on this page. See [[Hamilton's principle/Sources]].

== What the book did ==

Three things, and the third is the one that mattered most.

'''It made mechanics analytic.''' Lagrange stated in his preface that the work
contains no diagrams — a boast, and an accurate description of the method. Newton's
mechanics is geometrical; Lagrange's is algebraic, and can be carried through
without picturing anything.

'''It absorbed constraints.''' D'Alembert's principle, described at
[[Hamilton's principle/History/d'Alembert and virtual work]], eliminates the
constraint forces before any calculation. Lagrange combined that with the
variational method and produced the multiplier technique of
[[Hamilton's principle/Mathematics/Lagrange multipliers]].

'''It made the coordinates free.''' This is the decisive step. Newton's laws are
statements about components along particular directions, and changing to angles or
distances is laborious and error-prone. Lagrange's equations take the same form in
'''any''' coordinates. Choose whatever suits the problem, write one function,
turn the handle.

That freedom is what makes the method usable for a spinning top, a linked chain,
or a robot arm, and it is why the formalism survived when its competitors did not.

== The equation ==

The equation everyone calls Euler-Lagrange arrived through the correspondence
between the two men in the 1750s, when Lagrange was still in his twenties.

Euler had reached it earlier by a laborious limiting argument on a polygon. The
young Lagrange sent him a cleaner variational derivation — essentially the one on
[[Hamilton's principle/Mathematics/The first variation]] — and Euler adopted it,
gave it the name ''calculus of variations'', and let Lagrange publish first.

The conduct is worth noting beside Euler's conduct in the priority dispute at
[[Hamilton's principle/History/The Maupertuis-Koenig affair]]. In both cases he
declined a claim he could have pressed. In one case that looks like generosity to
a young mathematician; in the other it looks like deference to a superior.

== Least action, demoted ==

Lagrange treats the principle of least action, and the way he treats it is the
point.

For Maupertuis it was a fundamental truth about nature with a metaphysical
justification. For Lagrange it is a '''consequence''' — a result that falls out of
the mechanics, useful in some problems, not the foundation of anything.

That demotion is the correct assessment for its time. As a tool for finding the
motion, the principle only produces the differential equations, which were already
available by other routes.

Hamilton says so explicitly, and reports that Lagrange, Laplace and Poisson had
for that reason spoken lightly of its utility.<ref>Hamilton, ''On a General Method
in Dynamics'' (1834), §3. Verbatim. ESTABLISHED source.</ref>

The principle only became more than a curiosity when Hamilton found something else
to do with the action — vary the endpoints rather than the path, as on
[[Hamilton's principle/Mathematics/Action as a function of its endpoints]] — and
when Noether showed what its symmetries imply, at
[[Hamilton's principle/History/Noether 1918]].

== What Hamilton said about it ==

Hamilton's assessment is the one worth having, because he was both an admirer and
a critic.

He credits Lagrange with showing that the most varied consequences of the motions
of systems may be derived from one radical formula, and says the beauty of the
method so suits the dignity of the results as to make of the work a kind of
scientific poem.<ref>Hamilton, ''On a General Method in Dynamics'' (1834),
Introductory Remarks. Verbatim. ESTABLISHED source.</ref>

Then the limitation: the method gives the differential equations elegantly and
does not give their solution. For a system of attracting points, Lagrange's
formalism reduces the problem to integrating a large system of equations, and
supplies no way to do the integrating.

Hamilton's own method addresses exactly that gap.

== The two editions ==

A closing note on why this page insists on the split.

The 1811–15 edition is not a reprint. It is longer, reorganised, and altered by
Lagrange himself late in life, with the second volume published after his death.

Modern references cite whichever is convenient, and the fourth edition of 1888 —
a reprint of the second with Bertrand's notes — is the one most easily reached.

Any quotation from Lagrange therefore needs its edition named. This volume quotes
none, because it has opened neither.

== Where to go next ==

Jacobi's reworking of Hamilton's method is at
[[Hamilton's principle/History/Jacobi]].

== References ==

<references />

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	'''Neither has been opened for this volume.''' No equation and no wording is		'''Neither has been opened for this volume.''' No equation and no wording is
	attributed to Lagrange on this page. See [[Hamilton's principle/Sources]].		attributed to Lagrange on this page. See [[Hamilton's principle/Sources\|Sources]].

	== What the book did ==		== What the book did ==
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	'''It absorbed constraints.''' D'Alembert's principle, described at		'''It absorbed constraints.''' D'Alembert's principle, described at
	[[Hamilton's principle/History/d'Alembert and virtual work]], eliminates the		[[Hamilton's principle/History/d'Alembert and virtual work\|d'Alembert and virtual work]], eliminates the
	constraint forces before any calculation. Lagrange combined that with the		constraint forces before any calculation. Lagrange combined that with the
	variational method and produced the multiplier technique of		variational method and produced the multiplier technique of
	[[Hamilton's principle/Mathematics/Lagrange multipliers]].		[[Hamilton's principle/Mathematics/Lagrange multipliers\|Lagrange multipliers]].

	'''It made the coordinates free.''' This is the decisive step. Newton's laws are		'''It made the coordinates free.''' This is the decisive step. Newton's laws are
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	Euler had reached it earlier by a laborious limiting argument on a polygon. The		Euler had reached it earlier by a laborious limiting argument on a polygon. The
	young Lagrange sent him a cleaner variational derivation — essentially the one on		young Lagrange sent him a cleaner variational derivation — essentially the one on
	[[Hamilton's principle/Mathematics/The first variation]] — and Euler adopted it,		[[Hamilton's principle/Mathematics/The first variation\|the first variation]] — and Euler adopted it,
	gave it the name ''calculus of variations'', and let Lagrange publish first.		gave it the name ''calculus of variations'', and let Lagrange publish first.

	The conduct is worth noting beside Euler's conduct in the priority dispute at		The conduct is worth noting beside Euler's conduct in the priority dispute at
	[[Hamilton's principle/History/The Maupertuis-Koenig affair]]. In both cases he		[[Hamilton's principle/History/The Maupertuis-Koenig affair\|the Maupertuis-Koenig affair]]. In both cases he
	declined a claim he could have pressed. In one case that looks like generosity to		declined a claim he could have pressed. In one case that looks like generosity to
	a young mathematician; in the other it looks like deference to a superior.		a young mathematician; in the other it looks like deference to a superior.
Line 83:		Line 83:
	The principle only became more than a curiosity when Hamilton found something else		The principle only became more than a curiosity when Hamilton found something else
	to do with the action — vary the endpoints rather than the path, as on		to do with the action — vary the endpoints rather than the path, as on
	[[Hamilton's principle/Mathematics/Action as a function of its endpoints]] — and		[[Hamilton's principle/Mathematics/Action as a function of its endpoints\|Action as a function of its endpoints]] — and
	when Noether showed what its symmetries imply, at		when Noether showed what its symmetries imply, at
	[[Hamilton's principle/History/Noether 1918]].		[[Hamilton's principle/History/Noether 1918\|Noether 1918]].

	== What Hamilton said about it ==		== What Hamilton said about it ==
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	Jacobi's reworking of Hamilton's method is at		Jacobi's reworking of Hamilton's method is at
	[[Hamilton's principle/History/Jacobi]].		[[Hamilton's principle/History/Jacobi\|Jacobi]].

	== References ==		== References ==

	<references />		<references />