Lagrange and the Mecanique analytique
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
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.[1]
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.[2]
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.