Hamilton's principle/Chronology

A timeline of the subject, from the first variational principle in physics to the explanation of why such principles exist.

Dates are of first statement or first publication, as marked. Where this volume's source survey could not verify a claim, the entry says so; see Sources.

The seventeenth century

Chronology of the action principle.

date	event	page
1621	Snel derives the law of refraction and does not publish it	—
1637	Descartes publishes the refraction law, assuming light moves faster in a dense medium	Least time before least action
1656	Huygens invents the pendulum clock and begins looking for a curve of equal-time descent	the brachistochrone
1662	Fermat, Synthesis ad refractiones. Derives the same law from the opposite assumption, on the postulate that nature operates by the easiest and readiest ways	Least time before least action
1691	The hanging chain is shown to be a catenary and not a parabola	the catenary
1696	Johann Bernoulli poses the brachistochrone problem	the brachistochrone

The eighteenth century

The two long gaps in the subject's history.

date	event	page
1743	d'Alembert, Traité de dynamique. Reduces dynamics to statics by treating inertia as a force	d'Alembert and virtual work
15 Apr 1744	Maupertuis reads the Accord to the Académie. Action is speed times distance, with no mass. Printed 1748	Maupertuis and the 1744 memoir
1744	Euler, Methodus inveniendi. The first treatise on the calculus of variations; the action principle appears in its second appendix	Euler and the Methodus inveniendi
1746	Maupertuis extends the definition of action to include mass, in a Berlin memoir	Maupertuis and the 1744 memoir
1750s	Lagrange sends Euler a variational derivation of the Euler-Lagrange equation. Euler adopts it and names the calculus of variations	Lagrange and the Mecanique analytique
1751–53	The Maupertuis-Koenig affair. The Berlin Academy declares the Leibniz letter a forgery; the letter is now generally held authentic	the Maupertuis-Koenig affair
1786	Legendre proposes his condition and believes it sufficient. Lagrange finds the error	the Legendre condition
1788	Lagrange, Mécanique analytique, first edition. Mechanics as analysis, with no diagrams	Lagrange and the Mecanique analytique

The nineteenth century

date	event	page
1811–15	Lagrange's second edition, described by him as in many respects a new work	Lagrange and the Mecanique analytique
1833	Hamilton announces the dynamical principle in the Dublin University Review	Hamilton 1834 and 1835
1834	Hamilton, On a General Method in Dynamics. The characteristic function and the law of varying action. Corrects least to stationary in passing	Action as a function of its endpoints
1835	Hamilton's Second Essay introduces the function modern texts call the action	Hamilton 1834 and 1835
1837	Jacobi supplies the conjugate-point condition, closing the gap Legendre left	the Jacobi condition and conjugate points
1842–43	Jacobi's Königsberg lectures. Reduces Hamilton's two equations to one	Jacobi
1850	Foucault measures light as slower in a dense medium, settling the question against Descartes and Maupertuis, and for Fermat	Least time before least action
1866	Jacobi's Vorlesungen über Dynamik published, fifteen years after his death	Jacobi
later 19th c.	Weierstrass supplies the excess function and the theory of strong minima	the Weierstrass excess function

The twentieth century

date	event	page
1915	General relativity published. Energy conservation in it becomes contested	the Einstein-Hilbert action
26 Jul 1918	Noether, Invariante Variationsprobleme, presented to the Göttingen Scientific Society. Two theorems; the second settles the Hilbert assertion	Noether 1918
1926	Schrödinger constructs wave mechanics by completing Hamilton's optical-mechanical analogy	the optical-mechanical analogy
1942	Feynman's doctoral thesis on the principle of least action in quantum mechanics. Published 2005	Feynman 1948
1948	Feynman, Space-Time Approach to Non-Relativistic Quantum Mechanics. The sum over paths	the Feynman path integral
1971	Tavel's English translation of Noether, fifty-three years after publication	Noether 1918

Two long silences

The timeline has two gaps worth naming, because each is longer than it looks and each was ended by the same kind of act.

1662 to 1744, eighty-two years. Fermat's principle stood as an isolated result about light. Nobody extended a variational principle to matter until Maupertuis and Euler did so independently in the same year.

1834 to 1926, ninety-two years. Hamilton observed that mechanics and ray optics obey the same equation. Ray optics is the short-wavelength approximation to wave optics, so mechanics should be the short-wavelength approximation to something. Nobody pursued it, because there was no evidence of a wavelength and nothing to be waving. Schrödinger then completed the analogy by working it backwards.

The second silence is the more striking. The observation was published, the inference is short, and the mathematics needed was in place the whole time. What was missing was a reason to believe the missing corner existed.

A pattern in the corrections

Four moments in the timeline are corrections rather than discoveries, and they run in the same direction.

Lagrange corrects Legendre in 1786: a local condition is not sufficient.

Hamilton corrects the name in 1834: not least, but stationary.

Jacobi supplies the reason in 1837: minimality fails past a conjugate point.

Weierstrass supplies the rest later in the century: a path can lose to competitors close in position but not in slope.

Every one of them weakens a claim. The subject's history is largely the history of finding out that least was too strong a word, and the process took from 1786 to the end of the century.

Feynman's 1948 account then removes the word's basis entirely: nothing is minimised, because nothing is choosing.

Dates this volume could not verify

Several entries above rest on secondary sources rather than on documents this volume has opened.

Verified against ESTABLISHED sources: the 1834 and 1835 essays, and Noether's 1918 paper by way of its translation.

Everything else in these tables — including the Huygens, Bernoulli, Galileo, Legendre, Jacobi, Foucault and Schrödinger entries — is editorial. Dates are as reported by secondary literature and are not independently confirmed. See Sources.