Hamilton's principle/History/Maupertuis and the 1744 memoir

Maupertuis announced the principle of least action to the Académie Royale des Sciences on 15 April 1744, in a memoir titled Accord de différentes loix de la nature qui avoient jusqu'ici paru incompatibles. It is the first statement of an action principle for anything beyond optics, and it is the source of the name the subject still carries.

It is also the source of the word this volume spends most of its length correcting.

Provenance note

Three modernised statements attributed to their authors.

The memoir exists in a page-by-page transcription on French Wikisource, taken from a scan, and the transcription carries running text in the original eighteenth-century orthography.

Maupertuis's central sentence is corroborated below rather than transcribed. Four independent sources quote it with identical wording, which is strong evidence. It is not the same as having opened the memoir and read the sentence in it, and this volume does not pretend otherwise. See Sources.

A date worth getting right

Transcribed against corroborated.

The memoir was read on 15 April 1744 and printed in 1748, in the Académie's volume for 1744, at pages 417 to 426.

Citations give either date, and both are correct about different events. This volume cites the reading date, because the priority questions on the Maupertuis-Koenig affair turn on when the claim was made rather than when it appeared.

The problem he was solving

Maupertuis was not looking for a principle of mechanics. He was trying to settle an argument about light.

Descartes had derived the law of refraction from an assumption that light travels faster in a dense medium. Fermat had derived the same law from least time, which requires light to travel slower there. Two derivations, one law, incompatible assumptions.

The memoir's title says exactly this: an accord between different laws of nature which had until then seemed incompatible.

Maupertuis was a Newtonian and sided with the Cartesians on the speed question — he believed light goes faster in glass. So Fermat's least time was unavailable to him. He needed a different quantity to minimise, one that would give the right law under his assumption.

He worked backwards from the answer to the quantity. That is the honest description of the method, and it is why the choice can look arbitrary.

What he minimised

The quantity he arrived at he called the action.

In the 1744 memoir, action is the product of speed and distance.

Note what is missing. There is no mass in it. Mass entered in a separate memoir presented to the Berlin Academy in 1746, which extended the definition to mass times speed times distance and applied it to collisions.

Writing the 1744 quantity with a mass in it is an error, and a common one. It attributes to the first memoir a definition the second one supplied. This volume keeps the two apart, as it keeps Euler's own notation apart from its modern gloss on Euler and the Methodus inveniendi.

The statement

Maupertuis's formulation of the principle, in the memoir, is:

Lorsqu'il arrive quelque changement dans la nature, la quantité d'action, nécessaire pour ce changement, est la plus petite qui soit possible.

In English: when some change occurs in nature, the quantity of action necessary for that change is the least possible.

Corroborated, not transcribed. The wording above is identical across four independent sources. This volume has not opened the memoir to confirm it.

Two things are worth noticing about the sentence.

It is a claim about all of nature, not about light. The memoir arrives at the quantity through an optical problem and then states it universally. That leap is the memoir's real content.

The word "least" is doing real work. It is not a loose synonym for "stationary". Maupertuis meant a minimum, and his argument depended on it.

The metaphysics

The reason he meant it is that the principle was, for him, evidence of design.

He presented it as displaying the wisdom of the Creator: nature achieves its effects by the smallest expenditure, and that economy is a mark of the intelligence that arranged it.

This was not decoration around a physical result. It was the argument for taking the principle seriously in the first place. Maupertuis had no derivation. What he had was a quantity that gave the right optical law and a reason to believe nature would minimise something.

It is fair to say that the metaphysics is why he generalised the claim to all of nature, and unfair to say that the metaphysics is why the claim was right.

Why the metaphysics had to go

The principle survived. The reasoning for it did not, and the reason is the subject of this volume.

Nature does not minimise action. It makes it stationary. Stationary includes maxima and saddle points, and both occur in real physical situations — the maximum-time light path on Fermat's principle is the clearest case.

An argument from economy cannot survive that. A path that takes the longest time is not evidence of a Creator's thrift.

Hamilton made the correction in 1834, in passing, calling it the law of least, or as it might be better called, of stationary action.

The quantum account on the Feynman path integral then removes the last trace of purpose. Stationary action is where interference fails to cancel. Nothing is being economised and nothing is choosing.

What he got right

It would be easy to file Maupertuis as a man who reached a correct result by poor reasoning, and that is not quite fair.

He was wrong about the speed of light in a dense medium, and Foucault settled it against him in 1850. He was wrong that the principle expresses minimality. His grounds were metaphysical and his mathematics was thin — Euler, in the same year, had far more of it.

But he saw that a quantity of the dimensions of action might be the thing nature extremises, and that the claim should extend beyond optics to mechanics. Both are correct, and neither was obvious.

Euler, who had the mathematics, did not claim priority and defended Maupertuis's.

Where to go next

Euler's independent and better-founded statement, from the same year, is at Euler and the Methodus inveniendi. The priority dispute that followed is at the Maupertuis-Koenig affair.