Hamilton's principle/History/Hamilton 1834 and 1835

Hamilton's two essays on dynamics, published in the Philosophical Transactions for 1834 and 1835, introduced the characteristic function and the law of varying action. They are the source of the principle this volume is named for, and they are the only sources from which this volume quotes verbatim.

Both are ESTABLISHED on Sources: the full text of each was obtained with its mathematics intact, in editions transcribed from the original publication. What follows can therefore give Hamilton's own words rather than an account of them.

The situation in 1834

The empty corner of the optical-mechanical analogy.

Hamilton opens by placing himself against Lagrange. He credits the Mécanique analytique generously — Lagrange had shown that the varied consequences of the motions of systems of bodies may be derived from one radical formula, the beauty of the method so suiting the dignity of the results as to make of his great work a kind of scientific poem.

Then he states the limitation. Lagrange's method produces the differential equations of motion elegantly. It does not produce their solution. For $n$ attracting points the problem is the integration of $3n$ second-order equations, and Hamilton observes that for the solar system with the sun and ten known planets that is thirty ordinary differential equations, or sixty of the first order after a transformation of Lagrange.

His proposal is to replace that by the search and differentiation of a single function satisfying two partial differential equations of the first order and the second degree.

He is careful about what he claims for it. He does not promise a faster calculation, and says so: even if it should be thought that no practical facility is gained, an intellectual pleasure may result from the reduction of the most complex researches respecting the forces and motions of body to the study of one characteristic function, the unfolding of one central relation.

The correction to least action

The passage that gives this volume its terminology occurs in §3 of the first essay, and it is worth being exact about what Hamilton says and does not say.

He observes that Lagrange and others, having shown that the variation of the action integral vanishes when the extreme coordinates and the constant $H$ are given, deduced from that result only the well known law of least action. He then corrects the name in passing, writing of this well known law of least, or as it might be better called, of stationary action.

The correction is made without emphasis, as a matter of accurate naming rather than as a discovery. Hamilton does not argue for it and does not exhibit a counterexample. The mathematical content behind the correction — that stationarity is necessary and minimality is a separate and stronger claim — was supplied by Legendre, Jacobi and Weierstrass across the century, and is developed in this volume on the second variation, the Jacobi condition and conjugate points and the Weierstrass excess function.

He is blunter about the older principle's utility than modern accounts usually report. Applied to determine the motion, he says, it serves only to form the differential equations of the second order, which can always be otherwise found — and he notes that Lagrange, Laplace and Poisson had for that reason spoken lightly of the utility of the principle in the then-present state of dynamics.

His own law of varying action, by contrast, expresses by means of a single function not the mere differential equations of motion, but their intermediate and their final integrals.

That is the distinction the whole essay turns on. Varying the path with fixed endpoints gives the equations of motion. Varying the endpoints of the actual motion gives their solution.

The characteristic function

Hamilton defines $V$ as the accumulated living force, the integral of twice the kinetic energy along the true motion, and obtains the law of varying action as his equation (A.). Its coefficients give the final momenta, the initial momenta with opposite sign, and the time. The technical content is set out on Action as a function of its endpoints with his equations quoted.

A point of reading that trips almost every secondary account: Hamilton's $V$ is $\int 2Tdt$, not $\int Ldt$. It is the abbreviated action. The distinction is developed on Abbreviated action: Maupertuis versus Hamilton.

Where the idea came from

Not from mechanics. Hamilton had constructed the characteristic function for optical systems in his Theory of Systems of Rays, and the dynamics is a transposition of it. He says so in the first paragraph of the essay, noting that the dynamical principle is only another form of an idea already applied to optics, and that an intention of applying it to the motion of systems of bodies had been announced at the publication of that theory.

He then makes a claim about method whose reach he could not have measured: that the manner of calculation exemplified in optics and dynamics seems not confined to those two sciences, and may constitute, when matured by the future labours of mathematicians, a separate branch of analysis.

Ninety-two years later Schrödinger constructed wave mechanics by running that analogy in the direction Hamilton had no reason to run it. See the optical-mechanical analogy.

The second essay, 1835

The first essay's method required eliminations that Hamilton found burdensome. He opens the second essay by saying that many of them may be avoided by a general transformation introducing the time explicitly into a part $S$ of the whole characteristic function $V$, and that it is now proposed to fix attention on that part.

So $S$ — the function modern texts call the action — enters as a technical improvement to $V$, in the second paper, and not as the starting point. The modern presentation begins where Hamilton finished.

The second essay also carries the fundamental formula in the same form as the first, $\Sigma .m(x^{″}\delta x+y^{″}\delta y+z^{″}\delta z)=\delta U$, which is quoted on the first variation.

What Hamilton actually demonstrated

The 1834 essay does not stop at the general method. Hamilton works the two-body problem in detail, obtaining the plane orbit, the areal constant, the conic classification, Kepler's equation, and Lambert's theorem — the last as a corollary about which variables the characteristic function contains. That worked case is the subject of Central force and the Kepler problem, and it is his own argument for the method: not that it computes faster, but that it makes structural facts visible.

He also derives the seven classical first integrals from the invariance of $V$ under common translations and rotations of the whole system — a symmetry argument yielding conservation laws, eighty-four years before Noether's first theorem stated the general result.

Reception

The essays were admired and little used. The characteristic function is difficult to construct in practice, and the partial differential equations Hamilton offers in place of the ordinary ones are not obviously easier. Jacobi took up the method within a decade and reshaped it into the form that carries both their names; that is Jacobi.

The formalism became indispensable for a reason unavailable in 1834: it turned out to be the classical limit of quantum mechanics, and the structures Hamilton built for the solar system are the ones the new theory needed.

Provenance: every quotation on this page is verbatim from an ESTABLISHED source and is footnoted to its location in the essay. The account of reception, and the remark about Jacobi, are editorial; the Jacobi row of the survey on Sources is PROVISIONAL and nothing is attributed to him here.