View source for 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 [[Hamilton's principle/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 ==

[[File:OPL-HP-F-48.png|thumb|right|400px|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.<ref>Hamilton, ''On a General
Method in Dynamics'' (1834), Introductory Remarks. Verbatim. ESTABLISHED
source.</ref>

Then he states the limitation. Lagrange's method produces the '''differential
equations''' of motion elegantly. It does not produce their solution. For
<math>n</math> attracting points the problem is the integration of
<math>3n</math> 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.<ref>Hamilton,
''On a General Method in Dynamics'' (1834), Introductory Remarks. Verbatim.
ESTABLISHED source.</ref>

== 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
<math>H</math> 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.<ref>Hamilton,
''On a General Method in Dynamics'' (1834), §3. Verbatim. ESTABLISHED
source.</ref>

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
[[Hamilton's principle/Mathematics/The second variation]],
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]]
and
[[Hamilton's principle/Mathematics/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.<ref>Hamilton, ''On a General Method in Dynamics'' (1834), §3. Verbatim.
ESTABLISHED source.</ref>

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 <math>V</math> 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
[[Hamilton's principle/Mathematics/Action as a function of its endpoints]] with
his equations quoted.

A point of reading that trips almost every secondary account: '''Hamilton's
<math>V</math> is <math>\int 2T\,dt</math>, not <math>\int L\,dt</math>.''' It is
the abbreviated action. The distinction is developed on
[[Hamilton's principle/Mathematics/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.<ref>Hamilton, ''On a
General Method in Dynamics'' (1834), Introductory Remarks. Verbatim in substance.
ESTABLISHED source.</ref>

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.<ref>Hamilton, ''On a General
Method in Dynamics'' (1834), Introductory Remarks. Verbatim in substance.
ESTABLISHED source.</ref>

Ninety-two years later Schrödinger constructed wave mechanics by running that
analogy in the direction Hamilton had no reason to run it. See
[[Hamilton's principle/Applications/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 <math>S</math> of the
whole characteristic function <math>V</math>, and that it is now proposed to fix
attention on that part.

So <math>S</math> — the function modern texts call the action — enters as a
technical improvement to <math>V</math>, 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, <math>\Sigma\, . \, m(x''\delta x + y''\delta y + z''\delta z) = \delta
U</math>, which is quoted on
[[Hamilton's principle/Mathematics/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
[[Hamilton's principle/Applications/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
<math>V</math> under common translations and rotations of the whole system — a
symmetry argument yielding conservation laws, eighty-four years before
[[Hamilton's principle/Mathematics/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 [[Hamilton's principle/History/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.

<blockquote>
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
[[Hamilton's principle/Sources]] is PROVISIONAL and nothing is attributed to him
here.
</blockquote>

== References ==

<references />