Hamilton's principle/History/Jacobi

Jacobi took Hamilton's method within a decade of its publication and reshaped it into the form still used. He also supplied the missing piece of the sufficiency theory that Legendre had got wrong and Lagrange had noticed was wrong.

Two contributions, in different parts of this volume, from the same man.

Provenance note

The two long gaps in the subject's history.

The Vorlesungen über Dynamik are Jacobi's Königsberg lectures of 1842–43, edited by Clebsch and published in 1866, fifteen years after his death.

The German original is in the public domain and is reachable in at least three scans, including one from the University of Michigan's mathematics collection. The chapter on least action is the sixth.

An English translation appeared in 2009 and is under copyright.

Neither has been opened for this volume. No equation and no wording is attributed to Jacobi here. The split between a public-domain original and a copyrighted translation is the same situation as Noether's, described on Sources.

Reshaping Hamilton's method

Hamilton's construction of Action as a function of its endpoints required two partial differential equations, an initial one and a final one, because his characteristic function depends on both endpoints.

Jacobi's reworking needs one. He showed that a complete solution of a single equation suffices: given one, the motion follows by differentiation, with no integration required.

That result is Jacobi's theorem, and the equation is the one now called the Hamilton-Jacobi equation, treated at the Hamilton-Jacobi equation.

The simplification is not cosmetic. Hamilton's pair is hard to use. Jacobi's single equation, combined with separation of variables, is a practical method — and it is how the Kepler problem, the symmetric top and the other classically integrable systems are actually solved.

The names on the equation record a genuine partnership rather than a courtesy. Hamilton found the object; Jacobi made it usable.

The Jacobi condition

Jacobi's second contribution is in a different part of the subject and is, for this volume, the more important of the two.

Legendre had proposed in 1786 that a local condition — his own — was sufficient for a minimum. Lagrange found the error. For half a century nobody knew what was missing.

Jacobi supplied it in 1837: the conjugate point condition, treated at the Jacobi condition and conjugate points.

The idea is that neighbouring extremals leaving a common point can refocus, and that minimality fails once a path has run past the first place where they do. It is a condition on the whole journey, not on any point of it, which is precisely why every local test had missed it.

That result is the mathematical content behind this volume's insistence on the word stationary. Hamilton corrected the name in 1834; Jacobi supplied the reason in 1837, apparently without connecting the two.

Geodesics

Jacobi also worked on geodesics, including the difficult case of the ellipsoid, and on the curve enveloping the geodesics that leave a single point.

That envelope is the locus of conjugate points, so the two halves of his work meet there: the sufficiency condition and the geometry are the same subject. The connection is drawn at Geodesics.

Lectures, not a treatise

One thing about the Vorlesungen is worth stating because it affects how they should be read.

They are a record of lectures, taken down by a student and edited after Jacobi's death by someone else. They are not a book he wrote.

The mathematics is his. The arrangement, the emphasis, and in places the wording are Clebsch's and Borchardt's.

For a volume that cares about provenance, that matters: a quotation from the Vorlesungen is a quotation from an edited transcription of spoken lectures, which is a different kind of object from a quotation from Hamilton's own published essay.

Where this leaves the nineteenth century

With Jacobi the classical structure is essentially complete.

The variational machinery exists, from Euler and Lagrange. The reformulation in terms of endpoints and of phase space exists, from Hamilton. The single equation that integrates the whole problem exists, from Jacobi. The sufficiency theory is in place, from Legendre, Jacobi and — later in the century — Weierstrass, at the Weierstrass excess function.

What is missing is any explanation of why the action should be the thing that matters.

That arrives twice in the twentieth century: from Noether, who shows what the action's symmetries imply, at Noether 1918; and from Feynman, who shows why there is a stationary-action principle at all, at Feynman 1948.

Where to go next

Noether 1918.

Provenance: nothing on this page is quoted. The account of Jacobi's contributions is editorial summary; the Vorlesungen row on Sources is PROVISIONAL, and the 1837 attribution for the conjugate-point condition rests on secondary sources.