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	Jacobi took Hamilton's method within a decade of its publication and reshaped it		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		into the form still used. He also supplied the missing piece of the sufficiency
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	conjugate-point condition rests on secondary sources.		conjugate-point condition rests on secondary sources.
	</blockquote>		</blockquote>

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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 ==

[[File:OPL-HP-F-58.png|thumb|right|400px|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
[[Hamilton's principle/Sources]].

== Reshaping Hamilton's method ==

Hamilton's construction of
[[Hamilton's principle/Mathematics/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
[[Hamilton's principle/Mathematics/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
[[Hamilton's principle/Mathematics/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
[[Hamilton's principle/Applications/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
[[Hamilton's principle/Mathematics/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
[[Hamilton's principle/History/Noether 1918]]; and from Feynman, who shows why
there is a stationary-action principle at all, at
[[Hamilton's principle/History/Feynman 1948]].

== Where to go next ==

[[Hamilton's principle/History/Noether 1918]].

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

← Older revision		Revision as of 04:59, 16 September 2026
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	attributed to Jacobi here. The split between a public-domain original and a		attributed to Jacobi here. The split between a public-domain original and a
	copyrighted translation is the same situation as Noether's, described on		copyrighted translation is the same situation as Noether's, described on
	[[Hamilton's principle/Sources]].		[[Hamilton's principle/Sources\|Sources]].

	== Reshaping Hamilton's method ==		== Reshaping Hamilton's method ==

	Hamilton's construction of		Hamilton's construction of
	[[Hamilton's principle/Mathematics/Action as a function of its endpoints]]		[[Hamilton's principle/Mathematics/Action as a function of its endpoints\|Action as a function of its endpoints]]
	required '''two''' partial differential equations, an initial one and a final one,		required '''two''' partial differential equations, an initial one and a final one,
	because his characteristic function depends on both endpoints.		because his characteristic function depends on both endpoints.
Line 37:		Line 37:
	That result is Jacobi's theorem, and the equation is the one now called the		That result is Jacobi's theorem, and the equation is the one now called the
	Hamilton-Jacobi equation, treated at		Hamilton-Jacobi equation, treated at
	[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]].		[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation\|the Hamilton-Jacobi equation]].

	The simplification is not cosmetic. Hamilton's pair is hard to use. Jacobi's single		The simplification is not cosmetic. Hamilton's pair is hard to use. Jacobi's single
Line 57:		Line 57:

	Jacobi supplied it in 1837: the '''conjugate point''' condition, treated at		Jacobi supplied it in 1837: the '''conjugate point''' condition, treated at
	[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]].		[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points\|the Jacobi condition and conjugate points]].

	The idea is that neighbouring extremals leaving a common point can refocus, and		The idea is that neighbouring extremals leaving a common point can refocus, and
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	there: the sufficiency condition and the geometry are the same subject. The		there: the sufficiency condition and the geometry are the same subject. The
	connection is drawn at		connection is drawn at
	[[Hamilton's principle/Applications/Geodesics]].		[[Hamilton's principle/Applications/Geodesics\|Geodesics]].

	== Lectures, not a treatise ==		== Lectures, not a treatise ==
Line 102:		Line 102:
	that integrates the whole problem exists, from Jacobi. The sufficiency theory is		that integrates the whole problem exists, from Jacobi. The sufficiency theory is
	in place, from Legendre, Jacobi and — later in the century — Weierstrass, at		in place, from Legendre, Jacobi and — later in the century — Weierstrass, at
	[[Hamilton's principle/Mathematics/The Weierstrass excess function]].		[[Hamilton's principle/Mathematics/The Weierstrass excess function\|the Weierstrass excess function]].

	What is missing is any explanation of why the action should be the thing that		What is missing is any explanation of why the action should be the thing that
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	That arrives twice in the twentieth century: from Noether, who shows what the		That arrives twice in the twentieth century: from Noether, who shows what the
	action's symmetries imply, at		action's symmetries imply, at
	[[Hamilton's principle/History/Noether 1918]]; and from Feynman, who shows why		[[Hamilton's principle/History/Noether 1918\|Noether 1918]]; and from Feynman, who shows why
	there is a stationary-action principle at all, at		there is a stationary-action principle at all, at
	[[Hamilton's principle/History/Feynman 1948]].		[[Hamilton's principle/History/Feynman 1948\|Feynman 1948]].

	== Where to go next ==		== Where to go next ==

	[[Hamilton's principle/History/Noether 1918]].		[[Hamilton's principle/History/Noether 1918\|Noether 1918]].

	<blockquote>		<blockquote>
	Provenance: nothing on this page is quoted. The account of Jacobi's contributions		Provenance: nothing on this page is quoted. The account of Jacobi's contributions
	is editorial summary; the ''Vorlesungen'' row on		is editorial summary; the ''Vorlesungen'' row on
	[[Hamilton's principle/Sources]] is PROVISIONAL, and the 1837 attribution for the		[[Hamilton's principle/Sources\|Sources]] is PROVISIONAL, and the 1837 attribution for the
	conjugate-point condition rests on secondary sources.		conjugate-point condition rests on secondary sources.
	</blockquote>		</blockquote>