Hamilton's principle/Sources - Revision history

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This page states exactly which text in this volume is '''verbatim''', which is
'''editorial''', and which is '''reconstructed''', and what would be needed to
close each remaining gap. It is the volume's provenance record, and every page
that carries a provenance block points here.

The discipline is deliberate. A companion volume on Maxwell's ''On Physical Lines
of Force'' found, too late, that every reachable electronic copy of that paper had
lost its mathematics: equation numbers were attested but equation bodies had to be
reconstructed. This volume was planned to avoid repeating that, by verifying
source reachability '''before''' authoring rather than after.

== The three classes ==

[[File:OPL-HP-F-53.png|thumb|right|400px|The three provenance classes, and the empty one.]]

'''Verbatim''' text is transcribed from a source this volume has opened and
checked. It appears in quotation or is explicitly marked as verbatim in a
footnote. Every verbatim passage in this volume comes from a source tagged
ESTABLISHED below.

'''Editorial''' text is this volume's own prose: derivations performed here,
summaries, historical framing, and connective argument. Most of the volume is
editorial, and editorial passages that touch on history carry a provenance block
saying so.

'''Reconstructed''' text would be a source's content restated from secondary
literature without having opened the source. '''This volume contains no
reconstructed equations.''' Where a source could not be verified, its content is
paraphrased at the level of a summary, no equation is attributed to it, and the
page says so. That is the rule the Maxwell volume could not follow.

== Source survey ==

[[File:OPL-HP-F-54.png|thumb|right|400px|Four states an electronic source can be in.]]

Tags: ESTABLISHED, verified with the verification stated. PROVISIONAL, located
but not probed. UNCOMPUTED, work not performed, with its kind of block named.

{| class="wikitable"
! source !! edition targeted !! status !! reproduction
|-
| Hamilton, ''On a General Method in Dynamics'', 1834 || Wilkins edition of Phil. Trans. 124, 247–308 || ESTABLISHED || public domain; quoted verbatim
|-
| Hamilton, ''Second Essay'', 1835 || Wilkins edition of Phil. Trans. 125, 95–144 || ESTABLISHED || public domain; quoted verbatim
|-
| Noether, ''Invariante Variationsprobleme'', 1918 || Göttinger Nachrichten, 235–257 || ESTABLISHED, via the translation below || public domain; equations reproduced
|-
| Tavel translation of Noether, 1971 || Transport Theory and Statistical Physics 1(3) || ESTABLISHED || '''under copyright'''; paraphrased only
|-
| Euler, ''Methodus inveniendi'', Additamentum II, 1744 || Latin original, pp. 311–320 || PROVISIONAL || public domain
|-
| Euler, Additamentum II, English || Wikisource community translation || ESTABLISHED, with a notation caveat || CC-licensed, not public domain
|-
| Maupertuis, ''Accord de différentes loix de la nature'' || read 15 April 1744, printed 1748, pp. 417–426 || PROVISIONAL, statement corroborated not transcribed || public domain
|-
| Lagrange, ''Mécanique analytique'', 1788 || first edition || PROVISIONAL || public domain
|-
| Lagrange, ''Mécanique analytique'', 1811–15 || second edition || PROVISIONAL || public domain
|-
| Jacobi, ''Vorlesungen über Dynamik'', 1866 || Clebsch edition, German || PROVISIONAL || public domain
|-
| Jacobi, English translation, 2009 || Balagangadharan || not consulted || '''under copyright'''
|-
| Fermat, ''Synthesis ad refractiones'', 1662 || Latin, in the ''Œuvres'' on French Wikisource || PROVISIONAL, text-bearing but OCR-damaged || public domain
|-
| Feynman, 1948 || Rev. Mod. Phys. 20, 367 || UNCOMPUTED, block: absence || under copyright
|}

Thirteen rows, where the volume's plan began with nine. Four splits were forced
by the evidence and are described below.

== Splits forced during the survey ==

[[File:OPL-HP-F-55.png|thumb|right|400px|Three modernised statements attributed to their authors.]]

'''Lagrange, 1788 against 1811–15.''' These are not the same book. Lagrange
described the second edition in his own preface as, in many respects, a new work,
more extensive than the first. Quoting the 1811 text under an 1788 attribution
would be a provenance error of exactly the kind this page exists to prevent, so
the two carry separate rows and separate probes.

'''Euler, original against translation.''' The Latin original and the Wikisource
English are separate rows because their reachability differs and because a
translation is not the source.

'''Noether and Jacobi, original against translation.''' In both cases the
original is public domain and the modern translation is not. This produces a rule
the volume follows on every affected page:

<blockquote>
Where a source's mathematics is public domain and its available English is a
copyrighted translation, the volume reproduces the '''equations''' verbatim in
the author's notation and '''paraphrases''' the prose, marking every paraphrase as
such.
</blockquote>

[[Hamilton's principle/Mathematics/Noether's first theorem]] and
[[Hamilton's principle/Mathematics/Noether's second theorem]] both open with a
provenance note stating this. No English wording of Noether's is transcribed
anywhere in the volume.

== What is verbatim ==

[[File:OPL-HP-F-57.png|thumb|right|400px|Transcribed against corroborated.]]

All verbatim text in this volume is Hamilton's, from the two ESTABLISHED essays.
Twenty citations are marked in footnotes. The substantial ones:

{| class="wikitable"
! passage !! where quoted
|-
| the law of least, or as it might be better called, of stationary action || [[Hamilton's principle/Mathematics/Functionals and the space of paths]], [[Hamilton's principle/Mathematics/Action as a function of its endpoints]], [[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]]
|-
| the comparison paths as geometrically imaginable, dynamically impossible motion || [[Hamilton's principle/Mathematics/The first variation]]
|-
| equation (1.), the fundamental formula || [[Hamilton's principle/Mathematics/The first variation]], [[Hamilton's principle/Mathematics/Several dependent variables]]
|-
| equations (A.) to (E.), the law of varying action || [[Hamilton's principle/Mathematics/Action as a function of its endpoints]]
|-
| equations (F.) and (G.), the partial differential pair || [[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]]
|-
| equation (Y.), the canonical form || [[Hamilton's principle/Mathematics/Several dependent variables]]
|-
| the Kepler sequence (71.), (97.), (101.)–(107.), (A3.), (F3.), (S3.), (U3.) || [[Hamilton's principle/Applications/Central force and the Kepler problem]]
|-
| the optical origin of the method || [[Hamilton's principle/Applications/The optical-mechanical analogy]]
|}

Noether's '''equations''' — her identity (3), the divergence relations (12) and
(13), and the dependencies (16) — are reproduced in her notation on the two
Noether pages and on
[[Hamilton's principle/Mathematics/Several independent variables]]. Her English
is not.

== What is editorial ==

Everything else. In particular, every derivation in the mathematics tracks is
performed on the page where it appears and is not transcribed from any source.
Pages carrying editorial historical claims with no verified source behind them
each carry a provenance block; there are nineteen such blocks in the volume.

The named cases where a historical attribution rests on no verified source:

* Legendre 1786 and Jacobi 1837 on the sufficiency conditions, at [[Hamilton's principle/Mathematics/The Legendre condition]]
* Johann Bernoulli 1696 and Huygens 1659, at [[Hamilton's principle/Applications/The brachistochrone]]
* Galileo's parabola error and the 1691 solutions, at [[Hamilton's principle/Applications/The catenary]]
* Euler's and Lagrange's priority in the Euler-Lagrange equation, at [[Hamilton's principle/Mathematics/The Euler-Lagrange equation]]
* Schrödinger's 1926 construction, at [[Hamilton's principle/Applications/The optical-mechanical analogy]]
* Euler's equations for the rigid body, at [[Hamilton's principle/Applications/The rigid body]]

None of these attributions is load-bearing for any result in the volume. Each is
context, and each is marked.

== What is reconstructed ==

Nothing. This is the sentence the Maxwell volume could not write.

Where a source is unverified, the volume states a summary and attributes no
equation. The clearest case is
[[Hamilton's principle/Applications/The Feynman path integral]]: the postulate is
paraphrased from the paper's abstract, which is the only obtainable part, and no
equation on that page is attributed to Feynman.

== The Euler probe, and what it found ==

This row was tagged '''at risk''' rather than merely unprobed, because Wikisource
is exactly the source class that destroyed the Maxwell volume's equations: its
retrieval layer delivered prose intact while dropping the mathematics, so the
result read as complete and was not.

'''The probe cleared that risk and raised a different one.'''

The mathematics '''is''' present in the retrieved text of the Wikisource
translation. The momentum product and the time-integral form both appear. This is
not the Maxwell failure mode.

But the translation has silently modernised Euler's variable. The Euler Archive's
own description of E065 states his formulation with <math>v</math> defined as
'''half the square of the velocity''', so that the minimised quantity is

:<math>\int M\,ds\,\sqrt{v}</math>

The Wikisource translation renders the same statement with <math>v</math> meaning
'''the velocity''', giving <math>\int M v\,ds</math>.

The two agree as physics and disagree as attribution. Quoting the second as
Euler's formula would attribute modernised notation to him — the same class of
provenance error this page exists to prevent, arriving through a translation
rather than through a lossy scan.

'''Rule adopted.''' Euler's action integral is written in this volume as
<math>\int M\,ds\,\sqrt{v}</math> with his own definition of <math>v</math>
stated alongside it, and the modernised form is given only as a gloss, marked as
such. The Latin original at the Euler Archive remains the authority, and probing
it would close the row completely.

Two further notes. The Wikisource text is a community translation, released under
a Creative Commons licence rather than being public domain, so it is attributed
rather than treated as freely reproducible. And Wikisource is cache-only from this
volume's working environment: its pages could not be fetched directly, and the
probe was performed against indexed content. That is a limitation of the
environment, not of the source.

== Corroboration is not transcription ==

One row needs a distinction the other rows do not.

Maupertuis's central sentence is quoted identically by four independent sources.
That is strong evidence about the wording, and the volume quotes it. It is not the
same as having opened the memoir.

The companion Maxwell volume hit the same situation and recorded it the same way:
its famous velocity-of-light sentence was verified word for word against four
sources, and whether the printing italicised it was left unverified because nobody
had seen the page.

Corroborated passages are marked as corroborated wherever they appear. There is
one in this volume, on
[[Hamilton's principle/History/Maupertuis and the 1744 memoir]].

== A fourth source state ==

The survey began with three states a source could be in: text-bearing,
image-only, or stripped of its mathematics while its prose survived. The third is
the one that destroyed the companion Maxwell volume.

Fermat forced a fourth. His text is transcribed and retrievable, and it is
'''OCR-damaged''': words corrupted, and a Greek phrase reduced to characters that
are not Greek.

That state is not covered by the other three. It is not stripped, and it is not a
page image. It is legible and unreliable at the level of individual words — which
is precisely the level at which verbatim quotation operates.

Rule adopted: '''an OCR-damaged transcription supports paraphrase and does not
support quotation.''' Nothing is quoted from Fermat in this volume.

== Three notation traps found during the survey ==

Both are the same error in different clothes: a later convention attributed to an
earlier author.

'''Euler.''' His <math>v</math> is half the square of the velocity, so his integral
carries a square root. The available English translation modernises <math>v</math>
to the velocity and the root disappears. Recorded above.

'''Maupertuis.''' His 1744 action is speed times distance, with '''no mass'''. Mass
entered in a separate Berlin memoir of 1746. Writing the 1744 quantity with a mass
in it attributes the second memoir's definition to the first.

'''Fermat.''' His postulate is that nature operates by the easiest and readiest
ways. It names no quantity and asserts no minimum. ''Least time'' is a later gloss,
routinely attributed to him as his statement.

None of the three errors changes any physics. Both would put words in an author's mouth, which
is what this page exists to prevent.

== To close the remaining gaps ==

{| class="wikitable"
! row !! what would close it
|-
| Euler, Latin || open the Euler Archive scan (E065h) and confirm the <math>\sqrt{v}</math> form against the translation. The scan is a page image; expect transcription cost, not retrieval
|-
| Euler, English || '''resolved.''' Mathematics present; notation caveat recorded above
|-
| Maupertuis || '''partly resolved.''' The central sentence is corroborated identically across four independent sources but has not been read in the memoir itself. Opening the Wikisource ''Page:'' transcription would close it
|-
| Lagrange 1788 || open the Gallica facsimile. It is a page-image reproduction; expect no text layer
|-
| Lagrange 1811–15 || open the Gallica text-image volumes or the Internet Archive scans, which may carry OCR
|-
| Jacobi || open the German 1866 scan at Chapter 6, ''Das Princip der kleinsten Wirkung''. Three public-domain scans exist
|-
| Fermat || '''located.''' The transcription is OCR-damaged, so verbatim quotation needs a clean text: collate against a print edition of the ''Œuvres''
|-
| Feynman || institutional access to the paper, or the 1942 thesis published in 2005. This is the only row blocked by '''absence''' rather than by unperformed work
|}

Closing a row does not require rewriting the volume. Each affected page carries
its provenance block in one place, and verifying a source means replacing that
block with a citation, not revising a derivation.

== Pages whose content depends on an unresolved row ==

The history track was originally gated on this survey. It is now '''complete''',
written under a discipline that made the gate unnecessary: where a source is
unverified, the page attributes no equation and no wording to it, states a summary,
and carries a provenance block naming what is missing.

The pages formerly listed as blocked:
[[Hamilton's principle/History/Least time before least action]],
[[Hamilton's principle/History/Maupertuis and the 1744 memoir]],
[[Hamilton's principle/History/Euler and the Methodus inveniendi]],
[[Hamilton's principle/History/Lagrange and the Mecanique analytique]],
[[Hamilton's principle/History/Jacobi]] and
[[Hamilton's principle/History/Feynman 1948]].

All are written. Only
[[Hamilton's principle/History/Hamilton 1834 and 1835]] and
[[Hamilton's principle/History/Noether 1918]] rest on ESTABLISHED rows and carry
quotation; the rest quote nothing.

The lesson is worth recording. '''An unverified source does not block a page. It
blocks quotation from that page.''' The distinction was not obvious when this
survey was written and it saved the history track.

== Further reading, not treated ==

Named here so that no reader mistakes an omission for a claim of completeness:

* the '''direct method''' and the existence of minimisers. This volume proves conditions for a minimum and never that one exists; see the closing note on [[Hamilton's principle/Mathematics/The Weierstrass excess function]]
* '''constrained Hamiltonian dynamics''' for degenerate Lagrangians, flagged at [[Hamilton's principle/Mathematics/The Legendre transform]] and realised at [[Hamilton's principle/Applications/The electromagnetic Lagrangian]]
* '''non-holonomic''' dynamics, and the vakonomic-versus-dynamic distinction flagged at [[Hamilton's principle/Mathematics/Holonomic constraints]]
* the '''Laplace-Runge-Lenz''' symmetry of the Kepler problem, flagged at [[Hamilton's principle/Applications/Central force and the Kepler problem]]
* rigorous construction of the '''path integral''' measure, flagged at [[Hamilton's principle/Applications/The Feynman path integral]]

== Register ==

The mathematics, history and applications tracks take a technical register. The
plain-language track takes ISO 24495-1. This page takes the technical register.

← Older revision		Revision as of 04:58, 16 September 2026
Line 96:		Line 96:
	</blockquote>		</blockquote>

	[[Hamilton's principle/Mathematics/Noether's first theorem]] and		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]] and
	[[Hamilton's principle/Mathematics/Noether's second theorem]] both open with a		[[Hamilton's principle/Mathematics/Noether's second theorem\|Noether's second theorem]] both open with a
	provenance note stating this. No English wording of Noether's is transcribed		provenance note stating this. No English wording of Noether's is transcribed
	anywhere in the volume.		anywhere in the volume.
Line 111:		Line 111:
	! passage !! where quoted		! passage !! where quoted
	\|-		\|-
	\| the law of least, or as it might be better called, of stationary action \|\| [[Hamilton's principle/Mathematics/Functionals and the space of paths]], [[Hamilton's principle/Mathematics/Action as a function of its endpoints]], [[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]]		\| the law of least, or as it might be better called, of stationary action \|\| [[Hamilton's principle/Mathematics/Functionals and the space of paths\|Functionals and the space of paths]], [[Hamilton's principle/Mathematics/Action as a function of its endpoints\|Action as a function of its endpoints]], [[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton\|Abbreviated action: Maupertuis versus Hamilton]]
	\|-		\|-
	\| the comparison paths as geometrically imaginable, dynamically impossible motion \|\| [[Hamilton's principle/Mathematics/The first variation]]		\| the comparison paths as geometrically imaginable, dynamically impossible motion \|\| [[Hamilton's principle/Mathematics/The first variation\|the first variation]]
	\|-		\|-
	\| equation (1.), the fundamental formula \|\| [[Hamilton's principle/Mathematics/The first variation]], [[Hamilton's principle/Mathematics/Several dependent variables]]		\| equation (1.), the fundamental formula \|\| [[Hamilton's principle/Mathematics/The first variation\|the first variation]], [[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]]
	\|-		\|-
	\| equations (A.) to (E.), the law of varying action \|\| [[Hamilton's principle/Mathematics/Action as a function of its endpoints]]		\| equations (A.) to (E.), the law of varying action \|\| [[Hamilton's principle/Mathematics/Action as a function of its endpoints\|Action as a function of its endpoints]]
	\|-		\|-
	\| equations (F.) and (G.), the partial differential pair \|\| [[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]]		\| equations (F.) and (G.), the partial differential pair \|\| [[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation\|the Hamilton-Jacobi equation]]
	\|-		\|-
	\| equation (Y.), the canonical form \|\| [[Hamilton's principle/Mathematics/Several dependent variables]]		\| equation (Y.), the canonical form \|\| [[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]]
	\|-		\|-
	\| the Kepler sequence (71.), (97.), (101.)–(107.), (A3.), (F3.), (S3.), (U3.) \|\| [[Hamilton's principle/Applications/Central force and the Kepler problem]]		\| the Kepler sequence (71.), (97.), (101.)–(107.), (A3.), (F3.), (S3.), (U3.) \|\| [[Hamilton's principle/Applications/Central force and the Kepler problem\|Central force and the Kepler problem]]
	\|-		\|-
	\| the optical origin of the method \|\| [[Hamilton's principle/Applications/The optical-mechanical analogy]]		\| the optical origin of the method \|\| [[Hamilton's principle/Applications/The optical-mechanical analogy\|the optical-mechanical analogy]]
	\|}		\|}

Line 131:		Line 131:
	(13), and the dependencies (16) — are reproduced in her notation on the two		(13), and the dependencies (16) — are reproduced in her notation on the two
	Noether pages and on		Noether pages and on
	[[Hamilton's principle/Mathematics/Several independent variables]]. Her English		[[Hamilton's principle/Mathematics/Several independent variables\|Several independent variables]]. Her English
	is not.		is not.

Line 143:		Line 143:
	The named cases where a historical attribution rests on no verified source:		The named cases where a historical attribution rests on no verified source:

	* Legendre 1786 and Jacobi 1837 on the sufficiency conditions, at [[Hamilton's principle/Mathematics/The Legendre condition]]		* Legendre 1786 and Jacobi 1837 on the sufficiency conditions, at [[Hamilton's principle/Mathematics/The Legendre condition\|the Legendre condition]]
	* Johann Bernoulli 1696 and Huygens 1659, at [[Hamilton's principle/Applications/The brachistochrone]]		* Johann Bernoulli 1696 and Huygens 1659, at [[Hamilton's principle/Applications/The brachistochrone\|the brachistochrone]]
	* Galileo's parabola error and the 1691 solutions, at [[Hamilton's principle/Applications/The catenary]]		* Galileo's parabola error and the 1691 solutions, at [[Hamilton's principle/Applications/The catenary\|the catenary]]
	* Euler's and Lagrange's priority in the Euler-Lagrange equation, at [[Hamilton's principle/Mathematics/The Euler-Lagrange equation]]		* Euler's and Lagrange's priority in the Euler-Lagrange equation, at [[Hamilton's principle/Mathematics/The Euler-Lagrange equation\|the Euler-Lagrange equation]]
	* Schrödinger's 1926 construction, at [[Hamilton's principle/Applications/The optical-mechanical analogy]]		* Schrödinger's 1926 construction, at [[Hamilton's principle/Applications/The optical-mechanical analogy\|the optical-mechanical analogy]]
	* Euler's equations for the rigid body, at [[Hamilton's principle/Applications/The rigid body]]		* Euler's equations for the rigid body, at [[Hamilton's principle/Applications/The rigid body\|the rigid body]]

	None of these attributions is load-bearing for any result in the volume. Each is		None of these attributions is load-bearing for any result in the volume. Each is
Line 159:		Line 159:
	Where a source is unverified, the volume states a summary and attributes no		Where a source is unverified, the volume states a summary and attributes no
	equation. The clearest case is		equation. The clearest case is
	[[Hamilton's principle/Applications/The Feynman path integral]]: the postulate is		[[Hamilton's principle/Applications/The Feynman path integral\|the Feynman path integral]]: the postulate is
	paraphrased from the paper's abstract, which is the only obtainable part, and no		paraphrased from the paper's abstract, which is the only obtainable part, and no
	equation on that page is attributed to Feynman.		equation on that page is attributed to Feynman.
Line 218:		Line 218:
	Corroborated passages are marked as corroborated wherever they appear. There is		Corroborated passages are marked as corroborated wherever they appear. There is
	one in this volume, on		one in this volume, on
	[[Hamilton's principle/History/Maupertuis and the 1744 memoir]].		[[Hamilton's principle/History/Maupertuis and the 1744 memoir\|Maupertuis and the 1744 memoir]].

	== A fourth source state ==		== A fourth source state ==
Line 291:		Line 291:

	The pages formerly listed as blocked:		The pages formerly listed as blocked:
	[[Hamilton's principle/History/Least time before least action]],		[[Hamilton's principle/History/Least time before least action\|Least time before least action]],
	[[Hamilton's principle/History/Maupertuis and the 1744 memoir]],		[[Hamilton's principle/History/Maupertuis and the 1744 memoir\|Maupertuis and the 1744 memoir]],
	[[Hamilton's principle/History/Euler and the Methodus inveniendi]],		[[Hamilton's principle/History/Euler and the Methodus inveniendi\|Euler and the Methodus inveniendi]],
	[[Hamilton's principle/History/Lagrange and the Mecanique analytique]],		[[Hamilton's principle/History/Lagrange and the Mecanique analytique\|Lagrange and the Mecanique analytique]],
	[[Hamilton's principle/History/Jacobi]] and		[[Hamilton's principle/History/Jacobi\|Jacobi]] and
	[[Hamilton's principle/History/Feynman 1948]].		[[Hamilton's principle/History/Feynman 1948\|Feynman 1948]].

	All are written. Only		All are written. Only
	[[Hamilton's principle/History/Hamilton 1834 and 1835]] and		[[Hamilton's principle/History/Hamilton 1834 and 1835\|Hamilton 1834 and 1835]] and
	[[Hamilton's principle/History/Noether 1918]] rest on ESTABLISHED rows and carry		[[Hamilton's principle/History/Noether 1918\|Noether 1918]] rest on ESTABLISHED rows and carry
	quotation; the rest quote nothing.		quotation; the rest quote nothing.

Line 311:		Line 311:
	Named here so that no reader mistakes an omission for a claim of completeness:		Named here so that no reader mistakes an omission for a claim of completeness:

	* the '''direct method''' and the existence of minimisers. This volume proves conditions for a minimum and never that one exists; see the closing note on [[Hamilton's principle/Mathematics/The Weierstrass excess function]]		* the '''direct method''' and the existence of minimisers. This volume proves conditions for a minimum and never that one exists; see the closing note on [[Hamilton's principle/Mathematics/The Weierstrass excess function\|the Weierstrass excess function]]
	* '''constrained Hamiltonian dynamics''' for degenerate Lagrangians, flagged at [[Hamilton's principle/Mathematics/The Legendre transform]] and realised at [[Hamilton's principle/Applications/The electromagnetic Lagrangian]]		* '''constrained Hamiltonian dynamics''' for degenerate Lagrangians, flagged at [[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]] and realised at [[Hamilton's principle/Applications/The electromagnetic Lagrangian\|the electromagnetic Lagrangian]]
	* '''non-holonomic''' dynamics, and the vakonomic-versus-dynamic distinction flagged at [[Hamilton's principle/Mathematics/Holonomic constraints]]		* '''non-holonomic''' dynamics, and the vakonomic-versus-dynamic distinction flagged at [[Hamilton's principle/Mathematics/Holonomic constraints\|Holonomic constraints]]
	* the '''Laplace-Runge-Lenz''' symmetry of the Kepler problem, flagged at [[Hamilton's principle/Applications/Central force and the Kepler problem]]		* the '''Laplace-Runge-Lenz''' symmetry of the Kepler problem, flagged at [[Hamilton's principle/Applications/Central force and the Kepler problem\|Central force and the Kepler problem]]
	* rigorous construction of the '''path integral''' measure, flagged at [[Hamilton's principle/Applications/The Feynman path integral]]		* rigorous construction of the '''path integral''' measure, flagged at [[Hamilton's principle/Applications/The Feynman path integral\|the Feynman path integral]]

	== Register ==		== Register ==