Hamilton's principle/Sources

Hamilton's principle: map
Reading the volume How to read this volume Notation Glossary Equation index Figure gallery
Reading by level Start here: Comparing paths The mathematics: Functionals and the space of paths Stationary is not least: The Jacobi condition and conjugate points The applications: The brachistochrone The history: Hamilton 1834 and 1835
Sources Sources Chronology
Related Noether's theorems On Physical Lines of Force

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

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

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.

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

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:

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.

Noether's first theorem and 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

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:

passage	where quoted
the law of least, or as it might be better called, of stationary action	Functionals and the space of paths, Action as a function of its endpoints, Abbreviated action: Maupertuis versus Hamilton
the comparison paths as geometrically imaginable, dynamically impossible motion	the first variation
equation (1.), the fundamental formula	the first variation, Several dependent variables
equations (A.) to (E.), the law of varying action	Action as a function of its endpoints
equations (F.) and (G.), the partial differential pair	the Hamilton-Jacobi equation
equation (Y.), the canonical form	Several dependent variables
the Kepler sequence (71.), (97.), (101.)–(107.), (A3.), (F3.), (S3.), (U3.)	Central force and the Kepler problem
the optical origin of the method	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 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 the Legendre condition
• Johann Bernoulli 1696 and Huygens 1659, at the brachistochrone
• Galileo's parabola error and the 1691 solutions, at the catenary
• Euler's and Lagrange's priority in the Euler-Lagrange equation, at the Euler-Lagrange equation
• Schrödinger's 1926 construction, at the optical-mechanical analogy
• Euler's equations for the rigid body, at 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 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 $v$ defined as half the square of the velocity, so that the minimised quantity is

$\int Mds\sqrt{v}$

The Wikisource translation renders the same statement with $v$ meaning the velocity, giving $\int Mvds$.

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 $\int Mds\sqrt{v}$ with his own definition of $v$ 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 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 $v$ is half the square of the velocity, so his integral carries a square root. The available English translation modernises $v$ 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

row	what would close it
Euler, Latin	open the Euler Archive scan (E065h) and confirm the $\sqrt{v}$ 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: Least time before least action, Maupertuis and the 1744 memoir, Euler and the Methodus inveniendi, Lagrange and the Mecanique analytique, Jacobi and Feynman 1948.

All are written. Only Hamilton 1834 and 1835 and 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 the Weierstrass excess function
• constrained Hamiltonian dynamics for degenerate Lagrangians, flagged at the Legendre transform and realised at the electromagnetic Lagrangian
• non-holonomic dynamics, and the vakonomic-versus-dynamic distinction flagged at Holonomic constraints
• the Laplace-Runge-Lenz symmetry of the Kepler problem, flagged at Central force and the Kepler problem
• rigorous construction of the path integral measure, flagged at 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.