Hamilton, On a General Method in Dynamics (1834)

Companion source

Source: Trinity College Dublin, D. R. Wilkins; local typeset PDF (65 pages) and Plain TeX transcription .tex; retrieved 2026-09-08. Metadata: William Rowan Hamilton, “On a General Method in Dynamics; by which the Study of the Motions of all free Systems of attracting or repelling Points is reduced to the Search and Differentiation of one central Relation, or characteristic Function”, Philosophical Transactions of the Royal Society of London 124 (1834), 247–308. Transcribed in Plain TeX by David R. Wilkins, Trinity College Dublin, 2000, as stated in the TeX header. Extraction: the TeX file is the transcription; the PDF is its typeset form. Original page breaks are marked in the transcription where the editor recorded them. Rights: the 1834 text is public domain. The editor’s pages carry no licence statement; the repository already relies on the same editor’s Newton transcription under identical conditions.

Source digest

Hamilton defines the characteristic function \(V\) as the accumulated living force of the motion, \(V=\int_0^t 2T\,dt\), and proves the law of varying action: the variation of \(V\) with respect to the final and initial coordinates and the energy constant \(H\) is \(\delta V=\sum m(x'\delta x+y'\delta y+z'\delta z)-\sum m(a'\delta a+b'\delta b+c'\delta c)+t\,\delta H\). All equations of motion then follow from one function of initial and final positions and \(H\).

For this project, \(V\) is the fixed-energy action that Euler and Maupertuis extremize as \(\int v\,ds\), whereas the constant-force difference \(\Delta S\) of C001 is a fixed-duration comparison of the principal function introduced in the Second Essay. Keeping the two functionals apart matters for the action-value gap question Q1.

Passage anchors

• TeX line 252: section heading “Function of such Motion, and Law of varying Action”.
• TeX line 446: the phrase “the law of varying action” at the statement of the theorem.
• TeX line 201, footnote: Lagrange’s Mécanique analytique cited as the source of the equations Hamilton reduces to one function.
• Introduction: Hamilton credits Maupertuis, Euler and Lagrange with the principle of least action and announces the characteristic function as its extension.

Coverage and limits

Reading level passage for the introduction and the definition and law of varying action; metadata for the remaining sections. Formula transcription was checked only through the editor’s typeset PDF, not against the 1834 printing.