Hamilton, Second Essay on a General Method in Dynamics (1835)
Source: Trinity College Dublin, D. R. Wilkins; local typeset PDF (55 pages) and Plain TeX transcription
.tex; retrieved 2026-09-08. Metadata: William Rowan Hamilton, “Second Essay on a General Method in Dynamics”, Philosophical Transactions of the Royal Society of London 125 (1835), 95–144. Transcribed in Plain TeX by David R. Wilkins, Trinity College Dublin, as stated in the TeX header. Extraction: the TeX file is the transcription; the PDF is its typeset form. Rights: the 1835 text is public domain; editor’s transcription as for the first essay.
Source digest
Hamilton introduces the principal function \(S=\int_0^t(T+U)\,dt\), with \(U\) his force-function, as a function of the final coordinates, the initial coordinates and the time. Its partial derivatives give the final and initial momenta and the negative of the energy, and the equations of motion take the canonical form with \(H\) as the generating function. The essay states that the principal function, rather than the characteristic function of 1834, is the more convenient object for most applications.
For this project, the fixed-endpoint, fixed-duration action of C001 is a difference of this functional evaluated on comparison paths, and Branch A’s phase \(e^{iS/\hbar}\) uses the same \(S\). The 1834 characteristic function is the fixed-energy counterpart.
Passage anchors
- TeX line 151: “and to call it the Principal Function”.
- TeX line 173: the justification that principal functions serve “all the chief applications”.
- Opening paragraphs: the definition of \(S\) through \(T+U\) and the statement of its variation.
Coverage and limits
Reading level passage for the introduction and the
definition of the principal function; metadata for the
perturbation theory in the later sections. Formula transcription checked
through the editor’s PDF only.