Jacobi, Vorlesungen über Dynamik, lectures 6 and 7 (1842–43; 1866)
Source: Internet Archive, Toronto scan
vorlesungenber00jacouoft; local OCR excerpt.txt, retrieved 2026-09-08. The excerpt’s archive line range is recorded in the H06 batch note. Metadata: C. G. J. Jacobi’s Vorlesungen über Dynamik, gehalten an der Universität zu Königsberg im Wintersemester 1842–1843 und nach einem von C. W. Borchardt ausgearbeiteten Hefte, herausgegeben von A. Clebsch (Berlin: G. Reimer, 1866). Sechste Vorlesung, “Das Princip der kleinsten Wirkung”, p. 43; Siebente Vorlesung, “Fernere Betrachtungen über das Princip der kleinsten Wirkung”. Extraction: archive.org OCR of the 1866 Antiqua printing. Lecture headings are damaged in the OCR, so the boundaries were located from the table of contents and internal cross-references. Rights: public domain.
Source digest
Jacobi formulates the principle of least action at fixed energy as the stationarity of \(\int\sqrt{2(h-U)}\,ds\) over paths with fixed endpoints. In the first lecture he lists it as “das Princip der kleinsten Wirkung oder, wie es besser heissen sollte, des kleinsten Kraftaufwandes”. The sixth lecture opens by calling “kleinste Wirkung” a false rendering of “principe de la moindre action”, locates the principle’s value in the form it gives the differential equations and in the function that becomes a minimum when they hold, and warns that earlier times put exaggerated weight on the mere existence of that minimum; Euler’s “de motu projectorum” is cited as the first example. The seventh lecture continues the derivation, and the eighth treats Hamilton’s fixed-time integral.
For this project, the fixed-energy functional is the object whose comparison-path differences enter the action-value gap question Q1, and Jacobi’s warning about the existence of a minimum is the classical form of the distinction between C002’s vanishing infimum and a genuine gap.
Passage anchors
- Archive lines 576–577 (first lecture): the naming remark quoted above.
- Archive lines 164–170 (table of contents): the titles of lectures six to eight with their pages.
- Excerpt opening (archive lines from 2688): “Wir kommen jetzt zu einem neuen Princip … fälschlich der kleinsten Wirkung genannt”, through the sentence on Euler’s memoir.
- Archive lines 2787 and 2805: “so können wir das Princip der kleinsten Wirkung so aussprechen” and “In dieser wahren Form”.
Coverage and limits
Reading level passage for the naming remark and the
formulation statements; metadata for the derivations.
German OCR needs the scan for the displayed formulae.