The Principle of Least Action — The Foundation Beneath the Phase-Space Trilogy (Euler), yet Not Teleology.
The phase-space trilogy (Papers 76-82) all stood upon the Hamiltonian stage. Beneath that stage lies a single foundation: the principle of least action (Maupertuis -> Euler -> Lagrange -> Hamilton). From the stationarity of the action S=integral of L dt (delta S=0) follows the Euler-Lagrange equation, and from it Lagrangian mechanics -> (Legendre transform) -> Hamiltonian mechanics (the cotangent bundle T*M, Paper 76) -> Liouville (77) -> Noether (80) all grow; hence Paper 84 is the root of the trilogy. The crucial discipline: least action is rigorously equivalent to F=ma (not a deeper cause), and it is NOT teleology — delta S=0 is stationary (not necessarily a minimum), the Euler-Lagrange equation is local, and the apparent purpose dissolves into the interference of Feynman’s path integral (all paths contribute; only the stationary neighbourhood survives). Economy (elegance) is not purpose (distinct roots, Papers 58/48). Scope: no new mathematics; the ultimate “why does nature obey an action principle” is not solved (the path integral answers with interference, not purpose); “the Universe OS’s cost function” is a Layer-3 non-science lens (quarantined). No new prediction. Euler, the father of the calculus of variations, supplies the tool that supports the foundation.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Principle of Least Action [Paper 84]. Zenodo.
DOI: 10.5281/zenodo.21741231
ORCID: 0009-0006-8587-3450