Two Hundred Years After Hamilton: The Simple Axiom That Underlies Classical Mechanics

TL;DR

This paper demonstrates that the fundamental structure of classical mechanics can be derived from a simple additivity axiom of the principal function, offering new insights into classical and quantum theories.

Contribution

It introduces the additivity of the principal function as the core principle underlying all formulations of classical mechanics, unifying them under a simple axiom.

Findings

All Hamiltonian formulations derive from the additivity of the principal function.

The additivity axiom simplifies the understanding of classical mechanics.

Potential applications include new perspectives on symplectic geometry and quantum mechanics.

Abstract

In 1834-1835, Hamilton published two papers that revolutionized classical mechanics. In these papers, he introduced the Hamilton-Jacobi equation, Hamilton's equations of motion and the principle of least action. These three formulations of classical mechanics became the forerunners of quantum mechanics, but none of these is what Hamilton was looking for: he was looking for what he called the principal function, $S(q',q'',T)$, from which the entire trajectory history can be obtained just by differentiation. Here we show that all of Hamilton's formulations can be derived just by assuming that the principal function is additive, $S(q',q'',T)=S(q',Q,t_1)+S(Q,q'',t_2)$ with $t_1+t_2=T$. This simple additivity axiom can be considered the fundamental principle of classical mechanics and shows that analytical mechanics is essentially just a footnote to the problem of finding the shortest path between two points. The simplicity of the formulation could provide new perspectives on some of the major themes in classical mechanics including symplectic geometry, periodic orbit theory and Morse theory, as well as giving new perspectives on quantum mechanics. Moreover, it could potentially provide a unified description of different areas of physics, leading to insight for example, into the transition from deterministic dynamics to statistical mechanics.