Geometric and physical interpretation of the action principle

TL;DR

This paper provides a geometric interpretation of the stationary action principle in classical mechanics, linking it to the area enclosed by possible evolutions and clarifying its physical and geometric foundations.

Contribution

It offers a novel geometric and physical interpretation of the action principle based on three foundational assumptions, unifying Hamiltonian and Lagrangian mechanics.

Findings

Provides a geometric account of stationary action

Clarifies the physical meaning of the action principle

Unifies Hamiltonian and Lagrangian frameworks

Abstract

We give a geometrical interpretation for the principle of stationary action in classical Lagrangian particle mechanics. In a nutshell, the difference of the action along a path and its variation effectively ``counts'' the possible evolutions that ``go through'' the area enclosed. If the path corresponds to a possible evolution, all neighbouring evolutions will be parallel, making them tangent to the area enclosed by the path and its variation, thus yielding a stationary action. This treatment gives a full physical account of the geometry of both Hamiltonian and Lagrangian mechanics which is founded on three assumptions: determinism and reversible evolution, independence of the degrees of freedom and equivalence between kinematics and dynamics. The logical equivalence between the three assumptions and the principle of stationary action leads to a much cleaner conceptual understanding.