Duality of the Principle of Least Action: A New Formulation of Classical Mechanics

TL;DR

This paper introduces a dual formalism for Lagrange multipliers to derive core equations of analytical mechanics from a convex action function, revealing a fundamental connection to the calculus of variations.

Contribution

It presents a novel dual formulation that derives key mechanics equations solely from the convexity of the action, without relying on traditional dynamical assumptions.

Findings

Derivation of Hamilton-Jacobi equation from convex action

Generation of canonical transformations via the formalism

Reinterpretation of analytical mechanics as a calculus of variations problem

Abstract

A dual formalism for Lagrange multipliers is developed. The formalism is used to minimize an action function $S(q_2,q_1,T)$ without any dynamical input other than that $S$ is convex. All the key equations of analytical mechanics -- the Hamilton-Jacobi equation, the generating functions for canonical transformations, Hamilton's equations of motion and $S$ as the time integral of the Lagrangian -- emerge as simple consequences. It appears that to a large extent, analytical mechanics is simply a footnote to the most basic problem in the calculus of variations: that the shortest distance between two points is a straight line.