A dialectical view of the variational principles of mechanics

TL;DR

This paper proposes a philosophical framework based on dialectical opposites to unify and motivate variational principles in physics, including an axiomatization of quantum mechanics and derivations of classical principles.

Contribution

It introduces a dialectical perspective to the variational principles of mechanics, offering a unified axiomatic approach and extending to quantum mechanics.

Findings

A dialectical model of motion based on opposite elements.

Derivation of classical variational principles from the dialectical framework.

An axiomatization of quantum mechanics within this perspective.

Abstract

The aim of this work is to provide a possible philosophical motivation to the variational principles of physics in general and a possible way to unify the axiomatizations of mechanics theories. The leitmotif of this work is a dialectical view of the world: any smooth motion in nature is the consequence of the interplay and dynamical balance between pairs of opposite elements of the motion. We stress that the opposite property here differs from the common sense of opposing elements (of conflicting appearance but of same nature) such action/reaction forces between two bodies. The opposite elements here are incompatible and mutual exclusive in nature, with contrasting and complementary characteristics such as active/passive, disorder/order, curvature/flatness, etc. The dynamical balance is defined by an invariant variational relationship between all the pairs of opposite elements active in the considered motion. From this idea, an axiomatization of quantum mechanics is proposed in addition to the derivation of several existing variational principles of mechanics.