A Local Gauge-Covariant Formulation of Classical Dynamics

TL;DR

This paper introduces a gauge-covariant, relational formulation of classical dynamics that evolves without predefined laws or external time, recovering familiar equations as coarse-grained limits.

Contribution

It develops a novel framework where dynamics emerge from local gauge-invariant mismatch measures, without assuming an action principle or fixed background geometry.

Findings

Recovers diffusion, Navier-Stokes, and Maxwell equations in certain limits.

Defines a gauge-invariant quadratic measure of state incompatibility.

Proposes a dynamics driven by relaxation of local mismatch measures.

Abstract

Classical dynamical laws are conventionally formulated as closed evolution equations defined on fixed geometric backgrounds and a global time parameter. We develop a formulation in which neither prescribed evolution laws nor an external clock are assumed a priori. Grounded in the principles of conservation, locality of interaction, and independent local frame freedom, the framework treats spatial geometry as a relational structure that may evolve together with the state. We introduce a notion of local incompatibility defined as the covariant difference between neighboring states under a dynamical transport connection. Because the transport relations are not fixed, restoring compatibility requires the joint adaptation of both state variables and transport geometry. We show that locality, gauge covariance, and coercivity strongly restrict the admissible form of this incompatibility and lead to a simple, globally additive, gauge-invariant quadratic measure of mismatch. Admissible dynamics are then defined as the asynchronous, finite-rate relaxation of this measure, without assuming a predefined action principle. A global time description appears only as an effective coarse-grained limit of this local relaxation process. In appropriate limits, the resulting compatibility-restoration dynamics recovers familiar continuum equations, including diffusion, incompressible Navier--Stokes, and the Amp\`ere--Maxwell relation. In this sense, dynamics arises from the coupled evolution of state and transport geometry toward local gauge consistency. The formulation provides a constructive framework in which effective physical laws emerge from local relational constraints.