On Helmholtz-Hodge decomposition of inertia on a discrete local frame of reference

TL;DR

This paper introduces a local reference frame approach to mechanics, utilizing Helmholtz-Hodge decomposition to reformulate inertia and eliminate fictitious forces, with implications for continuum mechanics and conservation laws.

Contribution

It presents a novel formalism replacing inertial frames with local frames and applies Helmholtz-Hodge decomposition to express inertia, enabling a reformulation of continuum mechanics without fictitious forces.

Findings

Inertia expressed as gradient and curl of potentials.

Discrete equation of motion conserves angular momentum and energy.

Variables depend only on length and time units.

Abstract

The notion of inertial reference frame is abandoned and I replaced it by a local reference frame on which the fundamental law of mechanics is expressed. The distant interactions of cause and effect are modeled by the propagation of waves from one local reference frame to another. The derivation of the equation of motion on a straight segment serves to express the proper acceleration as the sum of the accelerations imposed on it, in the form of an orthogonal local Helmholtz-Hodge decomposition, in one divergence-free and another curl-free contribution. I wrote the inertia term in the form of a gradient of a scalar potential and a dual curl of a vector potential. The adopted formalism opens the way to a reformulation of the material derivative in terms of potentials and allows me to remove the fictitious forces from continuum mechanics. The discrete equation of motion, invariant by rotation at constant angular velocity, is used to conserve the angular momentum per unit of mass, in addition to the conservation of energy per unit of mass and acceleration. All the variables in this equation are expressed only with two fundamental units, those of length and time.