Modeling of the long-time asymptotic dynamics of a point-like object

TL;DR

This paper introduces a new framework for modeling the long-time asymptotic behavior of point-like objects, generalizing Newton's law and providing asymptotic differential equations that clarify the interpretation of complex equations like the Lorentz-Abraham-Dirac equation.

Contribution

The paper presents four novel concepts including LTAD-models and asymptotic differential equations, offering a new perspective on point-like object dynamics without solving their equations of motion.

Findings

LTAD-models can be constructed without solving the PO's equation of motion.

Asymptotic differential equations are equivalent to LTAD-models, not the original equations.

Interpretation of the Lorentz-Abraham-Dirac equation as an asymptotic differential equation.

Abstract

We introduce four original concepts: First, the point-like object (PO) specified as a classical extended real object whose response to an external force is aptly specified solely by the trajectory of a single point, whose velocity eventually stops changing after the cessation of the external force. Second, the dynamic models of an PO that generalize the Newton second law by the explicit modeling of PO-acceleration by nonlinear functions of the external force. Third, the long-time asymptotic dynamics of an PO (LTAD) modeled by polynomials in time-derivatives of the external force (by LTAD-models). To make LTAD-models we do not need to know the PO equation of motion. Given the PO equation of motion, without solving it, we can calculate the appropriate LTAD-models, but not vice verse. Fourth, the asymptotic differential equations about the LTAD. They are equivalent to the LTAD-models, but not to the PO equation of motion. To resolve the conceptual controversy surrounding the relativistic Lorentz-Abraham-Dirac equation, we interpret this equation as an asymptotic differential equation about the LTAD of an electrified PO, and not as a differential equation of motion for an electrified PO. Keywords: Point-like; asymptotic dynamics; cyclic motion; differential equation; Lorentz-Abraham-Dirac equation