Yank and Hooke's constant group theoretically

TL;DR

This paper explores the mathematical structure of a specific Lie group extension related to Aristotle's group, revealing orbit characteristics linked to Hooke's constant and yank, and interpreting these in physical systems with damping and force interactions.

Contribution

It provides a novel group-theoretic analysis of the second central extension of the (1+1) Aristotle Lie group, identifying orbit properties and their physical interpretations.

Findings

Identified four orbits on the dual of the second central extension.

Characterized the generic orbit by Hooke's constant and yank.

Linked the orbit properties to physical systems with damping and force.

Abstract

We study the second central extension of the (1+1) Aristotle Lie.We find that the first central extension admit four orbits on the dual of second central extension of the (1+1) Aristotle Lie group.The generic orbit is characterised by a Hooke's constant k and a yank y.If the physics of the orbit is studied with respect the evolution in time,it represents an elementary system with internal energy U in a posotion-momentum under the conjugation of a Hooke's force and a damping one proportional to the velocity as in particle mechanics.If the physics of the orbit is studied with respect the evolution in space, it represents an elementary system with an internal momentum P under the conjugation of a kind of Hooke's force and a damping one proportional to a slowness, slowness usually used in time travel waves.