Affine symmetry in mechanics of collective and internal modes. Part I. Classical models

TL;DR

This paper introduces a classical model of collective and internal modes in mechanics that is invariant under affine transformations, emphasizing both kinematics and dynamics, and relates it to integrable lattice dynamics.

Contribution

It presents a novel affinely-invariant model that extends previous kinematic approaches by incorporating affine invariance into the dynamics.

Findings

Affinely-invariant geodetic models can describe elastic vibrations.

The model relates to the dynamics of integrable one-dimensional lattices.

It advances the understanding of symmetry in mechanical systems.

Abstract

Discussed is a model of collective and internal degrees of freedom with kinematics based on affine group and its subgroups. The main novelty in comparison with the previous attempts of this kind is that it is not only kinematics but also dynamics that is affinely-invariant. The relationship with the dynamics of integrable one-dimensional lattices is discussed. It is shown that affinely-invariant geodetic models may encode the dynamics of something like elastic vibrations.