Six New Mechanics corresponding to further Shape Theories

TL;DR

This paper introduces six new shape mechanics theories based on configuration space geometry, extending classical shape theories with conformal, affine, and supersymmetric variants, and discusses their potential applications in probability and statistics.

Contribution

The paper develops six novel shape mechanics theories using invariant groupings, expanding the framework of classical shape theories with new geometric and supersymmetric models.

Findings

Introduction of conformal and affine shape mechanics.

Comparison of supersymmetric shape mechanics with GR-based Background Independence.

Outline of potential applications in probability and statistics.

Abstract

A suite of relational notions of shape are presented at the level of configuration space geometry, with corresponding new theories of shape mechanics and shape statistics. These further generalize two quite well known examples: --1) Kendall's (metric) shape space with his shape statistics and Barbour's mechanics thereupon. 0) Leibnizian relational space alias metric scale-and-shape space to which corresponds Barbour-Bertotti mechanics. This paper's new theories include, using the invariant and group namings, 1) $Angle$ alias $conformal$ $shape$ $mechanics$. 2) $Area ratio$ alias $affine$ $shape$ $mechanics$. 3) $ Area$ alias $affine$ $scale$-$and$-$shape$ $mechanics$. 1) to 3) rest respectively on angle space, area-ratio space, and area space configuration spaces. Probability and statistics applications are also pointed to in outline. 4) Various supersymmetric counterparts of -1) to 3) are considered. Since supergravity differs considerably from GR-based conceptions of Background Independence, some of the new supersymmetric shape mechanics are compared with both. These reveal compatibility between supersymmetry and GR-based conceptions of Background Independence, at least within these simpler model arenas.