Relational Quadrilateralland I: The Classical Theory

Abstract

Relational particle mechanics (RPM) models bolster the relational side of the absolute versus relational motion debate. They are additionally toy models for the dynamical formulation of General Relativity (GR) and its Problem of Time (PoT). They cover two aspects that the more commonly studied minisuperspace GR models do not: 1) by having a nontrivial notion of structure and thus of cosmological structure formation and of localized records. 2) They have linear as well as quadratic constraints, which is crucial as regards modelling many PoT facets.I previously solved relational triangleland classically, quantum mechanically and as regards a local resolution of the PoT. This rested on triangleland's shape space being S 2 with isometry group SO(3), allowing for use of widely-known Geometry, Methods and Atomic/Molecular Physics analogies. I now extend this work to the relational quadrilateral, which is far more typical of the general N -a-gon, represents a 'diagonal to nondiagonal Bianchi IX minisuperspace' step-up in complexity, and encodes further PoT subtleties. The shape space now being CP 2 with isometry group SU (3)/Z3, I now need to draw on Geometry, Shape Statistics and Particle Physics to solve this model; this is therefore an interdisciplinary paper. This Paper treats quadrilateralland at the classical level, and then Paper II provides a quantum treatment.

On the Conceptual Issues Surrounding the Notion of Relational Bohmian Dynamics

“…33 See Anderson (2007Anderson ( , 2014a for two fully worked-out models involving this formalism. Earlier work on "Barbour-Bertotti" models include, notably, Gergely (2000); Gergely and McKain (2000).…”

Six new mechanics corresponding to further shape theories

“…Models along the above lines rather quickly acquire complexity. Indeed [49,50] considered the K-shaped clustering of three relative Jacobi coordinate vectors presentation of quadrilaterals as axe configurations. The underlying shape space in this case is CP 2 : Fig 6.b), and the relational space is C(CP 2 ).…”

“…This figure concentrates on the two ratio coordinates β and χ, suppressing the further variety in relative angle coordinates φ and ψ (β, χ, φ ψ are Gibbons-Pope type coordinates, see e.g. [49]). Furthermore, how good the 'best fit' is can be assessed in substantially geometrically general cases by making set of relational objects out of primed and unprimed vertices (Fig 8.a), to which the corresponding notion of Shape Statistics is to be applied.…”

“…It is not that different [10] for metric RPM with and without scale! This is due to relative angular momentum (and relative dilational momentum [70], and mixtures [71,49]) having the same mathematics as angular momentum. Moreover, the quantum metric shape quadrilateral [50] did produce a more unusual and distinctive combination of features of the Periodic Table and of Gell-Mann's eightfold way.…”

Origin of structure in the universe: quantum cosmology reconsidered

“…Ie shift variables are being eliminated without taking partners away with them, which reflects that the equations being reduced cannot all be first class. So, whereas (25) being algebraic parallels how relational particle models [24,29] deal with their analogue of the thin sandwich -elimination of translation, rotation and optionally dilatation corrections -however, there the constraints are all first-class, so each auxiliary eliminated is accompanied by the kinetic line element itself losing one dimension. In a startling turn of events, this is not emulated by the SIC thin sandwich equations, indicating that the pieces of M i are not themselves first class despite coming from an object M i which in its original unsplit form is widely known to be first-class.…”