Shape Derivatives

TL;DR

This paper develops shape and scale derivatives within relational geometry, providing new tools for analyzing geometric invariants and solving related differential equations, with implications for physics and geometry.

Contribution

It introduces shape(-and-scale) derivatives by Taylor-expanding geometric invariants, applicable across various geometries and dimensions, including derivations of the Schwarzian derivative.

Findings

Defined shape derivatives for multiple geometries in 1D and higher dimensions.

Solved ODEs for constant and zero derivative cases.

Formulated PDEs for higher-dimensional geometries and obtained solutions.

Abstract

Shape Theory, together with Shape-and-Scale Theory, comprise Relational Theory. This consists of $N$-point models on a manifold $M$, for which some geometrical automorphism group $G$ is regarded as meaningless and is thus quotiented out from the $N$-point model's product space $\times_{I = 1}^N M$. Each such model has an associated function space of preserved quantities, solving the PDE system for zero brackets with the sums over $N$ of each of $G$'s generators. These are smooth functions of the $N$-point geometrical invariants. Each $(M, G)$ pair has moreover a `minimal nontrivially relational unit' value of $N$; we now show that relationally-invariant derivatives can be defined on these, yielding the titular notions of shape(-and-scale) derivatives. We obtain each by Taylor-expanding a functional version of the underlying geometrical invariant, and isolating a shape-independent derivative factor in the nontrivial leading-order term. We do this for translational, dilational, dilatational and projective geometries in 1-$d$, the last of which gives a shape-theoretic rederivation of the Schwarzian derivative. We next phrase and solve the ODEs for zero and constant values of each derivative. We then consider translational, dilational, rotational, rotational-and-dilational, Euclidean and equi-top-form (alias unimodular affine) cases in $\geq 2$-$d$. We finally pose the PDEs for zero and constant values of each of our $\geq 2$-$d$ derivatives, and solve a subset of these geometrically-motivated PDEs. This work is significant for Relational Motion and Background Independence in Theoretical Physics, and foundational for both Flat and Differential Geometry.