Double Affine Bundles

TL;DR

This paper develops a comprehensive theory of double affine bundles, extending the geometric framework of double vector bundles, with applications in Analytical Mechanics for a coordinate-free description of systems.

Contribution

It introduces the concept of double affine bundles, compares different approaches, and explores their relations to phase spaces and contact structures, filling a gap in geometric mechanics.

Findings

Established a formal theory of double affine bundles

Compared approaches and studied duality and models

Connected double affine bundles to phase spaces and contact structures

Abstract

A theory of double affine and special double affine bundles, i.e. differential manifolds with two compatible (special) affine bundle structures, is developed as an affine counterpart of the theory of double vector bundles. The motivation and basic examples come from Analytical Mechanics, where double affine bundles have been recognized as a proper geometrical tool in a frame-independent description of many important systems. Different approaches to the (special) double affine bundles are compared and carefully studied together with the problems of double vector bundle models and hulls, duality, and relations to associated phase spaces, contact structures, and other canonical constructions.