The Haar measure in solid mechanics

TL;DR

This paper highlights the importance of the Haar measure in solid mechanics, providing practical methods for its computation and application in mechanical scenarios, thus bridging mathematical theory and mechanics practice.

Contribution

It introduces practical approaches for applying the Haar measure in solid mechanics, including computation methods and examples for invariant theory and sampling on group orbits.

Findings

Computed Haar measure for 2D and 3D orthogonal transformations

Demonstrated applications in invariant theory and sampling

Provided accessible methodologies for mechanicians

Abstract

This article addresses the overlooked but crucial role of the Haar measure in solid mechanics, a concept well-established in mathematical literature but frequently misunderstood by mechanicians. The aim is to provide practical insights and methodologies for the application of the Haar measure in various mechanical scenarios. The article begins with an introduction to the Haar measure, underlying its significance and a theoretical foundation for its definition and formulas. Moving beyond mathematical abstraction, the core of the article relies on practical applications, including the computation of the Haar measure for the orthogonal transformations of the 2D and 3D space under several commonly used parametrisations, an application about invariant theory in mechanics and sampling on a group orbit. These applications are presented with practical examples, enabling mechanicians to integrate the Haar measure into their research seamlessly. The article caters to a broad audience, with sections designed for both introductory comprehension and in-depth exploration. Our goal is to equip mechanicians with a valuable tool, bridging the gap between mathematical theory and practical applications in the field of mechanics.