The Haar measure in solid mechanics

Abstract

This article addresses the overlooked but important 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, the uniform integration measure over a group, in various mechanical scenarios. This 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 focuses on practical applications : the computation of the Haar measure for the orthogonal transformations of the 2D and 3D space under several commonly used parametrizations, an application about invariant theory in mechanics, and the uniform sampling on a group orbit. These applications are presented with practical examples, enabling mechanicians to integrate the Haar measure into their research seamlessly. This article caters to a broad audience, with sections designed for both introductory comprehension and in-depth exploration. Our goal is to justify the origin of formulas used in mechanics that are usually taken as granted and clarify the underlying concepts in order to improve potential applications.