Theory and Application of Augmented Dimensional Analysis

TL;DR

This paper introduces augmented dimensional analysis, a novel algebraic approach that explicitly accounts for all relations among quantities, improving classical methods and connecting to matroid theory.

Contribution

It develops a new algebraic theorem for quantity functions, clarifies and relaxes classical  theorem prerequisites, and links dimensional analysis with matroid theory.

Findings

Provides a complete algebraic framework for dimensional analysis.

Restates the  theorem with explicit and relaxed conditions.

Connects dimensional analysis to matroid theory, revealing combinatorial structures.

Abstract

We present an innovative approach to dimensional analysis, referred to as augmented dimensional analysis and based on a representation theorem for complete quantity functions with a scaling-covariant scalar representation. This new theorem, grounded in a purely algebraic theory of quantity spaces, allows the classical \pi theorem to be restated in an explicit and precise form and its prerequisites to be clarified and relaxed. Augmented dimensional analysis, in contrast to classical dimensional analysis, is guaranteed to take into account all relations among the quantities involved. Several examples are given to show that the information thus gained, together with symmetry assumptions, can lead to new or stronger results. We also explore the connection between dimensional analysis and matroid theory, elucidating combinatorial aspects of dimensional analysis. It is emphasized that dimensional analysis rests on a principle of covariance.