Dimensional analysis with constraints

TL;DR

This paper introduces a linear-algebraic framework for dimensional analysis that efficiently handles systems with multiple or implicit constraints by transforming the problem into a linear structure in logarithmic variables.

Contribution

It provides a novel algebraic method to identify independent dimensionless quantities and eliminate redundancies without trial and error, especially in complex constrained systems.

Findings

Simplifies dimensional analysis for systems with multiple constraints

Enables algebraic elimination of redundant variables

Illustrated with classical drag force problem

Abstract

We develop a linear-algebraic framework for dimensional analysis in systems with constraints, particularly when variables are numerous or related by implicit relations so that direct elimination is impractical. By expressing both dimensional relations and constraints in logarithmic variables, the problem is reduced to a linear structure. This formulation yields a simple count of independent dimensionless quantities and, more importantly, a purely algebraic procedure to eliminate redundant ones without trial and error. The method is especially effective for systems with implicit or multiple constraints, and is illustrated with the classical drag force problem.