A linear-algebraic formulation of dimensional analysis with constraints

Summary · qwen2.5:32b

The article presents a linear-algebraic approach to dimensional analysis that incorporates constraints using logarithmic variables, where dimensional transformations and constraints are represented as subspaces whose intersection characterizes independent dimensionless quantities. This method streamlines redundancy elimination in complex systems with predefined relationships among variables. An example provided involves applying the technique to analyze drag force, demonstrating its utility in simplifying physical relations under constraints.

Suggested post angle

Exploring a linear-algebraic approach to dimensional analysis could be useful for optimizing real-time rendering in game development and simulations.

Excerpt

arXiv:2603.21527v2 Announce Type: replace Abstract: Dimensional analysis, especially Buckingham's $\pi$ theorem, reduces the number of variables by rewriting a relation in terms of dimensionless quantities. When variables are tied by definitions, constitutive laws, or other constraints, however, eliminating variables in advance can be awkward. We formulate dimensional analysis with constraints as linear algebra in logarithmic variables. Dimensional transformations and constraints are represented by subspaces, the effective number of independent dimensionless quantities is characterized by their intersection, and a matrix representation yields a systematic redundancy elimination procedure. Examples from falling motion, drag force, and stock-market indicators illustrate the scope and limitations of the method.