Dimensional analysis with constraints

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.

Figures (1)

• Figure 1: A schematic picture of the space of logarithmic variables ${\bm y}=\ln {\bm x}$. ${\cal M}={\bm \psi}^{-1}({\bm 0}_\ell) \subset {\mathbb R}^n$ is the constraint manifold and $T_{\bm y}{\cal M}=\ker J$ is its tangent space at a point ${\bm y} \in {\cal M}$, where $J=D{\bm \psi}$ is the Jacobian matrix. The ambient space admits the orthogonal decompositions ${\mathbb R}^n = \ker A \oplus \mathop{\mathrm{im}}\nolimits A^\top$ and ${\mathbb R}^n = \ker J \oplus \mathop{\mathrm{im}}\nolimits J^\top$, where $A \in {\mathbb R}^{m \times n}$ is the dimension matrix of the system. The number $d_{\rm eff}$ of admissible dimensionless quantities is given by the dimension of $\ker A \cap \ker J$, depicted as the intersection of $T_{\bm y}{\cal M}$ and $\ker A$ (dimensionless direction). For scale invariant constraints, the scaling direction $\mathop{\mathrm{im}}\nolimits A^\top$ lies in the tangent space, i.e., $\mathop{\mathrm{im}}\nolimits A^\top \subseteq \ker J$.