Dimensional Analysis and the Pi Theorem

  title={Dimensional Analysis and the Pi Theorem},
  author={W. D. Curtis and J. David Logan and Willard A. Parker},
  journal={Linear Algebra and its Applications},
On Buckingham’s Π-Theorem
A more "complete" version of the Pi-theorem: DRAFT
The traditional Pi-theorem tells us that for any dimensionally invariant relation there exists a full set of independent dimensionless "Pi groups" which can be used to nondimensionalise the relation.
On Buckingham's $\Pi$-Theorem
Roughly speaking, Buckingham's $\Pi$-Theorem provides a method to "guess" the structure of physical formulas simply by studying the dimensions (the physical units) of the involved quantities. Here we
On the proof of the Π-theorem in dimension theory
The 100-year anniversary of the proof of one of the brightest and most universal theorems in mechanics and physics, the Π-theorem in dimension theory is approaching. In connection with this
Dimensional Analysis of Matrices State-Space Models and Dimensionless Units [Lecture Notes]
Physical dimensions and units, such as mass (kg), length (m), time (s), and charge (C), provide the link between mathematics and the physical world. It is well known that careful attention to
What Are the Physical Dimensions of the A Matrix?
This article extended results of the work of Hart (1995) by determining the dimensional structure of a matrix under which standard operations involving the inverse, powers, exponential, and eigenvalues are valid.
Weak Solutions, A Priori Estimates
The fundamental laws of continuum mechanics that can be interpreted as infinite families of integral identities equivalent to systems of partial differential equations give rise to the concept of
Practical Guide to the Symbolic Computation of Symmetries of Differential Equations
A computational approach to finding symmetries and computer algebra programs to compute the usually very large system of determining partial differential equations and a computer algebra algorithm that at least automatically solves most of these equations and in simple cases provides a complete solution.
The Amazing Power of Dimensional Analysis: Quantifying Market Impact
This note complements the inspiring work on dimensional analysis and market microstructure by Kyle and Obizhaeva [18]. Following closely these authors, our main result shows by a similar argument as
Dynamic Dimensional Analysis
This chapter develops the concept of dimensional analysis to be used in model-prototype similarities, nondimensionalization, and simplification of equations for the most general coverage. The goals


Dimensional Analysis and the Buckingham Pi Theorem
Working from a simple example of a dimensional transformation, the essential elements are identified, and an abstract prototype transformation is defined. With the aid of careful definitions and a
Similarity methods for differential equations
1. Ordinary Differential Equations.- 1.0. Ordinary Differential Equations.- 1.1. Example: Global Similarity Transformation, Invariance and Reduction to Quadrature.- 1.2. Simple Examples of Groups of
The air wave surrounding an expanding sphere
  • G. Taylor
  • Physics
    Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
  • 1946
When the surface of a sphere vibrates in any assigned manner the spherical sound waves which are propagated outwards can be represented by wellknown formulae provided that the motion is such that