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Dimensional analysis using toric ideals: primitive invariants.

Mark A Atherton1, Ronald A Bates1, Henry P Wynn2

  • 1Department of Mechanical Engineering, Brunel University, London, United Kingdom.

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Summary
This summary is machine-generated.

This study introduces toric ideal theory and Graver bases for dimensional analysis, offering a unique and complete set of dimensionless invariants beyond traditional methods. The approach enhances the discovery of physical invariants across various scientific domains.

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Area of Science:

  • Physics
  • Algebraic Geometry
  • Dimensional Analysis

Background:

  • Classical dimensional analysis uses basic units to derive quantities like force.
  • The Buckingham method is a standard approach for finding dimensionless groups.
  • Toric ideal theory offers a novel algebraic framework for dimensional analysis.

Purpose of the Study:

  • To apply toric ideal theory and Graver bases to dimensional analysis.
  • To find a unique and complete set of primitive dimensionless invariants.
  • To demonstrate the method's efficacy using diverse scientific examples.

Main Methods:

  • Utilizing toric ideal theory from algebraic geometry.
  • Employing the Graver basis for a unique primitive basis.
  • Applying computer algebra to derive integer matrices and identify invariants.
  • Revisiting textbook examples and introducing new case studies.

Main Results:

  • The Graver basis provides a unique, maximal set of primitive dimensionless invariants.
  • This method yields a more complete set of invariants compared to the Buckingham approach.
  • Named invariants were identified in examples from convection, windmills, electrodynamics, and the hydrogen atom.
  • Computer algebra facilitates the computation of both simple and Graver bases.

Conclusions:

  • Toric ideal theory and Graver bases offer a powerful, systematic approach to dimensional analysis.
  • The method enhances the discovery and understanding of fundamental physical invariants.
  • This algebraic framework provides a rigorous foundation for generating dimensionless groups.