Dimensional Analysis

DA

Image Credit: Michael Taylor

If we believe that the natural laws of physics are encoded in all observational data, the question then is, how do we extract them? This is a problem that is a recurring theme in my work. In 2008 I was reading through the inspiring and radical papers of Stephan Rudolph and co-workers on knowledge representation and dimensional analysis and. while studying Buckingham’s Pi-Theorem, it appeared that there is a way to generalize the theory. The problem has three parts. First, you have to combine dimensional variables into dimensionless quantities according with a similarity transform. Stephan had shown how to do this for some well-defined cases. We built on this to define the general case. Secondly, the problem is to then address the task of inverting the resulting non-square matrix back into dimensional space using an inverse similarity transform. This is not only hard, but the matrix algebra required to do it needed inventing! Fortunately, I came into contact with some experts in pure mathematics in Valencia who came up with a way to do it. The third part of the problem is then how to break the degeneracy associated with forming dimensionless groups. The trick, it seems, is to refer back to data. But this is an ongoing challenge…

Peer-Reviewed Articles & Conference Papers:

  1. Taylor, M., Diaz, A.I., Jodar-Sanchez, L.A., Villanueva-Mico, R.F. (2008) A matrix generalisation of dimensional analysis: new similarity transforms to address the problem of uniqueness. Advanced Studies in Theoretical Physics, 2(20):979-995. [PDF] [link]
  2. Taylor, M., Diaz, A.I., Jodar-Sanchez, L.A., Villanueva-Mico, R.F. (2007) 100 years of dimensional analysis:  new steps toward empirical law deduction. [arXiv]

 

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