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Dimensional analysis using NondimensionalizationTransform

I have a paclet named dimensional analysis. I want to work on improving it but I've struggled to understand the Buckingham Pi theorem and the idea of dimensional groups and nondimensionalization. I have read about dimensional analysis on the documentation for NondimensionalizationTransform and DimensionalCombinations. I have also read some of a book titled Dimensional Analysis by J. C. Gibbings published by Springer in 2011. I've made a Wikipedia edit based on the book mentioned above to the Wikipedia article on Kepler's laws.

I am trying to understand how the function NondimensionalizationTransform works.

For example, if you assume Newton's law of gravity holds, you can derive Kepler's third law that t^2 is proportional to r^3, that is the period of the orbit of a planetary body is proportional to the radius of the orbit of a planetary body. There are more details at https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Dimensional_analysis.

I am wondering how I could use NondimensionalizationTransform and DimensionalCombinations to help derive this result.

Introducing a physical constant allows for a parameter to be eliminated and new relationships to be formed. In this case, the existence of the gravitational constant leads to the formation of Kepler's law. I can use this with IncludeQuantities as a list like IncludeQuantities -> {Quantity[1, "GravitationalConstant"]} or as IncludeQuantities -> "PhysicalConstants".

I can also use a unit system like the Planck units by typing UnitSystem->"PlanckUnits".

I want to improve my paclet Dimensional Analysis and leverage the functionality of NondimensionalizationTransform and DimensionalCombinations. Dimensional analysis is interesting.

I am wondering how to find a dimensional combination of physical constants that has the same dimensions as any unit, for example a farad. I have a resource function named PlanckUnitConversion that uses Dispatch to for example use the defining constants of the Planck system to express units. I would like to generalize this to any set of input quantities and expected output units such as {c, h, conductance quantum} to express 1 J of energy.

POSTED BY: Peter Burbery
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