Buckingham-Pi for "algorithmic" non-dimensionalization

[sysidentify]

I would like to use the Buckingham-Pi theorem in order to "algorithmify" non-dimensionalization of existing equations. I can get things to work for very simple problems, but am running into issues with a harder example. I posted my question on physics.stackexchange.com the day before yesterday, but it has received little attention.

Could someone here comment, either here or there?

Alternatively, do you know of existing ways to "algorithmify" non-dimensionalization?

 

[Chestermiller]

The way I learned to do this kind of thing is not by use of the Buckingham Pi Theorem. The method I learned always works great. For each of the independent variables (in this case only the time t), define a dimensionless parameter such as τ = t / t₀, where t₀ is a characteristic time to be defined in terms of the other parameters in the problem. If there are spatial variables, such as x, define X = x/x₀.

In your problem, for example, substitute t = t₀τ into your differential equation, and reduce all the terms to dimensionless form (by multiplying or dividing the entire equation by whatever parameters necessary to do this). You will end up with groups involving t₀. Select one of the groups involving t₀, and choose t₀ such that that group is equal to unity. Substitute this expression for t₀ into all the other groups involving t₀. This will automatically generate the other dimensionless groups involved.

The choice of t₀ or x₀ can also come out of the boundary conditions of the problem, if that choice makes better sense.

In your problem, the dependent variable is already dimensionless. However, in most problems that is not the case. If the dependent variable is not dimensionless, you can do the same trick with that variable.

Chet

 

[Chestermiller]

This gives:
$$τ=\frac{mgLt}{b}$$
$$ε=\frac{m^2gL^3}{b^2}$$

Chet