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The "let's assume the result is proportional to a product of powers" ansatz always threw me off a bit. Why not additive, or including some function of the indep
by sieste 3y ago
The "let's assume the result is proportional to a product of powers" ansatz always threw me off a bit. Why not additive, or including some function of the independent variables? Is there a better justification than "it often gives the correct result?"
- Horffupolde 3y agoThat’s a good question. Addition and subtraction does not make sense because you can’t add non-homogeneous dimensions. And if they were, there’d be a simpler expression. As for a function more complex than multiplication, you can consider the general formula such as an exponential decay and transform it using logarithms into a multiplicative form.
- ykonstant 3y agoThis technique, known as Rayleigh's method, has a partial justification in Buckingham's π theorem: https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem
- ssivark 3y agoYou cannot add/subtract things which don’t have the same dimensions. Further, such formulae cannot forbid negative answers even though you’re trying to model strictly positive quantities. That naturally lends to multiplication. If you want to exponentiate, you can only use dimensionless quantities in the exponent (and that kind of dependence does pop up sometimes, in interesting places). But the exponential dependence means very low sensitivity for small values for the exponent, so (by Taylor approximation) you might get away with using a constant, or redefining variables (by subtraction/rescaling) to come back to the multiplicative format. In other words, you’re taking the log on both sides and doing dimensional analysis with the new variables.