Of the various formal methods of dimensional analysis, the description here is based on Buckingham’s method from 1914. Let \( q_1, q_2, \ldots, q_n \) be \( n \) variables and parameters involved in a particular problem or situation, so that there must exist a functional relationship of the form

Buckingham’s theorem states that the \( n \) variables can always be combined to form exactly \( (n - r) \) independent dimensionless parameter groups, where \( r \) is the number of independent dimensions. Each dimensionless parameter group is commonly called a “\(\Pi\)-group” or a dimensionless group. \(f(q_1, q_2, \ldots, q_n) = 0\) can be written as a functional relationship

The dimensionless groups are not unique, but \( (n - r) \) of them are independent and form a complete set that spans the parametric solution space of \( \phi(\Pi_1, \Pi_2, \ldots, \Pi_{n-r}) = 0 \quad \text{or} \quad \Pi_1 = \varphi(\Pi_2, \Pi_3, \ldots, \Pi_{n-r}) \)

The power of dimensional analysis is most apparent when \( n \) and \( r \) are single-digit numbers of comparable size so \( \phi(\Pi_1, \Pi_2, \ldots, \Pi_{n-r}) = 0 \quad \text{or} \quad \Pi_1 = \varphi(\Pi_2, \Pi_3, \ldots, \Pi_{n-r}) \), which involves \( n - r \) dimensionless groups, represents a significant simplification of \(f(q_1, q_2, \ldots, q_n) = 0\), which has \( n \) parameters

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