Editing Buckingham π theorem (section)

===The simple pendulum===
We wish to determine the period <math>T</math> of [[Pendulum (mechanics)#Small-angle approximation|small oscillations in a simple pendulum]]. It will be assumed that it is a function of the length <math>L,</math> the mass <math>M,</math> and the [[Standard gravity|acceleration due to gravity]] on the surface of the Earth <math>g,</math> which has dimensions of length divided by time squared. The model is of the form
<math display=block>f(T,M,L,g) = 0.</math>

(Note that it is written as a relation, not as a function: <math>T</math> is not written here as a function of <math>M, L, \text{ and } g.</math>)

Period, mass, and length are dimensionally independent, but acceleration can be expressed in terms of time and length, which means the four variables taken together are not dimensionally independent. Thus we need only <math>p = n - k = 4 - 3 = 1</math> dimensionless parameter, denoted by <math>\pi,</math> and the model can be re-expressed as
<math display=block>F(\pi) = 0,</math>
where <math>\pi</math> is given by
<math display=block>\pi = T^{a_1}M^{a_2}L^{a_3}g^{a_4}</math>
for some values of <math>a_1, a_2, a_3, a_4.</math>

The dimensions of the dimensional quantities are:
<math display=block>T = t, M = m, L = \ell, g = \ell/t^2.</math>

The dimensional matrix is:
<math display=block>\mathbf{M} = \begin{bmatrix}
1 & 0 & 0 & -2\\
0 & 1 & 0 &  0\\
0 & 0 & 1 &  1
\end{bmatrix}.</math>

(The rows correspond to the dimensions <math>t, m,</math> and <math>\ell,</math> and the columns to the dimensional variables <math>T, M, L, \text{ and } g.</math> For instance, the 4th column, <math>(-2, 0, 1),</math> states that the <math>g</math> variable has dimensions of <math>t^{-2}m^0 \ell^1.</math>)

We are looking for a kernel vector <math>a = \left[a_1, a_2, a_3, a_4\right]</math> such that the matrix product of <math>\mathbf{M}</math> on <math>a</math> yields the zero vector <math>[0,0,0].</math> The dimensional matrix as written above is in reduced row echelon form, so one can read off a kernel vector within a multiplicative constant:
<math display=block>a = \begin{bmatrix}2\\ 0 \\ -1 \\ 1\end{bmatrix}.</math>

Were it not already reduced, one could perform [[Gauss–Jordan elimination]] on the dimensional matrix to more easily determine the kernel. It follows that the dimensionless constant may be written:
<math display=block>\begin{align}
\pi &= T^2M^0L^{-1}g^1\\
    &= gT^2/L
\end{align}.</math>
In fundamental terms:
<math display=block>\pi = (t)^2 (m)^0 (\ell)^{-1} \left(\ell/t^2\right)^1 = 1,</math>
which is dimensionless. Since the kernel is only defined to within a multiplicative constant, if the above dimensionless constant is raised to any arbitrary power, it will yield another equivalent dimensionless constant.

In this example, three of the four dimensional quantities are fundamental units, so the last (which is <math>g</math>) must be a combination of the previous. 
Note that if <math>a_2</math> (the coefficient of <math>M</math>) had been non-zero then there would be no way to cancel the <math>M</math> value; therefore <math>a_2</math> {{em|must}} be zero. Dimensional analysis has allowed us to conclude that the period of the pendulum is not a function of its mass <math>M.</math> (In the 3D space of powers of mass, time, and distance, we can say that the vector for mass is linearly independent from the vectors for the three other variables. Up to a scaling factor, <math>\vec g + 2 \vec T - \vec L</math> is the only nontrivial way to construct a vector of a dimensionless parameter.)

The model can now be expressed as:
<math display=block>F\left(gT^2/L\right) = 0.</math>

Then this implies that <math>gT^2/L = C_i</math> for some zero <math>C_i</math> of the function <math>F.</math> If there is only one zero, call it <math>C,</math> then <math>gT^2/L = C.</math> It requires more physical insight or an experiment to show that there is indeed only one zero and that the constant is in fact given by <math>C = 4\pi^2.</math>

For large oscillations of a pendulum, the analysis is complicated by an additional dimensionless parameter, the maximum swing angle. The above analysis is a good approximation as the [[Small angle approximation|angle approaches zero]].