User:Egm6936.s09/Dimensional analysis tutorial

= References =

Web references:
 * google search for "dimensional analysis"
 * Wikipedia: Dimensional analysis Buckingham's Pi theorem Similitude
 * E. Buckingham, On Physically Similar Systems; Illustrations of the Use of Dimensional Equations, Physical Review, Vol.4, No.4, pp.345-376, 1914.
 * P.W. Bridgman (Nobel laureate), Dimensional analysis (free-access internet book), by Yale University Press, 1922.
 * H. Hanche-Olsen, Buckingham's Pi theorem, 2004.
 * The Buckingham Pi theorem in dimensional analysis, 2.25 Advanced Fluid Mechanics, MIT OpenCourseWare
 * Google for "Vaschy Buckingham dimensional analysis" EXCELLENT SEARCH !!
 * T.G. Dobre and J.G. Sanchez Marcano, Chemical engineering: Modelling, simulation, and similitude, Wiley-VCH, 2007. Google book, p.465.
 * Measurement theory and dimensional analysis: Methodological impact on the comparison and evaluation process, E. Coatanea, J. Vareille, B. Yannou.
 * Conceptual modelling of life cycle design: A modelling and evaluation method based on analogies and dimensionless numbers, E. Coatanea, doctoral dissertation, Helsinki University of Technology, 2005.
 * Dynamos, Section 4, p.16, by T. Alboussiere, edited by P. Cardin and Cugliandolo, Elsevier, 2008.
 * Talk slides: 1. History, Equations 2. Energetics 3. Dimensionless parameters 4. MHD (Magneto-Hydro-Dynamics) effects
 * L. I. Sedov, Similarity and dimensional methods in mechanics, Academic Press 1959. Google book.

Other references:


 * Nondimensionalization (French thesis) (Bucky wiki) Nondimensionalization (French thesis) (Bucky wiki)

= Motivation =

Dimensionless partial differential equations
Cairo and Feix, Similarity and rescaling, Extracta Mathematicae, Vol.13, No.1, pp.1-19, 1998.

Experiments in mechanics and physics
= Prelude to the Pi theorem =

This problem has been frequently used to illustrate the procedure to nondimensionalize relations describing physical phenomena.

Nondimensionalization procedure
There are essentially three steps ( Bridgman (1922), p.1 ):
 * list all quantities that a physical phenomenon depends on.
 * write the dimensions of these quantities.
 * combine these quantities into a functional relation that remains true regardless of the unit system used.

But the above is not a systematic procedure to obtain the dimensionless quantities. Bridgman (1922)

resorted to physical reasoning to find dimensionless quantities. Further below, a systematic mathematical method is presented to search for dimensionless quantities.

The swinging pendulum
Let $$\displaystyle F_i$$, with $$\displaystyle i=1,\ldots,m$$, denote the set of $$\displaystyle m$$ fundamental units; e.g., in the SI system, a set fundamental units could be: $$\displaystyle F_1 = T = second$$ for time, $$\displaystyle F_2 = L = meter$$ for length, $$\displaystyle F_3 = M = kilogram$$ for mass, as in the SI system ( see also Wikipedia ). Another set of fundamental units could be $$\displaystyle \hat F_1 = \hat T = second$$ for time, $$\displaystyle \hat F_2 = \hat L = centimeter$$ for length, $$\displaystyle \hat F_3 = \hat M = gram$$ for mass, as in the cgs system.

Next, let the $$\displaystyle n$$ physical quantities of a system be denoted by $$\displaystyle R_i$$, with $$\displaystyle i=1,\ldots,n$$, with the corresponding units denoted by $$\displaystyle [R_i]$$, with $$\displaystyle i=1,\ldots,n$$.

For the swinging pendulum, there are five physical quantities $$\displaystyle R_i$$, which are listed below together with their corresponding dimensional formula for the unit $$\displaystyle [R_i]$$ in terms of the fundamental units $$\displaystyle F_i$$ :

Nondimensionalization by reasioning
It is obvious that the period $$\displaystyle \overline{T} $$ depends on the following parameters:

-	The length of the pendulum $$\displaystyle l$$

-	The mass of the bob $$\displaystyle m$$

-	The gravity $$\displaystyle g$$

The dimensions of the quantities involved are as follows:


 * $$\displaystyle [\overline{T}]=T,\,\,\, [l]=L ,\,\,\, [m]=M,\,\,\, [g]=LT^{-2}$$.

Now we suppose to change the system of units so that the unit of mass is decreased by a factor of $$\displaystyle M$$, the unit of length is decreased by a factor of $$\displaystyle L$$ and the unit of time is decreased by a factor of $$\displaystyle T$$. Therefore, the ratio $$\displaystyle l/g$$ increases by a factor of $$\displaystyle T^{2}$$ or the quantity $$\displaystyle (l/g)^{1/2}$$ increases by a factor of $$\displaystyle T$$. Hence the ratio


 * $$\displaystyle \Pi = \frac{\overline{T}}{\sqrt {l/g}}$$

is invariant under a change of units. $$\displaystyle \Pi $$ is called a dimensionless number. Since it doesn't depend upon the variables $$\displaystyle (m,l,g)$$, it is in fact a constant. Consequently, from dimensional considerations alone we find that:
 * $$\displaystyle \overline{T} = const \times \sqrt{\frac{l}{g}} $$

Nondimensionalization by physical argument
The goal now is to find the functional relation for the time of swing in terms of the other physical quantities such that this relation continues to hold when we change the current fundamental unit system to a different fundamental unit system, such as going from the SI system to the CGS system or to the Customary US system.

Assume that such functional relation can be written as

\displaystyle t  = f (l, m, g, \theta) \Longrightarrow t  - f (l, m, g, \theta) =:  \Phi (R_1, \ldots, R_5) =  0 $$

Define the physically meaningful function $$\displaystyle \Phi$$ ( Hanche-Olsen (2004), p.3

) of the physical quantities $$\displaystyle R_1, \ldots, R_5$$:

\displaystyle \Phi (R_1, \ldots, R_5) :=  t   - f (l, m, g, \theta) =  R_1 -  f (R_2, R_3, R_4, R_5) =  0 $$
 * {| style="width:100%" border="0"

$$  \displaystyle \Phi (R_1, \ldots, R_5) :=  t   - f (l, m, g, \theta) =  R_1 -  f (R_2, R_3, R_4, R_5) =  0 $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (1)
 * }

An explicit expression for $$\displaystyle \Phi$$ can be found in Pendulum (Wolfram) (with $$\displaystyle \theta$$ written as $$\displaystyle \theta_{max}$$ ). The angular amplitude of swing $$\displaystyle \theta$$ here is a finite (large) angle, and not restricted to "small" angles.

Let $$\displaystyle \overline{T}$$ be the period of oscillation, then

\displaystyle t   = \frac{\overline{T}}{4} $$ We have


 * {| style="width:100%" border="0"

$$  \displaystyle t  \sqrt{\frac{g}{l}} =  \int\limits_{\alpha = 0}^{\alpha = \theta} \frac {     d \alpha }  {      \sqrt{\cos \alpha - \cos \theta} }  =   \frac{\overline{T}}{4} \sqrt{\frac{g}{l}} \Longrightarrow \overline{T} =  4   \sqrt{\frac{l}{g}} \int\limits_{\alpha = 0}^{\alpha = \theta} \frac {     d \alpha }  {      \sqrt{\cos \alpha - \cos \theta} } $$ See Pendulum (Wolfram).
 * style="width:95%" |
 * style="width:95%" |
 * style= | (3)
 * }

Comparing Eq.(3) to the expression for $$\displaystyle t$$ in Pendulum (Wolfram), it can be seen that the mass $$\displaystyle m$$ had cancelled out in Eq.(3); see Bridgman (1922) , p.2,

on how he argued to remove the mass $$\displaystyle m$$ based on dimensional reasoning, without a need to have the explicit expression for $$\displaystyle t$$.

In the case of small angles, the time of swing depends only on $$\displaystyle g$$ and $$\displaystyle l$$, i.e., the angular frequency $$\displaystyle \omega$$ and the period $$\displaystyle \overline{T}$$ of oscillation (not the dimensional formula $$\displaystyle T$$ ) are given by


 * {| style="width:100%" border="0"

$$  \displaystyle \omega =  \frac{2 \pi}{\overline{T}} =  \sqrt{\frac{g}{l}} \Longrightarrow \overline{T} =  2 \pi \sqrt{\frac{l}{g}} $$ See Pendulum small oscillations (Wolfram).
 * style="width:95%" |
 * style="width:95%" |
 * style= | (4)
 * }

Following Hanche-Olsen (2004) ,

let the numerical magnitude of a physical variable $$\displaystyle R_i$$ be denoted by $$\displaystyle \nu(R_i)$$, i.e., the function $$\displaystyle \nu$$ maps a physical quantity to its numerical value ( $$\displaystyle \nu$$ is Greek for "n", which is mnemonic for number ). Consider the time of swing $$\displaystyle R_1 = t$$; if the fundamental unit $$\displaystyle F_1$$ changes from hour to seconds, then the numerical value $$\displaystyle \nu(R_1) = \nu(t)$$ is multiplied by 3600. This example is a very special case such that the unit $$\displaystyle [R_1] = F_1$$. In general we can write
 * {| style="width:100%" border="0"

$$  \displaystyle R_i =  \nu(R_i) [R_i] =  \rho_i [R_i] \, \ {\rm with} \ \rho_i :=  \nu(R_i) $$ The notation $$\displaystyle \rho_i := \nu(R_i)$$ was introduced just to shorten the notation for the numerical value of $$\displaystyle R_i$$.
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (5)
 * }

The numerical value $$\displaystyle \nu(t)$$ of the time of swing would not change if we changed the unit $$\displaystyle [m]$$ of mass or the unit $$\displaystyle [l]$$ of length; Bridman (1922) , p.2.

Note: Is this assertion an experimental fact or an assumption on the nature of the function $$\displaystyle \Phi$$?

Bridgman (1922) , p.2,

argued that if the function $$\displaystyle \Phi$$ in Eq.(1) is to remain true when we change the (fundamental) unit $$\displaystyle [m]$$ of mass, which in turn change the numerical value $$\displaystyle \nu(m)$$ of mass, then the mass $$\displaystyle m$$ should not enter into the arguments of $$\displaystyle \Phi$$, since otherwise the numerical value of $$\displaystyle \Phi$$ would change, as a result of the change in $$\displaystyle \nu(m)$$. Thus, Eq.(1) becomes
 * {| style="width:100%" border="0"

$$  \displaystyle \Phi(t,l,g,\theta) = 0 $$ Thus, Bridgman (1922) , p.2,
 * style="width:95%" |
 * style="width:95%" |
 * style= | (6)
 * }

used dimensional analysis to remove the mass $$\displaystyle m$$ from the arguments of $$\displaystyle \Phi$$ without a need to know the explicit expression for $$\displaystyle \Phi$$.

Next, recall from Table 1 that

\displaystyle [l] = L = F_2 \, \    [g] = L T^{-2} = F_2 F_1^{-2} $$ Bridgman (1922) , p.2,

again argued that if the function $$\displaystyle \Phi$$ is to remain true when we change the fundamental unit $$\displaystyle [l]$$ of length and thus the numerical value $$\displaystyle \nu(l)$$ of length then the numerical value $$\displaystyle \nu(g)$$ of the acceleration of gravity should also change. As a result, $$\displaystyle l$$ and $$\displaystyle g$$ should appear together as the fraction $$\displaystyle l/g$$ or $$\displaystyle g/l$$ in the arguments of $$\displaystyle \Phi$$. Eq.(6), and thus Eq.(1), now becomes
 * {| style="width:100%" border="0"

$$  \displaystyle \Phi(t,l/g,\theta) = 0 \ \, \      {\rm or} \ \Phi(t,g/l,\theta) = 0 $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (7)
 * }

Indeed, Eqs.(3)-(4) were in the form of Eq.(7). On the other hand, Eq.(7) is not yet in dimensionless form.

Numerical value and unit of a physical quantity
Now let's use the above simple pendulum example to explain the notations and the corresponding meaning in Hanche-Olsen (2004) .

Bridgman (1922) , p.2,

wrote that "The dimensional formulas show how the various fundamental units determine the numerical magnitude of the variables".

The dimensional formula for the unit $$\displaystyle [R_i]$$ of a physical quantity $$\displaystyle R_i$$ indicates how the numerical value $$\displaystyle \nu(R_i)$$ of that quantity changes when the fundamental units in this dimensional formula change; Bridgman (1922) , p.2,

For example, the dimensional formula for acceleration $$\displaystyle g$$ is
 * {| style="width:100%" border="0"

$$  \displaystyle [g] = LT^{-2} = F_2 F_1^{-2} \ {\rm and} \ g  = \nu(g) [g] $$ Let the fundamental units for length and time be, respectively, $$\displaystyle [l] = F_2 = meter$$ and $$\displaystyle [t] = F_1 = second$$. Next consider a new set of fundamental units $$\displaystyle \widehat{[l]} = \hat F_2 = kilometer$$ and $$\displaystyle \widehat{[t]} = \hat F_1 = hour$$. We have $$\displaystyle \hat F_2 = 1000 F_2 $$ and $$\displaystyle \hat F_1 = 3600 F_1 $$, or $$\displaystyle F_2 = \hat F_2 / 1000$$ and $$\displaystyle F_1 = \hat F_1 / 3600$$, and
 * style="width:95%" |
 * style="width:95%" |
 * style= | (8)
 * }

\displaystyle g  = \nu(g) [g] =  \hat\nu(g) \widehat{[g]} $$

The old unit $$\displaystyle [g]$$ can be expressed in terms of the new unit $$\displaystyle \widehat{[g]}$$ as follows

\displaystyle [g] =  F_2 F_1^{-2} =  \frac{1}{1000} \hat F_2 (\frac{1}{3600} \hat F_1)^{-2} =  \frac{3600^2}{1000} \hat F_2 \hat F_1^{-2} =  (3.6)^2 \times 1000 \   \widehat{[g]} $$ Thus

\displaystyle g  = \nu(g) [g] =  (3.6)^2 \times 1000 \   \nu(g) \widehat{[g]} =  \hat\nu(g) \widehat{[g]} $$ and hence the new numerical value $$\displaystyle \hat\nu(g)$$ of $$\displaystyle g$$ in terms of the old numerical value $$\displaystyle \nu(g)$$ of $$\displaystyle g$$ is then given by

\displaystyle \hat\nu(g) =  (3.6)^2 \times 1000 \   \nu(g) $$

In other words,

\displaystyle g  = \nu(g) \   m/s^2 =  \hat\nu(g) \   km / hr^2 =  (3.6)^2 \times 1000 \   \nu(g) \   km / hr^2 $$

For example, the value of the acceleration of gravity is

\displaystyle g  = \underbrace{9.81}_{\nu(g)} \   \underbrace{m/s^2}_{[g]} =  \underbrace{ (3.6)^2 \times 1000 \times \underbrace{9.81}_{\nu(g)} }_{\hat \nu (g)} \   \underbrace{km / hr^2}_{\widehat{[g]}} $$

Alternative set of fundamental units
In the above, the traditional fundamental units that were used correspond to the physical quantities $$\displaystyle (R_1, R_2, R_3) = (t,l,m)$$. It is also possible to select from Table 1 a different set of (non-traditional) fundamental units, which are units of a different set of physical quantities. For example, we may want to select $$\displaystyle (R_2,R_3,R_4) = (l,m,g)$$, and take the corresponding units of these physical quantities to form a new set of fundamental units $$\displaystyle (F_2, F_3, F_4)$$. Let $$\displaystyle [R_4]=G$$ be the dimensional formula for acceleration of gravity $$\displaystyle g$$, then $$\displaystyle [R_1]= T$$ $$\displaystyle = \sqrt{L/G} = L^{1/2} G^{-1/2} = F_2^{1/2} F_4^{-1/2}$$ is the dimensional formula for time $$\displaystyle t$$ (which is now no longer a fundamental unit) in terms of the new fundamental units. In fact, the choice of the fundamental units is arbitrary; Hanche-Olsen (2004) , p.1.

Nondimensionalization by mathematical method
As a generalization of Eq.(8), the unit $$\displaystyle [R_j]$$ of the physical quantity $$\displaystyle R_j$$ can be written in terms of the fundamental units $$\displaystyle F_i$$, with $$\displaystyle i=1,\ldots,m$$ as follows
 * {| style="width:100%" border="0"

$$  \displaystyle [R_j] =  \prod_{i=1}^{i=m} (F_i)^{a_{ij}} \, \ {\rm with} \ j = 1, \ldots, n $$ where the powers $$\displaystyle a_{ij}$$ are rational numbers.
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 * style= | (15)
 * }

The fundamental units must satisfy the following independence condition:
 * {| style="width:100%" border="0"

$$  \displaystyle \prod_{j=1}^{j=m} (F_j)^{x_{i}} =  1   \Longrightarrow x_1 = \cdots = x_m = 0 $$ i.e., no single fundamental unit can be expressed as a product combination of the other fundamental units.
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 * style= | (16)
 * }

For the above two different sets of fundamental units, we have:
 * {| style="width:100%" border="0"

$$  \displaystyle T^{x_1} L^{x_2} M^{x_3} = 1 \Longrightarrow x_1 = x_2 = x_3 = 0 $$
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 * style= | (17)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle L^{x_2} M^{x_3} G^{x_4}= 1 \Longrightarrow x_2 = x_3 = x_4 = 0 $$
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 * style= | (18)
 * }

The dimension matrix is defined by
 * {| style="width:100%" border="0"

$$  \displaystyle \mathbf A  := \left[ a_{ij} \right]_{m \times n} $$ where $$\displaystyle i = 1, \cdots, m$$ is the row index, and $$\displaystyle j = 1, \cdots, n$$ the column index.
 * style="width:95%" |
 * style="width:95%" |
 * style= | (19)
 * }

For the swinging pendulum with the $$\displaystyle m=3$$ fundamental units $$\displaystyle (T,L,M)$$, and $$\displaystyle n=5$$ physical quantities in Table 1, the dimension matrix is
 * {| style="width:100%" border="0"

$$  \displaystyle \mathbf A  = \left[ \begin{array}{rrrrr} 1 & 0 & 0 & -2 & 0     \\      0 & 1 & 0 & 1 & 0      \\      0 & 0 & 1 & 0 & 0   \end{array} \right]_{3 \times 5} $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (20)
 * }

A combination of the physical quantities $$\displaystyle R_j$$ is simply a product of these quantities raised to some powers, i.e., $$  \displaystyle R_1^{\lambda_1} \cdots R_n^{\lambda_n} $$, with the unit of such combination being
 * {| style="width:100%" border="0"

$$  \displaystyle \left[ R_1^{\lambda_1} \cdots R_n^{\lambda_n} \right] =  \prod_{i=1}^{i=m} (F_i)^{a_{i1} \lambda_1 + \cdots + a_{in} \lambda_n} $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (21)
 * }

The combination $$  \displaystyle R_1^{\lambda_1} \cdots R_n^{\lambda_n} $$ is called dimensionless if its unit is equal to 1, i.e.,
 * {| style="width:100%" border="0"

$$  \displaystyle \left[ R_1^{\lambda_1} \cdots R_n^{\lambda_n} \right] = 1 $$
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 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (22)
 * }

which, as a result of Eq.(21), then implies that
 * {| style="width:100%" border="0"

$$  \displaystyle \mathbf A  \boldsymbol \lambda =   \mathbf 0 \, \ {\rm with} \ \boldsymbol \lambda =  \left\lfloor \lambda_1, \cdots , \lambda_n \right\rfloor^T $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (23)
 * }

In other words, the column matrix $$\displaystyle \boldsymbol \lambda$$ is in the null space of the dimension matrix $$\displaystyle \mathbf A$$.

For the swing pendulum, using the matlab "null" command

> null(A, 'r')

where the argument 'r' was used to obtain the results in rational numbers instead of real numbers, one can find the null space of $$\displaystyle \mathbf A$$ to be
 * {| style="width:100%" border="0"

$$  \displaystyle Null(\mathbf A)  = \left[ \begin{array}{rr} 2 & 0	 \\	 -1 & 0	 \\	 0 & 0	 \\	 1 & 0	 \\	 0 & 1     \end{array} \right] =:  N(\mathbf A) $$
 * style="width:95%" |
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 * style= | (24)
 * }

Clearly,
 * {| style="width:100%" border="0"

$$  \displaystyle \mathbf A   \cdot N(\mathbf A)  = \left[ \begin{array}{cc} 0 & 0      \\      0 & 0       \\      0 & 0    \end{array} \right] $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (25)
 * }

Thus, the column matrix $$\displaystyle \boldsymbol \lambda$$ can be one of the two columns of the null-space matrix $$\displaystyle N(\mathbf A)$$, i.e., there are two possible dimensionless combinations of the physical quantities $$\displaystyle R_j$$:
 * {| style="width:100%" border="0"

$$  \displaystyle R_1^{2} R_2^{-1} R_4^{1} =  \frac{t^2 g}{l} \, \ {\rm with} \ \left[ \frac{t^2 g}{l} \right] =  T^2 L^{-1} [g] =   1 $$
 * style="width:95%" |
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 * style= | (26)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle R_5^{1} =  \theta \, \ {\rm with} \ [\theta] =  1 $$
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 * style="width:95%" |
 * style= | (27)
 * }

The dimensionless quantity in Eq.(26) is consistent with the result obtained in Eq.(7) by Bridgman (1922) , p.2,

As noted above, Eq.(7) is not yet in dimensionless form, as the quantities that appear in the list of arguments are not dimensionless. On the other hand, the mathematical approach in Hanche-Olsen (2004)

offers a systematic method to find the dimensionless quantities, which can be used in Eq.(7) to obtain a fully dimensionless relation. Eq.(7) can now be put in dimensionless form using Eqs.(26)-(27) as follows:
 * {| style="width:100%" border="0"

$$  \displaystyle \Phi(t \sqrt{g/l}, \theta) = 0 $$
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 * style= | (28)
 * }

For example, the first part of Eq.(3) is already in dimensionless form, i.e.,
 * {| style="width:100%" border="0"

$$  \displaystyle t  \sqrt{\frac{g}{l}} =  \frac{\overline{T}}{4} \sqrt{\frac{g}{l}} =  \int\limits_{\alpha = 0}^{\alpha = \theta} \frac {     d \alpha }  {      \sqrt{\cos \alpha - \cos \theta} } $$ and Eq.(4) can be rewritten in dimensionless form as
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 * style= | (29)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \overline{T} \sqrt{\frac{g}{l}} =  2 \pi $$
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 * style= | (30)
 * }

Change of size in fundamental units
When the size of a fundamental unit changes, i.e., the relation between the old unit $$\displaystyle F_i$$ and the new unit $$\displaystyle \hat F_i$$ given by
 * {| style="width:100%" border="0"

$$  \displaystyle \hat F_i =   \frac {F_i} {x_i} \Longrightarrow F_i =  x_i \hat F_i \Longrightarrow [R_i] =  \nu(R_i) F_i =  \nu(R_i) \underbrace {     x_i \hat F_i }_{F_i} =  \underbrace {     \nu(R_i) x_i }_{ \hat \nu(R_i) } \hat F_i $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (35)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \hat \nu(R_i) =  \nu(R_i) x_i \, \    \hat \rho_i =  \rho_i x_i $$ where $$\displaystyle x_i$$ is a factor. For example, if the unit $$\displaystyle [l] = [R_2] = F_2$$ of length changes from kilometer to meter, then $$\displaystyle x_2 = 1000$$, or
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 * style= | (36)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle F_2 =  x_2 \hat F_2 \Longrightarrow \underbrace{km}_{F_2} =  \underbrace{10^{3}}_{x_2} \    \underbrace{m}_{\hat F_2} $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (36)
 * }

= The Pi theorem =

Developers: Rayleigh, Vaschy, Buckingham, et al.
Many scientists contributed to develop the Pi theorem, sometimes called "P theorem" ; a detailed historical account can be found in the chapter titled Dimensional exploration in Roche (1998) . The principal developers appear to be Rayleigh (1878) , Vaschy (1892) , and Buckingham (1914) .

Aime' Vaschy was probably a Frenchman, based on his first name and on the French journal in which his paper on dimensional analysis appeared, and based on the fact that a prominent French academician referred to the Pi theorem simply as the Vaschy theorem, ignoring the traditional attribution of the theorem to Buckingham. The Pi theorem was often referred to as the Buckingham Pi theorem, but it seems that since the 1960s , the contribution of Vaschy has been recognized, and the theorem referred to as the Vaschy-Buckingham theorem.

Vaschy is pronounced like "Vashy" in English; so many authors (including even French authors!) mispelled the original name of Vaschy as "Vashy" ; see, e.g., Greneche et al. (2005) , in which ironically these French authors only referred to the paper by Buckingham , but not the orginal paper by Vaschy (1892).

In The Buckingham Pi theorem in dimensional analysis, MIT OpenCourseWare, it was stated that F.M. White pointed out in his book Fluid Mechanics who were instrumental in developing the Pi theorem, in particular Lord Rayleigh (1877), A. "Vaschy" (1892), and D. Riabouchinsky (1911).

F.M. White, Fluid mechanics, McGraw-Hill, 2006. Google book, p.293; Chap 5, Dimensional analysis and similarity Amazon.com

F. Casarejos, J.F. Villas da Rocha, and R.M. Xavier, Fine Structure Constants in n-dimensional Physical Spaces through Dimensional Analysis, 2003.

A. Vaschy, Ann. Tel, Vol.12, p.25, 1892.

J. Fluid Mechanics Digital Archive (1968), 32 : 825-826. ... This last part covers dimensional analysis, internal .... The Buckingham theorem is now apparently the Vaschy-Buckingham theorem. ...

Presentation by Salencon A. Vaschy (1892). Annales télégraphiques, Vol.25-28, pp.189-211.

In T.G. Dobre and J.G. Sanchez Marcano, Chemical engineering: Modelling, simulation, and similitude, Wiley-VCH, 2007. Google book, p.465, stated that the first modern presentation of the Pi theorem was due to Vaschy.

M. Zlokarnik, Brief Historic Survey on Dimensional Analysis and Scale-up, Scale-Up in Chemical Engineering, 29 Jun 2006.

Finally: J.J. Roche, The mathematics of measurements, Springer, 1998, 422 pages. Google book, p.208, Chap 11. Whole history since Rayleigh, Vaschy, then Buckingham, with excerpts from original works.

Theorem statement
Consider the dimensional quantity $$\displaystyle a$$, ,p.27 defined as a function of the independent dimensional quantities $$\displaystyle {a_1},{a_2} \ldots ,{a_n}$$ by


 * $$\displaystyle a = f\left( {{a_1},{a_2} \ldots ,{a_k},{a_{k + 1}}, \ldots ,{a_n}} \right)$$

Let the first $$\displaystyle k$$ $$\displaystyle (k \le n)$$ of the dimensional quantities $$\displaystyle {a_1},{a_2} \ldots ,{a_n}$$ have independent dimensions $$\displaystyle($$the number of basic units should be larger than or equal to $$\displaystyle k)$$. Hence, the dimensions of quantities $$\displaystyle a,{a_{k + 1}}, \ldots ,{a_n}$$ can be expressed in terms of the dimensions of the paramaters $$\displaystyle {a_1},{a_2} \ldots ,{a_k}$$.

Let us introduce dimensions of the basic quantities:


 * $$\displaystyle \left[ \right] = {A_1},\,\,\,\left[  \right] = {A_2},\, \ldots ,\,\,\,\left[  \right] = {A_k}$$

The dimensions of the remaining quantities will be


 * $$\displaystyle \left[ a \right] = A_1^A_2^ \ldots A_k^,\,\,\,\left[ \right] = A_1^A_2^ \ldots A_k^, \ldots ,\,\,\,\left[  \right] = A_1^A_2^ \ldots A_k^$$

We now change the units of measurement of the quantities $$\displaystyle {a_1},{a_2} \ldots ,{a_k}$$ by factors of $$\displaystyle {\alpha _1},{\alpha _2} \ldots ,{\alpha _k}$$ respectively; the numerical values of these quantities and the quantities $$\displaystyle a,{a_{k + 1}}, \ldots ,{a_n}$$ in the new system of units, will be given by


 * $$\displaystyle a_1^' = {\alpha _1}{a_1},\,\,\,a_2^' = {\alpha _2}{a_2}, \ldots ,\,\,\,a_k^' = {\alpha _k}{a_k},$$


 * $$\displaystyle {a^'} = \alpha _1^\alpha _2^ \ldots \alpha _k^a,\,\,\,a_{k + 1}^' = \alpha _1^\alpha _2^ \ldots \alpha _k^{a_{k + 1}}, \ldots ,\,\,\,a_n^' = \alpha _1^\alpha _2^ \ldots \alpha _k^{a_n}$$

In the new system of units,


 * $$\displaystyle {a^'} = \alpha _1^\alpha _2^ \ldots \alpha _k^a = \alpha _1^\alpha _2^ \ldots \alpha _k^f\left( {{a_1},{a_2} \ldots ,{a_n}} \right) = f\left( {{\alpha _1}{a_1}, \ldots ,{\alpha _k}{a_k},\alpha _1^\alpha _2^ \ldots \alpha _k^{a_{k + 1}}, \ldots ,\alpha _1^\alpha _2^ \ldots \alpha _k^{a_n}} \right)$$

This relation shows that the function $$\displaystyle f$$ is homogeneous in the scales $$\displaystyle {\alpha _1},{\alpha _2} \ldots ,{\alpha _k}$$. Because the scales are arbitrary, we choose them in order to cut down the number of arguments in the function $$\displaystyle f$$. Let us assume:


 * $$\displaystyle {\alpha _1} = \frac{1},\,\,\,{\alpha _2} = \frac{1}, \ldots ,\,\,\,{\alpha _k} = \frac{1}$$

The numerical values of the parameters $$\displaystyle a,{a_{k + 1}}, \ldots ,{a_n}$$ are determined in this relative system of units by the formular:


 * $$\displaystyle \Pi = \frac{a},\,\,\,{\Pi _1} = \frac, \ldots ,\,\,\,{\Pi _{n - k}} = \frac$$

Where $$\displaystyle a,{a_1},{a_2}, \ldots ,{a_n}$$ are the numerical values of the quantities considered in the original system of units. It is obvious that the values $$\displaystyle \Pi ,{\Pi _1} \ldots ,{\Pi _{n - k}}$$ are independent of the choice of the original system of units of measurement since they have zero dimensions relative to the units $$\displaystyle {A_1},{A_2} \ldots ,{A_k}$$. Therefore, these quantities can be considered nondimensional.

Using the relative system of units, one can write:


 * $$\displaystyle \Pi = f\left( {1, \ldots ,1,{\Pi _1}, \ldots ,{\Pi _{n - k}}} \right)$$

Hence, the relation between the $$\displaystyle n+1$$ dimensional quantities $$\displaystyle a,{a_1},{a_2}, \ldots ,{a_n}$$ reduces to one between the $$\displaystyle n+1-k$$ quantities $$\displaystyle \Pi ,{\Pi _1} \ldots ,{\Pi _{n - k}}$$, which are a nondimensional combination of $$\displaystyle n+1$$ dimensional quantities.

Proof
J. B. Boyling, A short proof of the Pi theorem of dimensional analysis, ZAMP, Vol.30, No.3, May 1979.

= More examples =

Solid mechanics
A. Vaziri and L. Mahadevan, Localized and extended deformations of elastic shells, Proceedings of the National Academy of Sciences (USA), 105, 7913, 2008. Movie

$$\displaystyle \square \, w$$

Physics
On the use, by Einstein, of the Principle of Dimensional Homogeneity, in three problems of the Physics of Solids

Theorem of Buckingham Kepler's 3rd law.

= References =