DIMENSIONAL AND MULTIDIMENSIONAL ANALYSIS

"Every physical magnitude is supposed to be endowed with a dimension that characterizes its quality." (Hans Reinchenbach)

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Dimensional Analysis
Physical Magnitudes

In mathematics, numbers are considered "pure", without attributes. In physics, however, they are considered magnitudes, consisting of a pure number (or quantity) and a unit, which can be simple or composite. For example, 50 meters, 10 seconds, 30 km/hour, etc. The same physical quantity can be expressed in different ways, depending on the unit chosen. For example, 500 meters = 0.5 km = 50,000 cms. Physical quantities can be primary (length, time and mass) and secondary, such as velocity, acceleration, force and energy.

Magnitudes are handled following the laws of algebra, e.g..

(50 m)/(5 sec) = (10 m/sec)
(30 Km/hour) * (2 hour) = 60 Km

Physical dimensions

Each primary physical quantity has a dimension associated with it, which is symbolized by a letter (L: length, T: time, M: mass). Compound quantities have an associated dimensional expression, which is a monomial, i.e. a product of powers of the primary dimensions. For example, using Maxwell's notation ([x] to refer to the dimensions of a physical quantity x):

• Velocity: [v] = LT⁻¹.
• Force: [f] = [m][a] = MLT⁻².
• Energy: [e] = L²MT−2.

In general, the dimensional expression of a physical quantity is L^n₁·T^n₂·M^{< i>n₃}, siendo n₁, n< sub>2 and n₃ integers (positive, zero or negative).

Constants and angles have dimension 1 (they are dimensionless), as do the arguments and results of trigonometric, logarithm, and exponential functions. For example, the Reynolds number (R) is dimensionless: [R] = 1.

Dimensional Analysis follows the rules of algebra, except for addition and subtraction. For example: [m]+[m]=[m]= M and [m]−[m]=[m]= M.

Dimensional Analysis is a conceptual tool, of qualitative type, widely used in pure and applied science to help model physical phenomena in a simple way, and thus gain understanding by expressing the relationships between the dimensions of the quantities involved in such phenomena.


Magnitudes independent of each other

A set of quantities is said to be independent of each other if none of them can be expressed as a product of powers of the remainder. For example, density (&ro;), velocity (v) and force (f) are independent of each other: [ρ]= ML⁻³, [v]=LT⁻¹, [f]=MLT⁻².


Principles of Dimensional Analysis

Dimensional Analysis studies the physical dimensions involved in the equations that model a phenomenon. It is a simple and powerful technique based on the following two principles:

1. Principle of dimensional homogeneity.
   In any equation relating variables of physical quantities, the units on each side of the equation must be the same. This is the so-called "law of conservation of dimensions". For example, the equation
   
   
   s = v₀t + at²/2
   
   Has dimensional homogeneity. Indeed:
   
   
   [s] = L
   [t] = T
   [v₀] = LT⁻¹
   [a] = LT⁻²
   [½] = 1 (all constants have dimension 1)
   [t²] = T²
   [v₀t] = LT⁻¹T = L
   [½at²] = LT⁻²T² = L
   
   Dimensional homogeneity implies that the arguments of exponential, logarithmic, trigonometric, etc. functions must be dimensionless.
   
   
2. Principle of similarity or mathematical homogeneity.
   The concept of similarity is fundamental in geometry. This concept is generalized to include physical phenomena. It states that "All laws of physics must be invariant under changes of local or global units". This principle is very important when it comes to modeling physical phenomena by means of prototypes or small-scale models.
   
   A function between real variables f(x₁, ... , x_n) is mathematically homogeneous if substituting the variables x_i for x_{< i>i}' = λ_ix_i (the λ_i being arbitrary) it is verified that
   
   
   f(x₁’, ... , x_n') =
   φ(λ₁, ... ,λ_n)*f(x₁, ... ,x_n)
   
   Mathematical homogeneity can be conditional or unconditional, depending on whether the λ_i are independent or bound. It is shown [Palacios, 1964] that every unconditionally homogeneous function is a monomial.
   
   Substitutions of the type x_i' = λ_{< i>i}x_i represent unit changes. Therefore a physical equation of the form f(x₁, ..., x_n) = 0 must fulfill that f(x₁, ..., x_n) sea homogénea matemáticamente.

Applications of Dimensional Analysis

• Helps in the modeling of physical phenomena. The first step in modeling is the identification of the variables involved. From purely dimensional considerations of the variables, the equation relating the variables can often be established.
  
  
• Solving problems at a qualitative level, whose direct or quantitative solution involves great mathematical difficulties.
  
  
• Detection of errors in models.

Multidimensional analysis
Created by George Hart [1995], it is a generalization of linear algebra that incorporates ideas from Dimensional Analysis. The main idea is that scalars, vectors and matrices used in science and engineering contain magnitudes, i.e., numbers (real or complex) and units.

Traditional linear algebra assumes two things:

1. The elements are numbers, dimensionless. They are scalar quantities.
   
   
2. Operations (addition and product) are closed, that is, the result of an operation between numbers is another number.

But if magnitudes are considered, the operations are not closed. Magnitudes of different types do not form an algebraic field, because they are not closed under the sum.

For example:

1. "1 meter + 1 second" is an undefined operation.
   
   
2. If we have the matrix
   
   

   x=(		1m	1s	)
   		1s	1m

   
   then neither x² nor its determinant is defined.

According to Hart, when considering dimensioned quantities, the linear algebra that results from considering vectors and matrices with dimensioned quantities in their components is surprisingly interesting and rich with properties different from nondimensional ones.


Specification in MENTAL
Dimensional Analysis

Calling dim(x) the dimension of the variable magnitude x, the rules to apply in the calculation of the dimensional expressions are:

⟨( (dim(r) = 1) )⟩
(The dimension of a constant is 1)

⟨( (dim(x*y) = dim(x)*dim(y) )⟩

⟨( (dim(x÷y) = dim(x)÷dim(y) )⟩

⟨( (dim(x+y) = dim(x) )⟩
(x and y must be homogeneous)

⟨( (dim(x−y) = dim(x) )⟩
(x and y must be homogeneous)

⟨( (dim(−x) = dim(x) )⟩

Since it is satisfied, for example, that

(T+T = T)   (3*T = T)   (−T = T)

we have a type of non-diophantine arithmetic [See Applications - Mathematics - Non-Diophantine Arithmetic].

The dimensions of a physical quantity can also be represented as a sequence in which the components correspond to the exponents of the basic dimensions. For example, if the basic dimensions are L, T and M, we have for example:

( dim(v) = (1 −1 0) ) // (L^1)*(T^−1)*(M^0)

( dim(m*a) = (1 −2 1) ) // (L^1)*(T^−2)*(M^1)

Then the product of two magnitudes becomes vector sum and the division becomes subtraction:

dim(v*(m*a)) // ev. (2 −3 1)

dim(v/(m*a)) // ev. (0 −3 −1)

This has some analogy with logarithms.


Multidimensional Analysis

At the level of pure mathematics we are interested in geometric quantities. In geometry we speak of a right triangle of lengths 3, 4 and 5, without specifying the unit, because the unit belongs to the physical world and all right triangles of lengths 3, 4 and 5 are similar.

Geometric quantities can be expressed as r*(u^n), where r is a real number and u^n is a unit of qualitative type in Euclidean space of dimension n. Its physical dimension would be Lⁿ. For example, a segment of length 6 can be expressed as 6*u, to indicate that it is a linear (one-dimensional) quantity of length 6, which is independent of the particular physical unit chosen. And 6*(u^2) represent 6 square units in two-dimensional space.

Operations with geometric quantities are homogeneous and involve quantities that have the same unit, avoiding mixing quantities that are qualitatively different. For example:

(6*u + 6*(u^2) + 3*u + 2*(u^2)) // ev. (9*u + 8*(u^2))

(6*u ÷ 3*u) // ev. 2

(6*u ÷ 3*v) // self-evaluate

When Dimensional Analysis is applied in mathematics through the inclusion of units, mathematics is richer, more conceptual, more understandable, more intuitive, more qualitative, more humanistic, with more possibilities, uniting the two modes of consciousness: the analytical and the synthetic.

In Multidimensional Analysis, the square of

x=(		1m	1s	)
		1s	1m

is not defined. In MENTAL, on the other hand, it is. The result is:

x=(		m^2+s^2		2ms	)
		2m		s^2+m^2

Units, qualitative or not, are treated as if they were variables.

In general, expressions of the type r*v, where r is a real number and v is a variable, follow the laws of algebra. The variable can be: a) a unit of quantitative type; b) a unit of qualitative type; c) a fuzzy variable, such as young, tall, rich, and so on. In the latter case, r is a factor between 0 and 1 [see MENTAL Language - Expressions - Magnitudes].

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Addenda
A little history

Although certain ideas of Dimensional Analysis were implicitly present in the works of Galileo, Kepler and Newton, it is considered that Dimensional Analysis was formally born with Fourier. In his work "Analytical Theory of Heat" (1822) he states: "It is necessary to note that each magnitude, indeterminate or constant, has a dimension of its own, and that the terms of one could not be compared if they did not have the same exponent of dimensions".

But the great promoter of Dimensional Analysis was Lord Rayleigh, when he applied it in a methodical way, and with great success, in the resolution of problems of great mathematical difficulty. Rayleigh's method was improved and generalized by Buckinham.


Pi (or Vaschy-Buckinham) Theorem

It is the central result of Dimensional Analysis [see demonstration in Palacios, 1964]. If in a phenomenon we have n independent dimensional variables, then it can be expressed equivalently by a smaller number of dimensionless variables, simplifying the model (by reducing the number of variables) and making it unit independent.

If in a phenomenon we have n dimensional variables x₁, x₂, ... , x_n related by means of

f(x₁, ... , x_n) = 0

and these variables are expressed in terms of k dimensionally independent quantities, then this expression can be written equivalently by another relation between n−k dimensionless independent variables:

Φ(π₁, ... , π_{n−k) = 0}

The variables π_i are expressed as products of powers (monomials) of the original dimensional variables. They are invariant to unit changes and are known as similarity parameters. The relationship between these variables is independent of the size or scale of the system. If two physical phenomena are similar, they are described by the same function Φ.

Pi's theorem had already been announced by Aimé Vaschy in 1892, although without reference to zero-dimensional monomials.


Bibliography

• Barenblatt, G.I. Dimensional Analysis. New York: Gordon and Breach Science Publishers, 1987.
  
  
• Bridgman, Percy William. Dimensional Analysis. Yale University Press, 1963.
  
  
• Buckingham, Edgar. On physically similar systems; illustrations of the use of dimensional equations. Physical Review 4 (4): 345–376, 1914.
  
  
• Buckingham, Edgar. The principle of similitude. Nature 96, 396–397, 1915.
  
  
• Hart, George W. Multidimensinal Analysis: Algebras and Systems for Science and Engineering. Springer−Verlag, 1995.
  
  
• Hart, George W. The Theory of Dimensioned Matrices. Internet.
  
  
• Hilfinger, Paul N. An Ada Package for Dimensional Analysis. ACM Transactions on Programming Languages and Systems. vol. 10. no. 2, pp. 189−203, 1988.
  
  
• Langhaar, H. Dimensional Analysis and Theory of Models. John Wiley & Sons, London, 1962.
  
  
• Packhurst, R.C. Dimensinal Analysis and Scale Factors. Chapman & Hall, London, 1964.
  
  
• Palacios, Julio. Análisis Dimensional. Espasa−Calpe, Madrid, 1964.
  
  
• Szirtes, Thomas. Applied Dimensional Analysis and Modeling. Elsevier, 2006.
  
  
• Vaschy, Aimé. Sur les lois de similitude en physique. Annales Télégraphiques 19: 25–28, 1892.