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Dimensional analysis

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International Prototype Kilogram (IPK)
Credit: User:Greg L, https://en.wikipedia.org/wiki/File:CGKilogram.jpg, CC BY-SA 3.0 license

(The International Prototype Kilogram (IPK) is an artefact standard or prototype that is defined to be exactly one kilogram mass.)

Dimensional analysis is a practical method to verify the homogeneity of a physical formula through its dimensional equations, that is to say the decomposition of the physical quantities that it involves into a product of magnitudes of base: length, time, mass, electrical intensity, etc., irreducible to each other.

Dimensional analysis is based on the fact that only magnitudes of the same size can be compared or added; we can add one length to another, but we cannot say that it is greater or less than a mass. Intuitively, it is clear that a physical law cannot change, except in the numerical value of its constants, for the simple reason that it is expressed in other units. Vaschy-Buckingham’s theorem demonstrates this mathematically.

In fundamental physics, dimensional analysis makes it possible to determine a priori the form of an equation from hypotheses about the quantities that govern the state of a physical system, before a more complete theory comes to validate these hypotheses. In applied science, it is at the base of modeling by mock-up and of the study of scale effects.

Homogeneous formulas

In a physical formula, the variables are not “only” numbers, but represent physical quantities.

A physical quantity is a measurable parameter that serves to define a state, an object. For example, length, temperature, energy, speed, pressure, a force such as weight, inertia (mass), quantity of material (number of moles) … are physical quantities. A physical measurement expresses the value of a physical quantity by its relation to a constant quantity of the same species taken as the reference unit of measurement (standard or unit).

The quantity is then expressed by a rational number multiplying the unit of measurement. As a result, operations between physical quantities do not only concern numbers, but also units. These units within the physical formulas constrain the form that these formulas can take, because certain operations possible on simple numbers become impossible when these numbers are associated with units. These constraints are those that make a physical formula known as “homogeneous”:

  • multiplication (or division) is possible between all units, or with dimensionless constants, but it is practically the only operation allowed without restriction; the multiplication or division of physical quantities is also possible, and relates both to the numerical values ​​and to the units of these quantities;
  • the addition (or subtraction) of physical quantities of a different nature is meaningless; addition or subtraction of physical quantities of the same nature is possible provided they are expressed with the same unit;
  • with the exception of the exponentiation (a generalization of multiplication and division), a mathematical function (like the sine or the exponential) can only relate to “pure”, dimensionless numbers.

Such a control is automatable. As early as 1976, Michel Sintzoff remarked that the reliability of computational programs in physics can be increased by declaring physical variables as such, and by coding their dimensions afterwards for basic dimensions in a fixed order. It is then possible to verify during the compile time their dimensional homogeneity by symbolic evaluation. For this, we note in particular that:

  • the dimensions of the various quantities form a multiplicative group whose generators are the basic dimensions;
  • addition, subtraction, min / max combinations, assignment of quantities assume operands and results of the same dimension;
  • the size of the product result (or quotient) of two quantities is the product (or quotient) of their dimensions.

Units of the same nature

If the addition of units does not make sense, that of physical quantities of the same nature is possible, provided however that they are reduced to a common unit.

“Nature” and unity

Here we have to remember what exactly is a “unit of measurement”: a physical measure aims at expressing the value of a physical quantity by its relation to a constant size of the same kind. If, therefore, the “hour” is a unit of time measurement, it is because we can compare temporal magnitudes with the particular magnitude of “one hour”: any physical measurement only evaluates a ratio between two magnitudes of the same nature.

These units of measurement are themselves measurable physical quantities, so a number associated with a unit, and taking “one hour” or “one minute” as a reference is fundamentally an arbitrary choice. The arbitrariness of this choice can be frustrating because it does not capture the “nature” of a unit: although a measure is a number associated with a unit (which gives this measure its nature), one can in reality only make reports, and access numbers without dimension.

The idea of ​​a system of natural units responds to this idea of ​​eliminating the arbitrary part in measurement: if there is a natural unit “T” that can serve as a universal reference for measuring time, then the minute and hour can be described as respectively nT and sixty times nT. If unity is natural, we can then consider that “T” concentrates the essence of this magnitude and is its very nature, which makes a number change its nature and become a physical measure: the arbitrary unity that it is thus dissociated into an essential physical quantity, which gives it its “nature”, and a conversion factor specific to this unit, which supports all the arbitrariness.

In this approach, a measure of a physical quantity then conceptually involves three entities: a natural unit, which gives the “nature” of the measurement, a conversion factor that derives from the magnitude used as a practical unit, and a measured number representing the ratio between the measured quantity and the practical unit. That natural unity is not clearly defined (the only clearly natural unit is the speed of light) is of no practical importance. A conversion factor, if it must be calculated, always takes the form of a ratio between two measurements of the same nature, and does not depend on the exact value of the natural unit.

Physical formulas and magnitudes

Regardless of what the value of a natural unit should be, one can consider from this perspective that a physical expression translates operations on complex objects, associating a number, a unit, and a conversion factor.

  • There are numerical operations performed on numbers, on which concentrate the practitioner users of the formula. This is the practical interest of the formula.
  • On the other hand, there are simultaneous operations on quantities, which represent the “nature” of the physical measures involved – and this, regardless of the choice of a unit; this is what the theorist focuses on when he examines the “dimensional equation”.
  • Finally, there are operations on conversion factors that result from the choice of a system of potentially arbitrary units. This is what needs to be looked at when moving from one unit system to another. In a physical formula, this choice in reality only translates into a dimensionless conversion factor (thus, not changing the “nature” of the expression). And since this factor only reflects an arbitrary choice, we get into well-designed systems (like the metric system) to choose the units for the conversion factor to be “one”, and disappears from the formula.

The dimensional equation of a physical formula is an “equation of quantities”, which has the same form as the initial physical formula, but which does not take into account numbers, conversion factors, or dimensionless numerical constants: only magnitudes. It represents the phenomena measured by a symbol; for example, a time is represented by the letter “T“, a length is represented by the letter “L“. It is this formula which makes it possible to determine the dimension in which the result of a physical formula must be expressed, independently of the numbers resulting from the measurements.

Conversion factor

Physical equations link together physical quantities, therefore numbers and units, and potentially conversion factors depending on the choice of these units.

Dimensional equation

The dimensional equation is the equation that relates the dimension of a derived quantity to those of the seven basic quantities. In a dimensional equation, the dimension of the derived variable X is commonly noted [X].

The general form of a dimensional equation is:

[X] = LαMβTγIδΘεNζJη,

where L, M, T, I, Θ, N and J are the respective dimensions of the seven basic quantities (Length, Mass, Time or duration, Electrical intensity, Thermodynamic temperature, Amount of matter and Light intensity); and α, β, γ, δ, ε, ζ and η are the respective exponents of the seven basic quantities.

These are called ”dimensional exponents”. Such a dimensional exponent is a relative integer. It can be (strictly) positive, null or (strictly) negative. A dimensionless quantity, or magnitude of dimension 1, is a quantity for which all dimensional exponents are zero.

Thus, the dimension of a magnitude is the way in which it is composed from the seven basic dimensions.

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