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It is a formalization of Rayleigh's method of dimensional analysis. Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.

The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization , even if the form of the equation is still unknown. A statement of this theorem is that any physical law can be expressed as an identity involving only dimensionless combinations ratios or products of the variables linked by the law e.

If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and Buckingham's theorem would not hold. Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena. Vaschy in , [5] then in —apparently independently—by both A.

Federman [6] and D. Riabouchinsky , [7] and again in by Buckingham. More formally, the number of dimensionless terms that can be formed, p , is equal to the nullity of the dimensional matrix , and k is the rank. For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent. However, the choice of dimensionless parameters is not unique; Buckingham's theorem only provides a way of generating sets of dimensionless parameters and does not indicate the most "physically meaningful".

Two systems for which these parameters coincide are called similar as with similar triangles , they differ only in scale ; they are equivalent for the purposes of the equation, and the experimentalist who wants to determine the form of the equation can choose the most convenient one.

Most importantly, Buckingham's theorem describes the relation between the number of variables and fundamental dimensions. It will be assumed that the space of fundamental and derived physical units forms a vector space over the rational numbers , with the fundamental units as basis vectors, and with multiplication of physical units as the "vector addition" operation, and raising to powers as the "scalar multiplication" operation: represent a dimensional variable as the set of exponents needed for the fundamental units with a power of zero if the particular fundamental unit is not present.

Making the physical units match across sets of physical equations can then be regarded as imposing linear constraints in the physical-units vector space. The matrix can be interpreted as taking in a combination of the dimensions of the variable quantities and giving out the dimensions of this product in fundamental dimensions.

A dimensionless variable is a quantity with fundamental dimensions raised to the zeroth power the zero vector of the vector space over the fundamental dimensions , which is equivalent to the kernel of this matrix.

The dimensionless variables can always be taken to be integer combinations of the dimensional variables by clearing denominators. There is mathematically no natural choice of dimensionless variables; some choices of dimensionless variables are more physically meaningful, and these are what are ideally used.

It is sometimes advantageous to introduce additional base units and techniques to refine the technique of dimensional analysis See orientational analysis and reference [9]. These variables admit a basis of two dimensions: time dimension T and distance dimension D. The elements of the matrix correspond to the powers to which the respective dimensions are to be raised.

In linear algebra, the set of vectors with this property is known as the kernel or nullspace of the linear map represented by the dimensional matrix. In this particular case its kernel is one-dimensional. The dimensional matrix as written above is in reduced row echelon form , so one can read off a non-zero kernel vector to within a multiplicative constant:.

If the dimensional matrix were 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, replacing the dimensions by the corresponding dimensioned variables, may be written:.

Since the kernel is only defined to within a multiplicative constant, the above dimensionless constant raised to any arbitrary power yields another equivalent dimensionless constant. The fact that only a single value of C will do and that it is equal to 1 is not revealed by the technique of dimensional analysis. We wish to determine the period T of small oscillations in a simple pendulum. It will be assumed that it is a function of the length L , the mass M , and the acceleration due to gravity on the surface of the Earth g , which has dimensions of length divided by time squared.

The model is of the form. Note that it is written as a relation, not as a function: T isn't written here as a function of M , L , and g. The dimensional matrix as written above is in reduced row echelon form, so one can read off a kernel vector within a multiplicative constant:.

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:. 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. This example is easy because three of the dimensional quantities are fundamental units, so the last g is a combination of the previous. Note that if a 2 were non-zero, there would be no way to cancel the M value; therefore a 2 must be zero.

Dimensional analysis has allowed us to conclude that the period of the pendulum is not a function of its mass. 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. 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 angle approaches zero. Drinks cooled with small ice cubes cool faster than drinks cooled with the same mass of larger ice cubes.

The common explanation for this phenomenon is that smaller cubes have greater surface area, and this greater area causes greater heat conduction and therefore faster cooling. A simple example of dimensional analysis can be found for the case of the mechanics of a thin, solid and parallel-sided rotating disc.

There are five variables involved which reduce to two non-dimensional groups. The relationship between these can be determined by numerical experiment using, for example, the finite element method. From Wikipedia, the free encyclopedia.

Key theorem in dimensional analysis. Comptes Rendus. Philosophical Magazine. The Theory of Sound. Volume II 2nd ed. Journal of the Franklin Institute. Federman A. Computer Safety, Reliability, and Security. Lecture Notes in Computer Science. Berlin: Springer. Ramsay Maunder Associates. Retrieved 15 April Categories : Dimensional analysis Physics theorems. Namespaces Article Talk. Views Read Edit View history. By using this site, you agree to the Terms of Use and Privacy Policy.

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## Principles of Similitude

It is a formalization of Rayleigh's method of dimensional analysis. Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix. The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization , even if the form of the equation is still unknown. A statement of this theorem is that any physical law can be expressed as an identity involving only dimensionless combinations ratios or products of the variables linked by the law e. If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and Buckingham's theorem would not hold. Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena.

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