MATHEMATICS OF DIMENSIONAL ANALYSIS AND PROBLEM …
MATHEMATICS OF DIMENSIONAL ANALYSIS AND PROBLEM SOLVING IN PHYSICS
Decio Pescetti, Dipartimento di Fisica, Università di Genova, Italy
1. Introduction
As is well known, the qualitative methods, based on the application of the principles of dimensional homogeneity, continuity and symmetry, offer the opportunity for a truly fertile analysis of the physical systems prior to their complete mathematical or experimental study [1-3]. In problem solving, the qualitative methods enable us to deduce useful information about the dependence of a physical quantity (the unknown) on other relevant quantities (the data) [4-8].
The complete description of a real physical system would require a great number of parameters. We must select and take into account the important quantities and ignore those which have relatively small effects. A given physical quantity is expressed by a number followed by the corresponding unit of measurement. The radius of the Hearth is [pic], but it is also [pic]; from this point of view it has no meaning to speak of big or small numbers. The order of magnitude of the relative effect, on the value of a specified unknown X, of a neglected quantity can be expressed by proper dimensionless products (pure numbers) of the system’s characteristics. For instance, consider a damped simple harmonic oscillator. A body of mass m is acted on by an elastic restoring force F=-kx, where x is the deplacement from the equilibrium position and k is the force constant. What is the effect, on the period τ, of a viscous damping force, [pic], where r is a constant. As already remarked, to speak of small or big value of r has no meaning; a given damping constant of value, for instance, [pic], has also the value [pic]. As is well known, such an effect is expressed by the dimensionless product [pic], whose physical meaning is: the ratio between the maximum value of the damping force and the maximum value of the elastic restoring force.
We show that the origin of the information, on the problem’s solution, obtained by the qualitative methods, is made more transparent by making a clear distinction between the mathematical dimensional analysis results and the phenomenological assumptions and laws of nature relating such results to the physical world. Obviously, wrong phenomenological assumptions plus correct mathematical dimensional analysis will lead to erroneous results, as discussed in the comment to example 4 of section 4.
Let us remark that the examples of dimensional analysis given in introductory physics textbooks are likely to be misleading. The reader may be left with the impression that dimensional analysis is a routine procedure. The practice of dimensional analysis requires a great deal of insight and experience. For instance, such insight enters in a crucial way into the initial selection of variables to be included in the analysis. Often considerable penetration is required to recognize when a particular dimensional constant, such as the acceleration due to gravity, may be required. Failure to include a relevant variable or dimensional constant will lead to an incorrect result. Unfortunately, there is nothing in the nature of the mathematics of dimensional analysis to tell the practitioner that a crucial variable has been omitted.
In section 2 we discuss the mathematical bases of dimensional analysis. In section 3 we present a set of exercises in linear algebra, which are proposed as an help to understand the rational of the paper: the necessity of a clear distinction between purely mathematical results and physical assumptions. In fact, the answers of such exercises are involved in the qualitative solution of the physical problems discussed in section 4. Section 5 is devoted to concluding remarks.
2. Dimensional analysis and continuity principle
The dimensions of physical quantities are represented by vectors in an abstract finite-dimensional linear vector space, referred to as the dimension space. In Mechanics the dimension space has a basis of three elements: length L, mass M, time T. The extension to electromagnetism requires a basis of four elements (L, M, T, electrical current I). Finally, the extension to thermal phenomena demands a five element basis: (L, M, T, I, absolute temperature Θ). For instance, in the linear vector space of mechanics, the quantities length, mass, time, density, velocity, acceleration, force, viscosity and elastic force constant are represented by the vectors (1,0,0), (0,1,0), (0,0,1), (-3,1,0), (1,0,-1), (1,0,-2), (1,1,-2), (1,0,-1), and (0,1,-2) respectively.
The principle of dimensional homogeneity (PDH) states that in any legitimate physical equation the dimensions of all terms which are added or subtracted must be the same.
The problem description indicates that a physical quantity X (the unknown) is a function of other quantities [pic](the data):
[pic]. (2.1)
The PDH allows us to study a function of fewer arguments:
[pic] , (2.2)
where [pic] is a dimensionless product between X (elevated at the first power) and some of the data, and [pic] are a complete set of independent dimensionless products between the data themselves. One has (Buckingham’s theorem): m=n-r, where r is the rank of the matrix formed by the dimensional exponents of the data.
Substantially, the Buckingham theorem is the following theorem of linear algebra: if the rank of the matrix [pic] , associated with the n vectors [pic] is r ................
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