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 , but it is also ; 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 , of a viscous damping force, , 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, , has also the value . As is well known, such an effect is expressed by the dimensionless product , 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 ). 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 (the data): . (2.1) The PDH allows us to study a function of fewer arguments: , (2.2) where is a dimensionless product between X (elevated at the first power) and some of the data, and 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 , associated with the n vectors is r, such as For every set , there corresponds an unique dimensionless product between the data : A complete set of independent is formed by m+1 elements. The principle of continuity states that small causes produce small effects. In teaching physics, this principle is tacitly taken for granted. Usually, its statement is not explicitly given. Perhaps because there are exceptions [1,9], especially for complex systems (butterfly effect). In fact, arbitrarily small causes might produce finite effects, but this cannot be the rule. The idealizations of physics such as frictionless planes, inextensible string, massless pulleys etc. find their justification and validity in the continuity principle. Such idealizations imply that we may have experimental conditions in which the friction of the plane, the extensivity of the string, the mass of the pulley etc., have negligible effects, with respect to those of other causes and, of course, with respect to the finding of a well defined unknown. Let us consider, for the sake of simplicity, the particular case m=1 in eq. (2.2), . Let us suppose, without loss of generality, P<<1. We say that has an essential type dependence on P if . On the contrary, the dependence is non-essential if , where c is a dimensionless positive constant. The criteria for the evaluation of the orders of magnitude [7,8] are: a) The dimensionless constants c are of the order of unity. b) A dimensionless product P very different from unity is irrelevant to the solution of the problem. The criterion b) is satisfied under the following conditions: i) continuity of function ii) a non-essential dependence of on P; iii) a correct choice of the dimensionless product , that is in should appear the dominant data. By dominant data, with respect to the prediction of the unknown X, we mean a subset of s ≤ (dimension of the linear space) relevant parameters for which the phenomenology under study is preserved, even if all the other parameters are absent (in the sense that they are not influential).
3. Linear algebra exercises Exercise 1. Find the coordinates of the vector (0,0,1) relative to the basis (0,1,0), (0,1,-2), (1,0,1). Answer: (1/2, -1/2, 0). Exercise2. Find the coefficients h_{i} of a linear combination of the vectors (0,1,0), (0,1,-2), (1,0,0), (0,1,-1) such as: . Answer: . Exercise 3. Express the vector (0,0,1) as a linear combination of any set of no more than three of the following vectors: (0,1,0), (0,1,-2), (1,0,0), (0,1,-1). Answer; i) (0,0,1)=1/2(0,1,0)-1/2(0,1,-2); ii) (0,0,1)=(0,1,0)-(0,1,-1); iii) (0,0,1)=-(0,1,-2)+(0,1,-1). Let us remark that answer iii) is not independent of answers i) and ii). Exercise 4. Find the coefficients h_{i} of a linear combination of the vectors (0,1,0), (0,1,-2), (1,0,0), (0,1,0) such as: . Answer: . Exercise 5. Express the vector (0,0,1) as a linear combination of any set of no more than three of the following vectors: (0,1,0), (0,1,-2), (1,0,0), (0,1,0). Answer: (0,0,1)=1/2(1,0,0)-1/2(1,0,-2). Exercise 6. Find the coefficients h_{i} of a linear combination of the vectors (1,0,0), (1,0,0), (0,1,0), (1,0,-2) such as: . Answer: . Exercise 7. Express the vector (0,0,1) as a linear combination of any set of no more than three of the following vectors: (1,0,0), (1,0,0), (0,1,0), (1,0,-2). Answer: (0,0,1)=1/2(1,0,0)-1/2(1,0,-2).
Examples Example1. (Harmonic oscillator) A body of mass m is acted on by an elastic force , where x is the deplacement from the equilibrium position and k is the force constant.. The body is relaxed from the rest at an initial position x_{0} . The body executes an oscillatory motion. Find the period . Solution: The relevant parameters are m, k, x_{0} ; so . By PDH and answer of algebra exercise 1, one finds, , where c is a dimensionless constant. The period is independent of motion amplitude x_{0} . By criterion a) for the evaluation of the orders of magnitude, the constant c is of the order of the unity. As is well known, the exact value of c is 2. Example2. (Damped harmonic oscillator) Find the effect, on period of a viscous damping force Solution: The relevant parameters arem, k, x_{0}, r. So, . By PDH and answer of algebra exercise 2, one has . A proper choice of the constant h is h=1/2. In fact is the ratio between the maximum value of the damping force and the maximum value of the elastic restoring force. [cause: friction). By PDH and answers i), ii) and iii) of algebra exercise 3, one has: . Representation i) is correct. In fact, the problem of finding the period makes sense also in absence of the friction force. Representations ii) and iii) are senseless from the physical point of view. Let us remark that a complet set of independent dimensionless products P_{x} is formed by two elements. For instance: . Obviously, there are infinite other possibilities; for instance: . The correct choice of P_{x} is a matter of physics; it does not follow from the mathematics of dimensional analysis. In conclusion . The dependence of is non-essential . As is well known, the complete exact mathematical solution of the problem leads to: . Example3. (Harmonic oscillator with m_{spring} ) A body of mass m is placed on an horizontal air track. The body is attached to an horizontal spring of force constant k and massa m_{s}. The body is released from the rest at a position in which the spring is stretched by x_{0} . The body executes an oscillatory motion. Find the period . Solution: The relevant parameters are m, k, x_{0}, m_{s}. So, . By PDH and answer of exercise 4, one finds that there is the following dimensionless product between the data: , where h is a constant. A proper choice of h is h=-1. Then: P=m/m_{s} . . By PDH and answer of algebra exercise 5, one finds: and ii) . The dominant parameters are m, k, x_{0} . Reprensentation i) is correct. In conclusion, one has , where is a dimensionless undetermined function of P. There is a non-essential dependence of on P: . A complete exact mathematical analysis yields to: . Example 4. (Flexible chain) A flexible chain of total length l and mass m rests on a frictionless table with length x_{0} overhanging the edge. At time t=0 the system is released. Find time after which the entire chain leaves the table. Solution: The relevant parameters are: . So, . By PDH and answer of algebra exercise 6, one finds: . A proper choice of h is h=1. Then . By PDH and answer of algebra exercise 7, one finds: or . In conclusion, the correct representation is . The sliding time does not depend on the mass m of the chain. It appears quite natural, from the physical point of view, to predict the following special cases: a) , b) . Therefore there is an essential dependence of on P. The solution can also be written , where . The essential dependence on P is preserved. The parameters x_{0} , m and g are a set of dominant parameters, but condition ii) for the validity of criterion b) for the evaluation of the orders of magnitude is not satsfied. The function is an “universal” function. The function cannot be obtained by the qualitative methods. The function can be found: i) by the analytical solution of the equation of motion F=ma; ii) by numerical solution of the equation of motion F=ma; iii) experimentally. One finds: . Remark: [(wrong phenomenological assumptions)+(correct mathematical dimensionl ansalysis)] [erroneous (senseless) information on the problem solution]. Wrong assumption: the motion of the chain is periodic; wrong question: find period By PDH etc. Erroneous conjecture: . In conclusion: senseless result: period .
4. Conclusions The application of the qualitative methods in problem solving is put into execution according to the scheme: [(1) Phenomenological behaviour of the system] [(2) Identification of the unknown X and of the relevent parameters A_{1}, A_{2}, ..., A_{n}] [(3) linear algebra (Buckingham’s theorem)] [(4) Complete set of independent dimensionless products P_{1}, ..., P_{m}; complete set of P_{x} independent dimensionless products] [(5) Proper choice of P_{1}, ..., P_{m}; correct choice of P_{x}] [(6) Physical exploitation of the qualitative solution]. The mathematics of dimensional analysis is involved in steps (3) and (4) only. The preliminary step (1) requires sound information about the system’s phenomenological behaviour. Step (2) demands the knowledge of the physical relations that must be invoked to work out a complete quantitative study of the system. In step (5) the mathematical dimensional analysis results are coupled with the physics of the problem, and finally in step (6) an explicit interpretation of the qualitative solution is given. Let us remark that, when studying a physical system, the information obtained by the qualitative analysis might be increased by breaking the problem into subproblems, even if additional quantities must be introduced [5]. The notion of similarity is closely related to dimensional analysis and model building. One of the reasons that dimensional analysis is a useful method in physics is that many variables are combined into one or few dimensionless numbers (products). It is significant that these dimensionless products serve often as a quantitative measure of similarity. It is surprising that no introductory university algebra textbook mentions the dimension space of the physical quantities, as an example of linear vector space. We hope that this paper will contribute in filling the gap between mathematics and physics in the teaching/learning of this important topic. The sistematic resort to qualitative methods along the lines presented in this paper should be the source of a fertile reflection on the mathematical modeling of physical systems and on the model-reality relationship.
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