1985-Orientational analysis, tensor analysis and the group properties of the SI supplementary units—II

Orientational Analysis, Tensor Analysis and the Group Properties of the SI SuppLementary Units- II

by DONALD B. SIAN0

Exxon Research and Engineering Company, Corporate Research Science Laboratories, Clinton Township, Route 22 East, Annandale, NJ 08801, U.S.A.

ABSTRACT : An extension of the methods of dimensional analysis to include the orientations of physical quantities shows that a useful approach is to assign orientations to the supplementary units, radian and steradian, and to require that physical equations be orientationally as well as dimensionally homogeneous. Orientational symbols representing the intuitive orientational character are shown to form a noncyclic Abelian group with four members. The methods of dimensional analysis derivefiom the fact that the laws of physics must be independent of the units: orientational analysis derives from the fact that they must also be independent of the coordinate system, i.e. they are expressible as tensor equations.

I. Introduction

The precise nature of the SI supplementary units radian and steradian remains something of an enigma, and various conflicting proposals have been put forward over the years (l-4). It is clear that the physical variable, angle, requires a unit in the same sense that measurements of length require a specification of the unit (meter, foot, mile, etc.) before any meaningful comparison of sizes can be made. Conversion factors are required in the same way as well-two degrees and two radians are very different in magnitude. On the other hand, for checking equations and for dimensional analysis, it is convenient to suppress the difference between the units, such as meter, foot, mile, etc., and to denote them all by the common symbol L. Similarly, masses, measured in grams, slugs or poundals, are associated with a common symbol M. These dimensional symbols are useful only because of the requirement that physical equations are dimensionally homogeneous, i.e. both sides of the equation must have the same dimensions.

The enigma surrounding the unit radian involves the proper determination of its associated “dimension”. One problem is that if it has an associated dimensional symbol of the sort that lengths have in L, which we could denote as [KJ, then the series expansions of trigometric formations such as sin 0 z 8+ e3/3!, would be dimensionally inhomogeneous since [e] # [Q3 presumably holds just as does L # L3. It is clear that if angle has an associated “dimension” it is not of the same sort as L. The classical alternative (3,4) is to suppose that while 8 has units, it also has the same dimension as pure numbers (numerics), i.e. it is dimensionless. This has the

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advantage of making the series expansion of trigonometric functions homogeneous. However, the price paid for this assumption is that in those equations which we would like to check for correctness by dimensional considerations, the angles and trigonometric functions are left out of consideration.

Similarly, in solving problems in dimensional analysis which involve angles, no information can be gained about the dependence of the solution on angle. While it is self-consistent to assume that angles are dimensionless quantities, it may not be as useful an assumption as we might like. Furthermore, the classical alternative is unsatisfying to an intuition (which one might or might not have) that there is something fundamentally different about angles and numerics which should be reflected in their “dimensions” as well as their units.

Another possibility is to examine the definition of tan 0 as the ratio of two orthogonally related lengths and to recognize that for small angles tan 0 % 0 so the dimensions ofangle might be written as LX/L,, for example. In this treatment (5,6) L, and L, have mathematical properties like L and M and are not to be regarded as cancelling out or to be further “decomposeable”. In this system, L, is not the same quantity as L,, but they are taken to be independent in the same sense that L and M are. This is not a logically consistent possibility, for in the power series expansion of tan 0, two successive terms are (LX/LJ2”+’ and (LX/L,)2”+3, where n is an integer. Dimensional homogeneity is not preserved.

II. Page’s System of the Classical Alternative

A more interesting possibility has been put forward by Page (7,8,9), who shows that the “dimensions” of 13 are different from numerics because their algebra is different. He proposes that the dimensions of 8, symbolized by [O], have the property that [012 2 [l] where the latter quantity is the dimensional symbol associated with numerics and the symbol g denotes dimensional equality. He also (implicitly) assumes that Cl] CO] g [O] [l] e [KJ. From this, it follows that [O-J2 g Cl] and

co] 2n+1 A CO] so that the power series expansion of sin 8 and cos f3 are “dimensionally” homogeneous because they contain only odd or even powers, respectively. It follows then that sin 8 4 CO] and cos 0 g [l]. This is logically self- consistent and leads to some interesting consequences. It is easily seen that the two symbols Cl] and [O] together with their multiplication rule generate a finite group: every product of two elements belongs in the group, there is an identity element, every element has an inverse and associativity is satisfied. The dimensional symbols L”‘” where m and n are integers also form a group with multiplication as the operation : there is an identity element, Lo g [l], an inverse for every element, LAm/“, and closure and associativity are satisfied. In this case, however, the group has an infinite number of elements. The classical alternative, that angles have the same dimensions as numerics, can be put into similar terms. The “dimension” of 8 and numerics, both symbolized by [l] forms a (trivial) group with one element: [l] [I] % Cl] ; Cl] - l A [l], etc.

To avoid confusion, it is convenient to reserve the word “dimensions” for symbols such as L” (where n is a rational number) which form the infinite group, L, previously described. The symbols [O] and [l] which form either a one or two member group

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Orientational Analysis-11

will be referred to as “lower dimensions”. Another useful notation is that for equality in dimensions be symbolized by g while that for equality in lower dimensions be G, as we have already done. In order to be useful of course, we need to require that equations that represent physical situations be homogeneous in both their dimensions and lower dimensions.

The dimension and lower dimensions must be assigned to physical variables in a way that is consistent with their definitions. Thus, for example, in the definition of work as a dot product of force and displacement

W=F~l=Flcos0 (1)

we see that

and

W !? ML2T-2

w z [l] in both the classical alternative and in Page’s system, because they both assume M, I g [l]. On the other hand, torque is

z=Fxl=FIsin@ (2)

so

r g MI,2T-2

and

in Page’s system, while the classical alternative has z z Cl]. Thus, Page’s system makes the intuitively satisfying differentiation between work and torque, as well as angle and numeric, which the classical alternative does not make. There is also perhaps some improved utility as well in extending the ability to check equations and to derive new results by dimensional reasoning. However, there are some features that are to some extent paradoxical. For example, for an angle in the x-y plane,

tan 6 = &/lx (3)

where lY and 1, are the sides of the right triangle. Since

while

d cll d 1,/L = m = [l],

we see that (3) is not lower dimensionally homogeneous. This can be corrected only by assuming that a “hidden” lower dimension appears on the right hand of (3). This

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Donald B. Siano

could be, for example, sin 7112 which has the value unity and lower dimension of [S], i.e. tan 0 = (1,)/(1,) sin 7c/2 s [a], and lower dimensional homogeneity is restored. However, this will seem to some to be an ad hoc assumption. As another example, consider the definition of coefficient of friction, CL. It is defined as the ratio of the force acting parallel to the plane of sliding, to the normal force. Since forces have lower dimension Cl], we see at once that

p = FJFz A [l]/[l] g Cl].

Now, the angle at which a plane must be tilted to the horizontal in order to just make a block start sliding is given by elementary means as

tan 0 = p. (4)

The equation is not lower dimensionally homogeneous and therefore could not have been arrived at by dimensional reasoning. Again, a “hidden” lower dimension may be present.

ZZZ. Orientations

There is, however, another method which can be of some assistance in problems of this sort which we call orientational analysis. This method uses the properties of a group with four elements denoted as l,, l,, 1, and 1, (the identity element), rather than Page’s two. This group, called the Vierergruppe, is the only noncyclic group with four elements and has the multiplication rules l,l, = l,l, = l,, 1; ’ = l,, 1: = l,, and similarly for y and z subscripts, while a “cross product” rule holds : l,l, = l,l, = 1, and similarly for cyclic permutations of X, y and z. The appropriate orientational symbol is assigned to physical variables by examining their orientations in space (which has an orthogonal coordinate system). Thus, a length, velocity, acceleration or force in the x-direction has an associated orientational symbol 1,. A physical variable unoriented in space, such as numerics, mass, time, growth rate and energy, are assigned the symbol for orientationless quantities l,, the identity element. The orientation of derived physical variables such as kinetic energy are determined by assigning the correct orientation to the primitive variables and using the appropriate multiplication rules, i.e.

KE = 1/2mvZ + 1/2mvz + 1/2mvZ

211 12P1 1 00x 12P1 OOY 112 002

= o 10

where we have used g to mean “has the same orientations as”. Note that all the terms on the right hand side all have the same orientationless character, and the equation is therefore orientationally homogeneous. Returning to the definition of the tangent, we put l,, 2 1, and Z, g l,, thus

tan 8 = 1,/l, 2 1,/l, P 1:.

Since tan 6 = 8 for small 8,O 2 1, also. Thus, angles are taken to have an orientation in the direction perpendicular to the plane defined by their sides.

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Orientational Analysis-II

A body rotating about the z-axis has an angular velocity given by

o - dO/dz

where 8 is in the x-y plane. Thus 0 22 1, and so does dO/dt. This is the conventional vector sense of o as well. The homogeneity of the trigonometric functions is also satisfied and, if 8 is in the x-y plane,

sin8~11,;cos8~1,,

the distinction between torque and work are made as well. Work is orientationless, i.e. has the associated orientational symbol 1, since

W = F - I = FJ, g l,l, 2 1,

and

,r = F x 1 = FJ,, 2 l,l, g 1,

which is also the vector sense. The friction coefficient has an orientation which is determined from its definition,

i.e.

p= F,/F, g lJZ g 1, (5)

so it is perpendicular to the plane determined by the forces. Thus, in the problem of the block sliding down the plane, Eq. (4) is orientationally homogeneous because 8 clearly has orientation 1, as well.

This method of assigning orientational symbols to physical quantities is useful for checking equations for correctness. As an example, Amonton’s law states that the velocity of a vehicle, v, can be found by the length of the skid mark produced, d, when stopping it in a skid when the coefficient of friction is ,U :

v2 = 4dgp, (6)

where g is the acceleration due to gravity. If the velocity and length are in the x- direction, say, and g is in the z-direction, ,U will be in the y-direction (perpendicular to both of the two forces). Thus, since numerics are orientationless,

12 g 1 X 1 11 OXZY

p W,)

p l,l,

= O lo

and orientational homogeneity is preserved. Thus, if ,u had been raised to an even power, we would not have obtained orientational homogeneity and this should force us to reconsider the correctness of the equation.

Many examples of this sort may be given. Equations that represent physical situations are orientationally homogeneous provided that the orientations of all the physical variables can be assigned unambiguously in an orthogonal coordinate system. A simple case in which assignments cannot be made is provided by the

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Donald B. Siano

definition of the sine of an angle as the opposite side divided by the hypotenuse :

sin 0 = y/h. (7)

Since y and h are not orthogonal, the equation cannot be checked for orientational homogeneity. The orthogonality requirement is not a very strong limitation for most physical situations can be described this way or can be transformed to a form that satisfies it. However, one other important proviso must be made. Since fractional powers of orientational symbols are not defined (there is no element of the Vierergruppe that multiplied by itself gives l,), equations containing fractional powers must be cleared of them by raising both sides to the smallest power that does this. For example, a falling body in a gravitational field has a velocity u, after it has fallen a distance z given by

“z = J(s4.

This cannot be checked for orientational homogeneity, but

vf = gz

can be :

12 p 11 z z Z.

Equations which are free of fractional powers may be said to be in “normal form” when the lowest possible integral powers are used. Thus,

v; = g*z*

is not in normal form. While dimensionally and orientationally homogeneous, it is orientationally correct only in a trivial way and is useless for the purposes of orientational analysis. As another example, Eq. (7) can be written

sin 8 = y/4(x2 + y”)

or sin* 8 = y*/(x* + y*),

which is in normal form and is again seen to be orientationally homogeneous. Thus, orientational analysis ascribes a special status to equations in normal form.

These two rules, orthogonality of oriented physical variables and no fractional powers, are essential elements in the method of orientational analysis.

There is another interesting difference between Page’s proposal and that described here. Consider a triangle in the x-y plane that has one edge aligned with the x-axis. Let rI be the vector length of this side and r2 be the vector length of the other side as shown in Fig. 1.

According to Page’s system, the area is given by

a ._ = 1/2r, x r2 = 1/2r,r, sin 8 5 [l] [l] [l] [Q z [f3].

The method of orientations can handle only orthogonal variables and so r2 cannot be used. However, its projection on the y-axis, 2,, can be, hence

a xy = l/21,1, g l,l, g 1,.


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FIG. 1. A triangle with sides rl and rZ.

Thus, both Page’s system and the method of orientations are similar in that they give areas some group character that is not the same as the identity element. Similarly, the volume of a parallelpiped with sides rl, rz and r3 is

V = (rl x r2)*r3 = r1r2 sin 0 cos I3 g [19],

while the method of orientations again considers only three orthogonal lengths :

v = l,lylz g l,l,l, g 1,.

Thus, the results are now different. In the method of orientations, volumes are unoriented (orientation = lo) while in Page’s system, it has the same group character as angle and area. It therefore appears that the method of orientation has some advantages in that it is somewhat more in line with an intuition that has been nourished by vector methods, where area is taken to be a vector in the direction normal to the area, and volumes are scalars.

IV. The Supplementary Units

To return to the question of the nature of the unit “radian”, the SI has it that it is arbitrary whether it is to be regarded as a fundamental (like meter) unit or as a derived (like meter’) unit. The argument can be made that a radian is defined as the ratio of two lengths so it has dimensions like L/L = [l] (usually written as just 1). Thus, it is a “derived” unit and not a fundamental one-it is just another name for unity (more properly, the identity element). On the other hand, it is argued that it cannot be regarded as derived because angles cannot be compared to numerics-the unit, degrees or radian, must be specified when the size of an angle is reported. The confusion is resolved when it is recognized that “radian” is associated with the Vierergruppe rather than the one that M, L or T forms. The Vierergruppe is not a subgroup of the latter group and so it does not appear to be appropriate to say that the radian is a “derived” unit (in the sense that mz is derived from m). It is instead a “base” unit for oriented quantities which are associated with the Vierergruppe.


Donald B. Siano

This leads us to consideration of the units of the friction coefficient, i.e. should it be measured in units of radians as well? It is clear that the units of a quantity are to a large degree chosen as a matter of convenience and convention. The dimensions of a physical quantity, once it is clearly defined in operational terms is not, however. They are automatically fixed once the list of base dimensions (M, L, T or F, L, T, etc.) is agreed upon. That the dimension of force is MLT-’ is universally agreed upon, but this is not a convention. It is not possible that some other combination of the dimensional symbols will satisfy the requirements of consistency : to claim that the dimensions of force are M’LT- ’ is just wrong. On the other hand, to claim that units of force are Newtons is just as correct as to claim dynes or pounds. Units are arbitrary, dimensions are mandatory. Oriented dimensionless physical quantities may have units as well but they do not have to be expressed in units of radians if this is not convenient. Friction coefficient could be given in the units N/dyne, for example, if it was more convenient for some purpose, as well as the unit N/N. There would then be a conversion factor between the two kinds of units, just as there is between mph and kmph, but the orientation of the friction coefficient is 1, in either case, just as the dimensions of both velocities are LT-I.

The common choice for the units of the friction coefficient is usually taken as N/N (= dyne/dyne, etc.) mostly as a matter of convenience. There is one other reason for choosing N/N as opposed to N/dyne, however, It is generally true that the results of the dimensional analysis of a problem can yield order of magnitude estimates of the size of the dependent variable because the undetermined constant (like the number 4 in Amonton’s law) is almost always of order unity. If, however, for some reason, we had chosen to take the units of p to be N/dyne, our guess for v2 when 1, g, and p are known would be off by a factor of 105. In this sense, for this purpose, the more useful choice is N/N. However, the choice N/dyne cannot be said to be incorrect; it is merely less useful for some purposes.

V. Spherical and Cylindrical Coordinate Systems

The orientational symbols used so far are applicable only in Cartesian coordinate systems, but it is possible to define in an analogous way appropriate symbols that are used in other coordinate systems. In cylindrical coordinates (r, tI,z), the orientational symbols are l,, lBh l,, where leT is in the direction of increasing angle. The orientation of 8 is, of course, 1, as before. These three, together with an identity element form a group isomorphous with the Vierergruppe, so 11 = l,, l,l, = l,,, etc. An element of area is

da = r d8 dz g l,l,l, g 1, (8)

as expected. An element of volume is

du = I d8 dz dr g l,l,l,l, g l,, (9)

also as expected. An element of the circumference is

dc = I d0 g l,l, 2 l,, (10)

which is perfectly reasonable.


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Orientational Analysis--II

FIG. 2. The relationship between the various orientations and geometry in spherical coordinates.

In spherical coordinates (I, 0, c#J), the appropriate orthogonal set is (l,, 10T, l+r) with the same set of rules as before: 1: = l,, leTlOT = l,, etc. As shown in Fig. 2, r, r sin C/J de, and r d4 are the quantities which have the respective orientation of

l,,l,r and l&,. It is clear that C#J and the tangential 0 have the same orientation so 1, = I,,, but the orientation of 8 is a little more obscure: it can be chosen in a consistent way by consideration of the solid angle

dR = $ = sin C$ de d& (11)

The natural assignment of the orientation of dA is 1, so

dR g 1,/l; g l,l,l, 2 1,

or

1, g 1,. (12)

This may seem a little paradoxical at first, but it really is no more so than the requirement in cylindrical coordinates that 1, g 1,. Thus, an element of area is

dA = (r sin 4 d@(r d4) g 1,&r 2 1, (13)

= l,l,l,l,l, p 1,

which is consistent. Similarly, an element of volume is

dV = (r sin 4 dB)(r d+) dr g l,,l,,l, g 1,

p l,l,l,l,l,l, p l,l, p 1,

which is also consistent with (12).

(14)

This treatment also clarifies the relationship of steradian to radian. It is sometimes claimed that though the units are different, the “dimensions” of angle and solid angle are the same. The steradian is seen to be the unit associated with the orientational


Donald B. Siano

symbol I,, but because of (12), this is the same as 1, which has the associated unit of radian. In this sense, the “dimensions” of angle and solid angle are indeed the same, while their units are different.

VZ. Electromagnetic Quuntities

An interesting application of the method to spherical geometry is in the fundamental laws of electromagnetism. The electric displacement D, the electric field intensity, E, the magnetic induction, B, and the magnetic field intensity, H, are easily proven to have the orientations of their vector sense, and Maxwell’s equations are orientationally homogeneous. The permittivity, E,,, and the permeability, p,,, of the vacuum are orientationless. The role of the ubiquitous factor 47t which appears in various equations (we shall consider only the SI formulae) is of some interest. Perucca (10) has correctly stated that this factor is not a numeric but is a solid angle. As we have seen, solid angles are associated with the orientational symbol 1, which must therefore be taken into account when checking equations for orientational homogeneity. Coulomb’s law is usually written :

F - l qlq=f 4X&, r=

where i is a unit vector along the line connecting the charges q1 and q2.

If we take the origin of a spherical coordinate system at ql, then q2 has coordinates (I, 8,4) and the force on q2 is directed along r, and has a component along r of F,. The equation can be written

F = 1 4lcl2

’ G&rZ

This equation is orientationally homogeneous because

Another example is Biot-Savart’s law :

dB_~~Idlsin~B

47c 12 ’

where the geometry is shown in Fig. 3. Here we place the origin of the spherical coordinate system at the elementary length dl and are considering the magnetic field intensity, B, at the point (I, 8, I$). Clearly, dl has the same orientation as 8, and again we take the factor of 471 to be a solid angle. The magnetic field intensity has an orientation l,,, so the equation can be written as :

and is orientationally homogeneous :

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FIG. 3. Diagram representing the Biot-Savart law.

according to the rules previously given. These examples show that the right hand side of the component equations already contain the information on the direction of F and B, and in a sense, the unit vectors i and @are superfluous. They also show why the SI method of rationalization is decidedly superior to systems in which the 4as do not occur in these equations and wind up putting in their appearance in other equations.

VII. “‘Hi&Zen” Orientations

As in Page’s system, equations which appear to be orientationally inhomogeneous because of the neglect of implicit or “hidden” orientations sometimes occur. This might be considered to be a serious flaw in the method but they are rarely found, and can be avoided by some simple considerations.

A simple example involves the range, R, of a projectile fired at a velocity, uO, at an angle to the horizontal, 8 :

R = (2 vf, sin 8 cos Q/g,

which is orientationally homogeneous. However, the maximum range is given when t?=rr/4;sosin~cos8=1/2and

R, = (2 v$g) * l/2 = v;/g.

This equation is superficially inhomogeneous (assuming that u$ 2 1,) because the factor of l/2 actually has the orientation of sin 8 = 1, and so it not a numeric. Situations of this sort can be avoided by working with the more general equations rather than ones in which variables (especially angles) have been assigned particular values.

Another interesting example is found in the magnetic field intensity at a distance d from an infinitely long thin wire carrying a current I, i.e.

B =!!?f eT 2a d’

As previously discussed, the Biot-Savart law is given for spherical geometry and the


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factor of rc has the orientation 1, g 1,. If d has the orientation l,, the equation is orientationally inhomogeneous for it gives

This is due to an implicit factor of sin (742) which is obtained by evaluating an integral in the derivation of the equation. If the wire had a finite length, no problem arises because the angular factor is explicit. Another viewpoint on this is that for an infinitely long wire, the spherical geometry appropriate to the Biot-Sarart law should be replaced by the cylindrical geometry appropriate to the situation. In this case, the orientation of 7~ is clearly 1, and d g 1, so the equation is orientationally homogeneous because l,l, 2 l,l, g l,,.

VIII. Applications of Orientational Analysis

The use of orientations to extend the results of dimensional analysis is straightforward. First, the problem is set up and solved as far as possible, using only orthogonally related and oriented quantities as previously described. If there remain unknown exponents, the partial solution is checked for fractional exponents. If it does, they are cleared by raising both sides to the lowest integral power that removes them. Then orientational symbols are assigned to all the variables and the result of the requirement of orientational homogeneity is deduced. This will give some restriction on the nature ofthe exponents. The knowledge that exponents are usually small and the sign of the exponent (given by simple physical intuition) can often lead to a complete solution of the problem.

For example, Amonton’s law can be “derived” as follows. Assume the velocity of the vehicle is related to the skid mark length, mass of the car, the acceleration due to

gravity, 9, by

v = d”mbgcpe.

The assumption of dimensional homogeneity gives

LT-’ g L”Mb(LT-2)c(Lo) so

l=a+c;b=O;-l=-2c.

Thus, a = c = l/2 and

where e is undetermined and C is a numeric of order unity. This is as far as dimensional analysis can be taken. To use the requirement of orientational homogeneity, this equation must be squared to get rid of the fractional exponents, so

v2 = dgp’“.

Assigning orientations, v g l,, d 22 l,, g g l,, p g l,, gives

12 P 1 x 1 p XZY


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or

which can only be true if 2e + 1 is even, or 2e is odd. Physical intuition should reveal that as the friction coefficient is increased, while d

and g are held fixed, u must increase also. Thus, 2e is an odd integer greater than zero. The lowest possibility is 2e = 1 and this is the correct answer. If we did not know that the solution involved only one power, then a power series expansion may be assumed, i.e.

where the K, are constants. It is usually stated that dimensional analysis can reveal nothing about the nature of undetermined functions of dimensionless variables. However, orientational considerations has revealed that in this case, only odd powers of p can be involved. This method can be used in a large number of cases where dimensional analysis yields only partial solutions.

The method must be used with some care when cylindrical or spherical symmetry is present because the quantity rr, treated as a numeric in conventional dimensional analysis, often represents an angle in space with a corresponding orientation. It is readily seen by considering a few examples that reliable results in the case of cylindrical symmetry are only obtained if rc (or 0) is assumed to be part of the solution. For example, consider the moment of inertia, I, of a hoop with mass m and radius r.

I = m”rbzc,

ML2 2 M”Lb(Lo)‘,

so

Orientations give

l=a;2=b,

Z = mr27cc.

1 0 4 1 121’ 2 1’ Orz ZY

which implies c is even. Again, the correct answer is the lowest : c = 0. If we had used the area of the hoop rather than the radius, we obtain

I=mAti

and orientations give

1, g 1,1,1;

g 1c+l z

which is true only if c is odd. The correct answer is c = - 1. Thus orientations gave correct information in both cases but did not yield a completely determined solution. Another interesting way to view this example is to suppose that we


Donald B. Siano

neglected rr (or 0) from the start. Then dimensional analysis would give for the pair (m, I) that

I = mr2

and this equation is orientationally homogeneous. On the other hand, the pair (m, A) gives

l=mA

which is not orientationally homogeneous. As in dimensional analysis, this is a clue that some variable has been left out of consideration. A little reflection shows that this can only be rr (or 19) and we would go back and put it in, arriving at the same conclusion as before.

The fact that careful consideration must be given to the orientation of x, which dimensional analysis always lumps with the undetermined constant numeric (which we have always denoted as C), appears to limit the utility of orientational analysis in cylindrical or spherical coordinate systems. Viewed another way, however, we could perhaps justly claim a small advantage because of the increased knowledge about the constant-that is, whether x appears or not. It is universally claimed that dimensional analysis can yield no information about the value of the numerical constant. Being very like dimensional analysis in spirit, orientational analysis, however, does produce some additional information.

IX. Orientational Analysis and Tensor Analysis

The fundamental basis for the orientational analysis may be found in the fact that the laws of nature should be independent of the kind of coordinate system arbitrarily selected to describe them. That is, the equations of physics are tensor equations. A tensor equation between two quantities can only be true if they are both of the same order. Thus, a scalar (a tensor of zero-order) cannot be put equal to a vector (a tensor of first-order) because they transform differently. A second rule is that the only corresponding components of a tensor may be added or equated. For example, if ai, bi and ci are components of tensors, a valid expression is

ai+bi = Ci (16)

andi= 1,2or3,but

ai+bi = Ck (17)

where k # i is not. This is another way of stating that valid equations are homogeneous in their directional character (as expressed in their subscripts).

If we consider a tensor A, with components along the 1,2 and 3 axes of a,, a, and as, respectively, it can be seen that there are two parts to a tensor-magnitudes and references to their respective axes in the subscripts. It is useful for our purposes to concentrate attention on only the references to the axes and to ignore the magnitudes. We can use the symbol li to mean a reference to the ith axis, and use a different equality sign (g) when we consider equality between the references to axes of different quantities. Thus, we could consider Eq. (7) as containing only li (i = 1,2,

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3) and is therefore homogeneous in this symbol, while (17) is not. If there is no reference to the axes in the description of the physical variable, i.e. it is an invariant, or scalar, we will signify this by assigning it the symbol 1,. Higher order tensors can be assigned symbols that refer only to the axis involved in an analagous way.

Thus, a second-order tensor B has nine components with the corresponding orientational symbols lli, l,, . . . l,,, or, generally, lij (i,j = 1, 2 or 3). In order to simplify the discussion, we will consider only orthogonal, three-dimensional, coordinate systems so that the covariant and contravariant components of a tensor are the same and we will write all the equations in covariant form. The product of a scalar with a tensor gives a tensor of the same order, e.g.

aa, = aia = bi

or, in terms of orientational symbols,

l,li LL lJ, g 1,

so that it is apparent that 1, is an identity element. The inner product of two tensors is a tensor of lower rank than the sum of the

ranks by two. For the special case of two vectors, the inner product is therefore a scalar : that is

a,b, = cc.

The corresponding orientational equation is then

lili g 1,.

Since 1, is the identity, li is its own inverse, that is

(1%

1.7’ L2 1. I* (20)

Multiplication of two tensors yields a tensor of rank equal to the sum of the ranks of the two tensors, thus

aibj = Cij

has the orientational equation lilj = lij (21)

and as a special case lili = lii. (22)

Thus, the diagonal elements of a second rank tensor have the same orientational symbol as scalars. This is in agreement with the rules given previously for orientational symbols. For example, the inertia tensor is a tensor of rank two, which has components in Cartesian coordinates

I,, = s

p(y2 +z2) dI’

along the diagonal, and

Vol. 320, No. 6, pp. 285-302, December 198s Printed in Great Britain

Zxy= - s

pxydV

299

Donald B. Siano

as components off the diagonal. The method of assigning orientational symbols given previously has

I,, p l&l, p I&, g 1,

in agreement with (22). The off-diagonal terms would have, for example,

I, 2 l,l,l,l() p l,l, p 1,. (23)

This latter result cannot be deduced from the considerations given so far-we only have (21). This result can be deduced by a consideration of the second rank antisymmetric tensor associated with two vectors A and B as

cik = 1/2(aib, - a&). (24)

The dual of this tensor is a tensor of rank given by the number of dimensions minus the rank of the tensor, or 3 - 2 = 1, i.e. a vector. It has components

c: = l/d(g) (a& -a&,), etc., (25)

where g is the determinant of the metric tensor which has rank two and is symmetric. In general, it is defined in the expression for the square of the differential of arc length

ds2 = gij dxi dxj (26)

in terms of the differentials along the Xi axes of the chosen coordinate system. In orthogonal coordinate systems gij = 0 for i # j and so (26) becomes

ds2 = gii dxi dxi. (27)

Thus, because of (22) gii has the corresponding orientational symbol 1, and therefore, so does ,/g. Thus, Eq. (25) has the orientational equation

1, p (l,/l,)l,l, p 121, 2 l,l,

where the last inequality is required because the two terms in parentheses must have the same tensor character in order to be subtracted.

That is, a cross-product rule holds for the orientational symbols, provided we are concerned only with three-dimensional, orthogonal coordinate systems. Another correspondence with the previously deduced multiplication rules for the orientational symbols is that, since outer multiplication is associative, so are the orientational symbols :

li(ljlk) = (1,131,.

This completes the proof that the multiplication rules for the orientational symbols (previously derived by intuitive methods) follow from the requirement that the equations of physics must be expressible as tensor equations. The Vierergruppe is therefore deduced to be a consequence of the tensor character of physical equations.

The reason for clearing equations of fractional exponents may be also justified for the same reason. Consider as a simple example the tensor of rank one with components ai. Is a!l2 a tensor? By the definition of (covariant) tensors, ai in the


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coordinate system Xi can be related to Cri in another coordinate system Xi by the relation

ad iii = -aj

a.3

By taking the square root of both sides,

p - ad t - 4 > _ aj12

a2

we see that it does not transform like a tensor, and so ail2 is not a tensor. However, scalars, which have the transformation law

cr = a,

thus

proves that a l/2 is also a tensor of rank zero. This is entirely analogous to the rule for checking equations for orientational homogeneity which, as may now be under- stood, involves determining whether or not they are tensor equations.

The foregoing analysis makes clear the origin of the rules for orientational analysis, as well as its limitations. There are two: the coordinate system used to describe the physical situation of the problem under study must be orthogonal and in three dimensions. For problems in four or more dimensions, the cross product rule breaks down, while for non-orthogonal coordinate systems, g has off-diagonal components so that (25) cannot be used to deduce the cross-product rule.

X. Conclusion

We have seen that there are a number of similarities between dimensional analysis and orientational analysis : the requirement of dimensional and orientational homogeneity in physical equations, the role of group properties assigned to symbols that represent physical quantities and the reasoning used in the analysis of physical problems and checking equations. The origin of both techniques is to be found in invariant rules that physical equations must follow: dimensional analysis arises from the fact that the laws of physics must be expressible in a form independent of the units ; orientational analysis arises from the fact that the laws of physics must be expressible in a form independent of the coordinate system.

References

(1) C. H. Page, Am. J. Phys., Vol. 47, p. 78, 1979. (2) J. deBoer, Am. J. Phys., Vol. 47, p. 818, 1979. (3) A. Lodge, Nature, Lond., Vol. 38, p. 281, 1888. (4) J. C. Maxwell, “Encyclopaedia Britannica”, 9th edn., Vol. VII, p. 241, 1875-1889. (5) H. E. Huntley, “Dimensional Analysis”, Dover Publications, New York, 1967.


Donald B. Siano

(6) C. M. Focken, “Dimensional Methods and Their Applications”, Edward Arnold, London, 1953.

(7) C. H. Page, J. Res. mtn. hr. Stand., Vol. 65B, pp. 227-235, 1961. (8) C. H. Page, IEEE Trans. Educ., Vol. E-10, pp. 7&74, 1967. (9) C. H. Page, J. Res. natn. Bur. Stand., Vol. 79B, pp. 127-135, 1975.

(10) E. Perucca, La Ricerca Scientijica, Vol. 30, Suppl. 12, pp. 220&2211, 1960.


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1985-Orientational analysis, tensor analysis and the group properties of the SI supplementary units—II