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JO URNAL OF RESEARCH of the National Bureau of Standards-B. Mathematics and Mathematical Vol. 66B, No. 3, JUly- September 1962 as a Fourth Fundamental Quantity

Jacques E. Romain*

(April 18, 1962)

The ad\:antage of co nsidering anglc as a. fourth fu ndamental quantity of a~1 d physles IS stressed, a nd a n alternative approach is suggested to introduce an angular dimensIOn m such a \\"ay t hat t he physical laws, in a ny form, are dimensionally homogeneous. A few examples are descn bed to s how how so me eq uations of mathematics and mechanics should be revised to put t he new point of vic II' into p ractice. I, Introduction co mplex exponential formalism is used in physics i~ stead of trigonometric functions, Of course, the Chester H. Page 1 r ecently wrote in this JOUl'na] dIfficulty can be disposed of by stating that "this is ,l most interesting study about "physical entities and an illustration of the fact that equations used in their mn,thematical I' epl'esentn,tion ," One or the pllysics need not be dimensionally homogeneous ideas in that paper, which will be of intcrest h ere, is wh en terms

W= - 1) The appeamnce of in an equ ation can often be traced to the relation is clilll err sionally nonhornoge rrc ous, and that therefore dimensionfll homogell city is Hot preserved when the s= ()R (1) ·Prcscnt address, OO Il (, l' al Dynamics, Vort " ·ol'th. rr cxas. I C. H . Page, J. Research ;.rBS 65 8 (M atb. and _Vl aLh. Phys.) No, 4 227-23,; (1961). ' between an arc s of a circle, its radius R and the ~ J. Pa l acio ~ , Analyse dimcn sionncllc (Gauthier-Villars, Va ris, 1960). 3 \V. C. MIchels and A. L. Patterson, Elements of ~\Io clc rn l'hysics (Van central angle () measured in . The nece sity Xostl'and, l-'rincctoll , N.J., 195 1L namely sectioll 13.4. 'F. H aberli, Schwriz. Arch. ange\\,. \\'iss. T echn , 1 5,343 (1949). of dimensional homogeneity of both sides of eq (1) ' P . and I). E. Spe ncer,.T. Fran klin InsL. 24-8,495 (l949). (in which sand R are lengths) seems to imply that Ii Internationa.l O r g: ~ln i zHLio n for Standardization Doc rSO/TC 1<) 131 E, D eeember, l954. , . -, () is dimensionless. 97 It has been tried (see footnote 6) to a,void contra­ eq (1). The reader should back to the elemen­ diction by taking into consid eration two kinds of tary demonstration of eq (4); the general scheme is: "a,ngles": the "geometrical angle" which is the usual (1 ) show that t he length of an arc e of a circle lies one, a, quan tity related to the portion of the plane between those of two line segments which are pro­ contained between two concurrent straight lines, and portionftl to sin e and tan e; (2) divide by the radius t he " analytical angle" which is the e to be found in H: the two segments yield sin e and tan e, while t Il e eq (1). But the so-called "all alytical ft ngle" is no arc becomes siR, or < e> on accoun t of eq (3). Thus, new notion indeed: it is plainly the measure of the the true form of eq (4), valid for a,ny unit of angle, is angle in radians. This can be seen immediately from eq (1) , which originates in a theorem statin g that in IU110-)o(sin e) /= 1. (5) a circle two arcs are proportion al to t heir cen tral angles. When one of the ftrcs is equal to tbe radius A similar equation would be obtained with tan e its central angle is one , an d the proportion in stead of sin e. becomes The equations showing how to replace the sine s= (e/1 racl )R , (I a) ancl tangent o[ a smftll angle by t he angle itself follow in the correc:t form which is the correct form of eq (1), with e a true "geometrical" angle. (6) How is it t hat one should think himself obliged to particularize the measures of angles under the naJne As a consequence, every equation derived from eq of "analytical a,ngles," while not feeling any need (3) or (5) should be recalcula,tecl. The details of whatsoever of "anaJytical lengths," "analytical t lte calculations are left to t he reader. " or "analytical "? The explanation , Here are a few results: and the reason why mechanics could be developed (a) D erivatives of trigo nometric Junctions: without recognizing the fundamen tal dimension of angle, is that angles often ftppear in mechanics d (s in x) /dx= (1/1 md) cos x, (7) t hrough their measure in radians, which indeed can be seen to be derived from eq (1a) . Something simi­ or lar happens in other chapters of physics: [or instance d(sin x) = cos x cl< x> , (8) it is well-known that the dimensional analysis in is less complete if the further funda­ and anftlogous expression s for the derivatives or cos mental dimension of solid angle is not taken into x, ti1l1 x, . . .. An appli cfttion of eq (7) which will account; actually what is here being said of angles be of further use is can be extended to solid angles. The basic angular quantity will be taken here to d(sin wt )/clt= cos wt. (9) mean the ordinary "geometrical" angle, cer tainly the most primftry notion in any study of angles. Trig­ (b) By co mparing t he expansions of ei X to onometric functions will be defined in the elemen­ those of cos x, sin x derived from eq (7), namely tary way as ratios of lengths, i.e., dimensionless quan ti ties. This is simpler than star tin g from power expansions, and has the advantage of putting sin and cos on an equal footing. Since the denominator (1 rad) ,,·ill often appear, I shall write for the sake of brevity, for any quftntity Q: 0/(1 md) =' (2) one readily obtains [or Euler's eq uation the dimen­ sion ally homoge ll eous form T his should not be mistaken Jor t he dimensionless meftsure of Q: it is so only for angles. n'Vhen Q is (10) an angle e, < e> is simply the "analytical" ftlter­ native to the "geometrical" angle e. ) Thus eq (Ja) which will be used in the form will be written (3) From eq (10) follows 3. Angular Dimension in Mathematics cosh i < x>=cos x; sinh i< x>=i sin x. (11) Restrictions are met in mathematics about rtngles, at least on two occasions: these are eq (1), an d the (c) When remembering eq (7) anel compu ting follo\ying: agalll the following integml, one gets (4) (a2_ y2) - 1/2dy= (I /1 rad) sin- 1 (yja) C bot~l of which are only true wIlen e is measured in J + rachans. Equation (1) has just bee n dealt with and = T quantities of t he Sl1m e is, to this writer 's feeling, canceled by the neeessity dimens ion; w has the dimension 8]'- 1, if 8 denoLes or either introducing u in t he co mplex nUlllbers or the angular dimension) : the arguments of tri gono­ retaining inhomogen eous equaLions whcrcver co m­ metri c runctions are angles, but those functions plex numbers are invol ved in t he polm form . t hemselvcs ar e pure dimensionless numbers, as well The only irreducible difrerence b eL ween Lhe t \\·o as the arguments or hy perbolic fun ctions alld t he approach es boils down to t he rac L LhaL in Pltge's exponents. 'Vith those eq uations ill mind no schem e the dimension of a ngle is ingled ouL by difficult:v can be exp erien ced in setting up the equa­ t he requirement that [8]2 = [J ]' while a,ll foul' dimen­ tions or m ec ha ni cs. sions, as such , are put on a n cqual roo Lin g in t he B efore s witching to m ecJlanical equalio ns, l eL it b e present approach. ( In ract, JlOwever, Lhe ex pli ciL pointed out that homogeneity can easily b e restor ed referen ce to the radian , or to u , also singles ouLthe in Page's schell1 e 1'01' the equation a ngular dimension in both systems.) Adillittedly, t his is essentially a maLter of tasLe.

4. Angular Dimension in Kinematics a lld more gener ally for t he polar expression o[ any complex number , " 'h erein the sa me problem occurs. The basic appeanwce of a ngles inkinem aL ics is . ' VJlitt n eeds be done is simply assigning th e t hrough a ngular velocity. The mu tu al dependence dimension or a ngle to tlte co mplex "i". Since [OF or linea,l' a nd a ngular in a = [1] (Page's notation), therc is no conflict with is obtain ed by differentiating eq (3 ) wi t h respcct lo Lill1 e, a nd puLlillg v for ds/dt and w for dO /dt: as a quanlity cquation; a nd as sin 1', in tlmt sch elll e, l' = H. (J 2) also h as Lhe dimens ioll of a ngle, thaL would m ak e both lhe exponcn t ix a nd thc product i sill :r dimen­ This equaLion is dimensionall y homogeneous, for sionless. the dimensional formu la of w is 8T-'; t hus Lhat The UllU SWll fe}ltme is t hat in a ny cOJllplex or < w> is ]'- 1. number (L + bi, a is a llullleric but bhas l he dilll ensiOJl L et us n ow invcsLigllte a fc\\' eo nsequences of eq (12). From eq (12) t he dilll ensionally corrcct or angle (or more generally (L has the dimensions or the modulus a nd b those or Jll od ulus a ngle). expressions can be deduced 1' 01' the rolaliolml ki netic The appearance or a n angular dimension in complex energy o[ a solid: numbcr r epresentation would not seelll incongruous if one r cmembers that t he n a tural geom cLri c rcprcsen­ tation of complex numbers is in a planc. But t he feature jus t alludcd to implies a fUlth er consequence. II" ho mogeneity is to be prcservcd ror Lhe cenLrifugal fo ree on a particl e: tJu'oug hout, t he imagin a ry p ar t b of any complex number a+ bi must have t he dimension o[ a ngle, Fc= m< w>zH, even when b= 1. This implies in troducing som e arbitrary " unit o[ imagill ary p art," say u, with t he a nd for Lh e law of : dimens ion of a ngle, bY lllea ns of which th e co mplex nu mber (a+ i) can be written (a+ ui). It is t hen obvious t hat t he use o[ u is co mpletely equivalent The a ngular velocity vecLor Q of a rOLating solid to putting is defined by t he equation j = 'U.i v= Q X r . (13) ( j dilllension] ess) a nd \\Ti ting It is usually consider ed to have a length equal to the value of th e a ngular velocity w. The climen­ a+ bi= a+ (b/u)j. sional homogeneity of eq (13) (where the cross­ 1\ow t he dimensionless b/u is exactly tile expression product symbol is dimensionless) shows that the defined byeq (2) . In other words, in the suggested tru e relation, when a ngle is considered a fundamen tal " homogenizecl Page's formalism ," t he equations quantity, is involving complex numbers go over in to a form (14) equivalent to t hat of the present approach. with the dimell sional formula T- I. This dimen­ The compa riso n b etween Page's schem e a nd th e sional formula is found to agree with the p articular present proposal can t hus be summarized as follows: case of t he vortex vector of a fluid: In Page's scheme the relations d (sin 1')/dx= cos x; eiT = cos x+ i sin x me qua II ti ty eq ua lions, \\'l! ereH s in the present in which no angular dimension appea rs. 99 Binet's formula expressing the radial into - k in eq (17) and keeping eq (18) as a definition of a poin t obeying the law of areas reads of w; it reads, thanks to eq (ll):

x= xo cosh < w> t+(vo/ t. where A is the constant of . Equations (13), (14) show that the relation between And, finally, the radius of curvature R of a curve the () torque 9)( on the axis causing a gyro­ in a point is defined, in terms of the infiJlitesimal arc scope to precess, th e I about that ds and the corresponding angular d() axis, a~d ~ h e angular velocities of rotation and of of the tangent, by preceSSIOn IS 9)c = I , (15) where 9)( has the conventional dimension. which is a direct application of eq (3). Equation (15) is responsible for the formulation of the law of distribution of tension in a string con­ 5 . Angula r Dimension in Dynamics strained to lie on a surface with friction, in terms of the angle () of rotation of the tangent: The appearance of angles in dynamics may derive from many of the above equations: From eq (6) comes the proper form of the equation yielding the shear angle: The same eq (15) ,vould be involved in the study of the motion of a point missile in the air with friction. < f >=2T(I+ rr) jE (16) The dimensional homogeneity shows that the rela­ tion between torque, angle of rotation, and th e (where rr is Poisson's ratio and E Young's modulus), torsional constant in a twisted wire is and th e physical expression of the non diagonal com­ ponents €ij of the infinitesimal strain tensor of a continuous medium. in terms of the shear angles relevant to the directions i, j: which can also be seen theoretically to proceed from eq (16 ). The expression for the torsional period €ij=' follows with the correct dimension: Also from eq (6) follows the -order approxima­ tion of the formula for the period of th e circular pendulum with amplitude (): 6 . Conclusion

The ~Lbove survey does not claim to be anything Equation (9) leads to the expression of the equa­ like an exhaustive study of the introduction of an tion for the harmonic motion of a point on a straight angular dimension in mechanics. It is just a collec­ line under a central attractive - kr proportional tion of examples to show that no disadvantage can to : arise from its systematic use. Actually, this writer taught rational mechanics for five years with this x= (vo/

; J. E. Romain, Pllblications dc l'C" Jl tyersitc de l'Etat, Elisabcthville (Ka­ tanga), '{o. 3,1962. and has the correct dimension of a time. The analogous motion under a central repulsive force + kr is deduced from eq (17 ) by changing k (Paper 66B3- 78)

100