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(1)Notes on the Concept of Mass in Classical Dynamics By. Jun FUJIMURA (Received April 30, 1988) Abstract: Investigations are made for the inertial mass concept relating to the inertial reference system. With the postulate that there exist some conserving physical quantities, and that the conservation law is invariant under the transformation between inertial reference systems, the functional form representing the conserving quantities are studied. These quantities are restricted for scalar and vector ones, and it is assumed that they are some analytical functions of velocity of the free moving particle. From above mentioned postulates, the procedure Ieads us to the analysis of these universal functions of velocity u, which appear to be scalar constants in both of the case of Galillean and Lorentzian transformations. These scalar constants which should concern to the quantities characterizing the material property of the subject bodies in motion, are interpreted as inertial masses of the bodies.. It must be insisted here, that these constants are not equal for scalar and vector quantities properly, because they are defined mutualy independently relating to the conserving quantities of scalar and vector. However, though this is the case for the constants in Galillean transformations, it appears that there exist relations between these constants in the case of Lorentzian transformations. Therefore, it is able to reduce these ' (three) constants only to one with the aid of the condition concerning to transformation property, in the Lorentzian case. Thus, the concept of mass is well established, if it is considered in the framework of Lorentzian transformations strictly. However, with the concept of correspondence principle, which means that the limiting value of corresponding quantities should be in agreement with ones of Galillean case in the limit of c-)Foo, where c is the velocity of light, it is possible to obtain the mass concepts in Galillean case also. This result, giving rightly the velocity dependence formulae of mass in special relativity, indicates that the concept of inertial mass could be well defined. and confirmed only in the framework of special relativity.. 1. Introduction. The concept of mass in classical mechanics may be one of the most import-. ant concepts, representing the property of matter in relation to mechanical motion. Its meaning, however, has not ever been clarified throughly, in the process of the formation and development of mechanics during the long history of physics. Indeed, it is veiled in a mist even in this present stage, as Max * Department of Physics, Yokohama National University..

(2) 26 J. FuJIMuRA Jammer has stated:') "One has to admit that in spite of the concerted effort of physisists and philosophers, mathematicians and logicians, no final clarification of concept. of mass has been reached. The modern physist may rightfully be proud of his spectacular achievements in science and technology. However, he should always be aware that the foundations of his imposing edifice, the basic notions of his discipline, such as the concept of mass, are entangled with serious uncertainties and. perplexing diMculties that have as yet not been resolved." In fact, excepting the preliminary stage of statical mechanics, it was necessarily required to establish a strictly defined concept of mass, in the developing process of statical mechanics to dynamics, especialiy for lifting up the kinematics. to dynamics which treats "the motion of matter" where the concept of mass has a decisive role as representing material aspects of bodies, Isaac Newton, who was conscious of this point, still remained･to give the definition of mass as "the quantity of matter expressed by the density multiplied the volume."2) This is a very vurgue definition in its context, because it must be reduced to. the definition of density such as "the quantity of mass contained in unit volume". This is clearly an argument in a circle, as is pointed out by E. Mach.3) And thus, it is more reasonable to understand the definition of mass by I. Newton in his equation of motion, known as the 2nd law of motion, as is stating the quantitative relation between accerelation and force, or, in other words,. the propotional constant of them. However, E. Mach, through his critical and penetrating analysis, finaly concluded that this equation of motion is but a de-. finition formulae for the concept of force.`) This conclusion means that this one equation contains definitions of two quatitities simultaneously, the definition. of mass and definition of force, which is just a tortology.. In this way one cannot find any consitent definition of inertial mass in the. Newtonian system of mechanics; it may rather reasonable to accept the Newtonian mechanics as one in which no logical definition of inertial mass is given. but a phenomenological introduction of mass concept is given. On the other hand, for the gravitational mass, we have rather clear meaning or definition, which concerns to the coupling strength of gravitational interactions or strength of source for the gravitational field. There are significant differencies in their. quality between the clarifing of meanings of them, and it might be the reason that I. Newton has separated the two kinds of mass,. Anyhow, in this situation, Mach stated that the concept of mass should be defined by another physical law, which is independent on the 2nd law, the equa-. tion of motion. This was a very epochmaking remark to the wholei structure of classical mechanics, and might be accepted as one of the most important pointing out among Mach's critical considerations of mechanics. For this purpose, he took the 3rd law of I. Newton, the action-reaction principle, as the law to define the mass concept. Though it seems rather curious to make use of.

(3) Notes on the Concept of Mass in Classical Dynamics 27 the 3rd law, which seems rather unfamiliar to mass concept at a glance, he accomplished his program successfully. Using the action-reaction principle, he eliminated the mutual force interacting for particles in the equation of motion for two particles, and was able to define the mass concept in its ratio, by the ratio of accerelations of these two bodies.. Although this program to define the mass concept seems to be consistent in its context, there remain some problems. Indeed, according to Mach's program, the foundation of mass concept depends solely on the two body problem, implicity on the interactions of them, and not on the behaviour of single body. As is usualy. noted, the Newtonian mechanics is constructed on the basis of the concept of mass point, and is realized essentially as the mechanics of single particle. Therefore,. it may be preferable to construct the whole system of mechanics on the concept of single particle, to recouncil this contradiction. Moreover, because of the definition of mass in its ratio, it may be necessary to find an unit mass, the meaning of which must be universal. This means necessarilly that there exist some universal constants concerning to this unit mass. Probably it forced him to proceed to the proposition of the so-called "Mach's principle" in which the total mass of whole universe is expected to be taken into account, giving some tendencies to mysterious thoughts. Another point that is more serious concerns to the role of the lst law. If the whole system of Newtonian mechanics were constructed only with the 2nd law and 3rd law, then what would be the identity of the lst law? This question is oftenly given the answer by interpreting the lst law as a special case of the 2nd law, when there exists no external force. Indeed, it is not difficult to find such an explanation in several textbooks, where the role of the lst law is treated as very insignificant one. If this is the case, the concept of "inertial". mass may become very unclear, because the concept of mass does not relate to the inertia of bodies in any way.. To authors' opinion, the lst law of Newton, which define the concept of inertial systems, should be the foundation for constructing the whole system of Newtonian mechanics. If the concept of mass were defined on the basis of the lst law, but not of the 3rd law, it would be more consitent to adopt it as the. "inertial mass" with its naming as well as its meaning. It may be not preferable to deny the concept of single particle or isolated system in Newtonian mechanics, as far as we take the concept of absolute space-time frame-work as an intuitive form of I. Kant, which is defined indendently on the existence of matter. The role of the 3rd law should be reserved for the treatment of two body problems or many body problems in their interactions. In this note, investigations as a trial are devoted to this purpose and intention..

(4) 28 J. FuJIMuRA 2. Conservation Law and Intertial System. The clue to the formulation of mechanics may be given by the conservation law of some physical quantities; such as energies, momenta, etc. As early as in 19th century, J.R. SchUtz has indicated that all the system of mechanics could be constructed if the requirements of invariancy of energy conservation law is established throughly under the Galillean transformations of reference sys-. tem, the contents of which he called "the priciple of absolute conservation of energy".5). Though his considerations were not yet completely, they were very important because of his remark to the invariance property of conservation law and its fundamental role for all the logical structure of mechanics. This direc-. tion of approach was followed by A. Einstein, E. Poincar6 et al., in the establishing process in special theory of relativity.. At present, it is usual to understand a physical law as a conservation law of a physical quantity or an invariance problem under a transformation of co-. ordinate system. It seems rather reasonable matter to construct the whole system of mechanics under the assumption that there exist some conserving quantities and that the conservation laws of these qantities are invariant under. the transformations among the reference systems which are dynamicaly equivalent each other. In other words, if a physical quantity is a constant of motion in a reference system, then it must be a constant of motion in another reference system also, as far as these reference systems are dynamicaly equivalent. This is the essential fact of conservation law, and the set of these reference systems. is called "inertial system" in mechanics, Thus the postulate existing in the foundation of dynamics is, "In inertial systems, a conservation law is well established equivalently.". (Postulate A) '. Investigations for these reference systems has started with the famous study by G. Galillei, who had stated it as the reference systems moving each other with constant velocities, and had been fo!lowed by many authors to the fruitful results of Isaac Newton, the law of inertia (the lst law of motion). This principle, the so-called Galillei's principle of relativity, was characterised by Galillean. transformation in its analytical expression, was replaced in later days by Lorentz. transformation in the prominent work of special theory of relativity by A. Einstein. But there remains without change the fundamental property of "moving. each other with constant velocity" for inertial system. The quantity that characterize the inertial system is the velocity, a quantity which is kinematical. or of space-time property, and do not concern to the material properties of bodies. These are the space-time relations in the coordinate systems but are independent to the material world. The second postulate for constructing the logical dynamics, then, should be given in the requirement; "In inertial system, any two reference systems move with constant velocity.

(5) Notes on the Concept of Mass in Classical Dynamics 29 each other." (Postulate B) The concept of velocity, though not containing the concept of material, is implicitly supposing the subject of motion. Thus the conversion to dynamics. from kinematics, must be searched for the consideration of some physical quantities concerning to materials taking into accounts the subject of motion. In other words, it is expected to obtain some physical quantities concerning to matter, if considerations are made for the postulate A under the contemplation of postulate B; this procedure might be the way to deduce the concept of mass.. As usual, assumptions are made below that the time is uniform, space is unifrom and isotropic, with the measure of Euclid, and transformations between reference frames are linear one. That is, notingtwo reference systems by the symbol S and S', with the space-time variable x, t and x', t' respectively, the transformation relations are given for Newtonian and Minkowskian space time, with the characteristic velocity of two reference systems v,. dx'=dx-vdt dt' == dt. (2. 1). for Galillean transformations, and. '. v dx'=dx+(r-1)](v･dx)-7vdt v. 1. where. dt'=-ri, (v･dx)+rdt (2.2) 1 r= vi-Zil. for Lorentz transformations.. These are regarded as the definition of the inertial system. Now the assumptions are made that there exist some physical quantities 9 which are conserved in S system. The conservation law for 9 should be written. d9 dt. =O. (2.3). According to postulate A, the coresponding quatities in S' system must be conserved, thus. d9, dtt. =O (2.4). (2.3) and (2.4) are restrictions for 9 and have a role in decision of the functional. form of them. On the other hand, 9, being considered as a mechanical quantity, should be composed of two quantities; one, the quantity relating to the material bodies which behaves as a subject of motion, and the other, the kinematical quantity which concerns to the space-time. For the,latter, it is reasonable to take up.

(6) 30 J. FuJIMuRA the velocity, which is defined by. clx u= dt. . (2.5). Because of the uniformity and isotropy of space-time, 9 could not contain the variable x and t explicitly, Thus we can regard the quantity 9 as afunction of only one variable u. (Here, we are treating a free particle, and the concept of force is necessarily not introduced.). 9=9(u). (2.6) Now we restrict ourselves to consider the quantity 9 as vector 2 and scalar 9o.6) Writing. 8= u2 (2.7) we can express the quantities 2 and 9o as. 2== f(6)u (2. 8). 9o=g(6) (2.9) where f(8) and g(g) are scalar functions with only one variable 8. They must be universal functions, because they have meanings of universality as expressing. the conservative quantity. And so, if we turn to another reference system S', the relations. 2'=f(ei)ut (2.8i). 96=g(e') (2.g') must hold; with the notation. 8X= u2. (2.7') We assume below that the universal functions f(6) and g(8) are, continueous and differenciable single valued one, as far as necessary.. 3. Mathematical Relations for Conservative Quantity. From the relations (2.3) and (2.4), which is the direct results of postulate. A, we can write for 2 and 9o,. d2(u) d2'(u'). dt= dtt =O (3.1). d9o(u) d96(u'). dt == dtt ==O･ (3,2). And then, substituting the relations (2.8), (2,8'), (2.9) and (2.9'), we obtain im-. mediately.

(7) Notes on the Concept of Mass in Classical Dynamics 31 u･(dfd(tg) u)+f(e)(.･ ddut ). =.i.( dil;') .i)+f(ei) (u'. dd"t') (3.3) and. dg(8) dg(6'). dt == dti･ (3･4). Noting the relation. d d6d dt dtdg' We can deform the equations (3.3) and (3.4) in the forrn. {eF@+ -ll- f(6)} d,6,. -{6iF(6')+ -l}- f(g)} d,6,l (3,s). G(e) d,g, -C(6･) d,il (3.6) where the definitions are made for F(6) and G(8) as. F(6)=. df(8) de. (3.7). G(8) =. dg(g) de. (3.8). As the functions f(e) and g(e) are universal functions, the functions F(e) and G(6) defined by (3.7) and (3.8) are also universalfunctions. Thus (3.5) and (3.6) are giving the relations to be satisfied by universal functions deduced from postulate A, and if the relations between de/dt and de'/dt' are given, we can determine the function f(6) and g(6), solving the functional equations (3.5) and (3.6). The relations between d6/dt and d6t/dt' are given by the postulate B, the transformation relations of coordinate system of (2.1) and (2.2).. Here it must be remarked that there appear's no relation between these two. universal functions f(e) and g(6), as is seen above, to which the concept of mass concerns. They are determined by equations (3.7) and (3.8) independently, thus indicating that the mass concepts are necessarily different for scalar and vector concervlng quantltles.. 4. Determination of Universal Function. I.. -Galillean TransformationUsing the relation (2.1) for Galillean Transformation, we obtain immediately,. it'=u-v (4.1).

(8) 32 J. FuJIMuRA Here we choose two reference system S and S' as its characteristic velocity v is perpendicular to u. This choice does not break generality in determining the universal functions. With the notation. rp=v2 (4.2). we get. d6t d6. et=6+ ,7 (4. 3) dt, == dt (4･4). and the equations (3.7) and (3.8) becomes respectively,. 1. 6F(e)-(g+rp)F(6+rp)+ll{f(6)-f(e+rp)}==O (4.5). G(6)-G(6+rp)=O (4.6) with the condition d6/dt)FO. These are the functional equations to determine f(e) and g(e).. Setting 6= O in these formula,. 2rpF(rp)==f(O)-f(rp) (4.7). G(rp)=G(O) (4.8). with. df(rp) - dg(rp) F(rp)= drp , G(rp)v drp . We can easily integrate (4.7) and (4.8), (see Appendix) and get. f(n)=f(O) (4.9) g(rp)=G(O)rp+g(O) (4.10) or, with the variable 6,. f(e)=f(O) (4.11) g(e)=G(o)e+g(o). (4.12) Here, f(O), g(O) and G(O) are constants which concern to the material pro-. perty of moving bodies. Writihg them as. f(O)=mi, g(O)=m,, G(O)=m,, (4.13) we get the expression for f(u2) and g(za2),. f(u2)=m, (4.14) and for 2 and 9o,. g(u2)==m2+m2u2 (4.15). 2= miu (4. 16) 9o= m2+m3u2. (4.17).

(9) Notes on the Concept of Mass in Classical Dynamics. 33. As is seen immediately, these corespond to the momenta and energies of moving bodies; in classical mechanis, the constants are tentatively chosen as. 1. M3='2-Mi, M2=O (4.18). and one calls mi as "inertial mass". Evidently, there is no reason to inquire the relation (4.18) for constants nz's, as far as we restrict ourselves to the consideration of Galillean transformation. of reference system. However, it might be very remarkable that the universal functions f, g and G appear as scalar constants and that the conserving quantities. 2 and 9o are actualized with some reasonable expressions of momentum and energy of free particles, as are given by (4.16) and (4.17).. 5. Determination of Universal Function. II.. -Lorentzian TransformationThe situation is not so different in the case of Lorentzian transformations. We may utilize the relation (2.2) instead of (2.1) for the functions (3.7) and (3.8).. Equivalently, it is more conveniant to use the relation. i- ", 6'-(i-", rp)(i-2, 6)/[i-", (v･u)] (5.i) which is given directly from the expression of line element for Minkowskii space, where e, 6' and rp mean. 8==u2, e'=u'2, 77=v2. (5.2) As before, we choose the reference systems S and S' to have the relation v±u, and we obtain. 1-zl,i6'=r'2(1-z},d6) (5.3) and. det H,d8 dtt==r dt. (5.4). where r= (1- z},. rp)"i2. With these relations, (3.7) and (3.8) become. 1. eF(8)-r'38tF(8t)+-ii{f(e)-r-3f(gt)}=O (5,5) and. G(6)==r-3G(6'). (5.6) These are equations to determine the universalfunctions f(6) and g(e) with the relation (5,3)..

(10) 34 J. FuJIMuRA Taking e==O in (5.5) and (5.6), and taking into account that e'=rp in this case, we obtain. 2rpF(rp)= r3f(O)-f(rp) (5.7). G(rp)=r3G(O) (5.8) Integrating these equations (see Appendix),. f( rp)= Vlf-/ O-,i, rp (5 9). g(rp)= zii2-G;-),, (s io) and from (5.10) we get immediately the relation. g(O)=2c2G(O) (5.10'). and then,. g(rp)= vlg-(tle1 rp (s lo") Changing the variable T to 8, we get finally,. f(6) -. f(o) Vi-Ei,6. (5.11). g(6)== Vlg-(O-,i,-6 (s 12) This results indicate that two universal functions f(e) and g(6) are the same function except a constant multipling factor. Setting the constants f(O), g(O) and G(O) as. f(O)=pti, g(O)=pt,, G(O)=pt, (5.13) with the relation. pt,=2c2pt,, (5.14) we get the expressions for the conserving quantity 2 and 9e,. 2=uf(u2)=vfi."s.,e (s ls) 9o=g(u2)=viptlge,, (s i6) Here we define a new vector quantity 2', which is equal to 2 with a constant factor k.. 2t== le2 (5.17).

(11) Notes on the Concept of Mass in Classical Dynamics 35 f(o) g(O). . (5. 18). le =. The pt's, the quantity concerning to mass concept, thus become with only one constant mo,. pti= pt2=moc2 (5.19) and. 1. pt3= -2 m, (5.20). giving for conservating quantities 2' and 9o, as. '. 2,= vMi C.2 !%, (s. 21) moc2. 90= vl.ge,2･ (s.22). It is clearly that these formula for 2' and 9o are corresponding ones to the formula of momentum and energy in special relativity, with rest mass mo.. 6. The Concept of Mass -Concluding Remarks--. In this short note, we have introduced the concept of mass as a quantity conserning to material property when some quantities are conserved under the transformations between inertial systems. This procedure depends solely on the conservation !aws in inertial systems, but not on the 3rd law of mechanis as is. done by E. Mach. It does not depend also on the 2nd law, which describe the motion of particles under the action of external forces. Therefore, it may be regarded as one to be free from the concept of mutual interactions and be adaptable as a definition for free single particle, thus confirming the concept of "inertial mass".. As is seen in 3, the definitions are mutualy independent for vector and scalar quantities. It means that the concept of mass for vector quantity (relat-. ing to momentum) and that for scalar quantity (relating to energy) must be different ones as far as there exists no relation between 2 and 9o. Indeed, as is shown in the case of Galillean transformation, three constants mi, m2 and m3 are mutualy independent and are given by the formula (4.16) and (4.17) as. 2=m!u 9o=m2+m3u2. On the other hand, in the case of Lorentzian transformation, where the concepts of space and time are mutualy related, three constants pti, pt2 and pt3 are not independent but the relation (5.14) holds; among them.

(12) 36 J. FuJIMuRA pt,=2c2pt, and we have the relations for 2 and 9o, with two constants. 2= ptiU Vi-y,2 pt2 9o== v,-y,2' Moreover, it is possible to reduce these two constants to only one, mo, if we transfer 2 to 2', by multiply a constant factor to 2, as is done in (5.17). In this case of Lorentzian transformation, we may successfully define an inertial mass concept, which is common in both quantities of vector and scalar. We may thus conclude that the concept of inertial mass could be defined well in the framework of special theory of relativity. And on this basis, if the corresponding principle is applicable, where the assumptions that (5.16) and (5.17) could transfer to (4.16) and (4.17) in the limit. of u2<c2,. pti -D･ mi=moc2 (6,1). pt2.m2=moc2 (6.2) 1. pt, ---i> m,= -2 m, (6.3) it may be possible to certify the mass concept uniquely in the framework of Newtonian mechanics, It must be remarked that there is no reason to require the relation (4.18) in the framework of Newtonian mechanics, and this situation would probab!y have caused the defect of factor 1/2 in the definition of "vis viva" by G.W. Leipnitz and succeding dispute among the schools of him and of Descartes.7)8) Another point to be added is the comment concerning to the deduction of the formulae of velocity dependence of mass in special relativity. As is well known, the formulae. m(u) = Mo Vi-Y,-2. (6.4). is usually deduced by the procedure given by G.N. Lewis and R.C. Tolman,9) where the collision problem of two bodies are elegantly investigated on the basis of momentum conservation law. As is immediately realized, this procedure depends on the consideration of interacting problems of two bodies, thus exists. on the line of E. Mach. Therefore, it may be unsatisfactory from the viewpoint requiring the mass definition far the free single particle.. In this note, investigations are made on the mass concept, postulating the conservation law and its invariancy under the transformations between inertial.

(13) Notes on the Concept of Mass in Classical Dynamics 37 systems, and thus being independent on the mutual interactions. We believe that this way of defining the mass concept may be more consistent than the usual one, and the situations concerning to mass concept could be improved.. Appendix 1. The Integration of (4.7). With variable x, the eguation to be solved is given by. 2x. df dx. =f(O)-f(x), x>O (Al.3). and the integration is immediately obtained as. C. f(X)== f(O)+ vl- (Al.2) where C is an integration constant. Considering the limiting case x-->O, then f(x)->f(O), we get. C=O (Al.3). and then. f(x)=f(O) (Al.4) 2. The Integration of (5.7). The equation to be solved is, with variable x,. 2x. df dx. =r3f(O)-f(x), x>O (A2.1). where. 1 r= vi-ig,-' We can change the form of the right hand side of Equation (A2.1) with the relation. r2=1+ il,l r2 (A2.2) dr -1 3. dx - 2c2 r. , (A2.3). as. r3f(O)=7(1+ ", r3)f(O) -(r+2x ddr. )f(o) . And therefore the equation becomes. d. 2x d. {f(x)-rf(O)}==-{f(x)-rf(O)} (A2.4).

(14) J. FuJIMuRA. 38. which can be integrated immediately, with the result C f(x)==rf(O)+ vl-. (A2.5). In the limit x-->O, we get r->1 and f(x)-->f(O), then. c ==o.. (A2.6). Thus we get the result f(X)=V{(-O)i2'. (A2.7). Notes and references [1] [2] [3] [4] [5] [6] [7] [8] [9]. Max JAMMER: Concepts of Mass-in Classical and Modern Physics- (Harper and Row, New York, 1961), p. 224. Isaac NEwToN: Philosophiae Naturalis Prinpicia Mathematica (London, 1986). Ernst MAcH: Die Mechanik in ihrer Entwicklung-Historisch-Kritisch Dargestellt(Wissenschaftliche Buchgesellschaft, Darmstadt, 1982), p.239. Ernst MAcH: ibid., p. 240. J.R. ScHUTz: G6ttingen Nachrich. Math.-Phys. Klasse, 110 (1877). In general, other quantities such as 2nd rank tensor Tile should be considered. Studies of them concerning to angular momentum will appear in elsewhere. G.W. LEipNiTz: Acta Eruditorium (Leiptzig, 1686). R. DEscARTEs: Principia Philosophiae (Amsterdam, 1644).. G.N. LEwis and R.C. ToLMAN: Phil. Mag., 18, 510 (1909)..

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