Foundations of Newtonian Dynamics: An Axiomatic Approach for the Thinking Student 1
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1 Foundatons of Newtonan Dynamcs: An Axomatc Approach for the Thnkng Student 1 C. J. Papachrstou 2 Department of Physcal Scences, Hellenc Naval Academy, Praeus 18539, Greece Abstract. Despte ts apparent smplcty, Newtonan mechancs contans conceptual subtletes that may cause some confuson to the deep-thnkng student. These subtletes concern fundamental ssues such as, e.g., the number of ndependent laws needed to formulate the theory, or, the dstncton between genune physcal laws and dervatve theorems. Ths artcle attempts to clarfy these ssues for the beneft of the student by revstng the foundatons of Newtonan dynamcs and by proposng a rgorous axomatc approach to the subject. Ths theoretcal scheme s bult upon two fundamental postulates, namely, conservaton of momentum and superposton property for nteractons. Newton s laws, as well as all famlar theorems of mechancs, are shown to follow from these basc prncples. 1. Introducton Teachng ntroductory mechancs can be a major challenge, especally n a class of students that are not wllng to take anythng for granted! The problem s that, even some of the most prestgous textbooks on the subject may leave the student wth some degree of confuson, whch manfests tself n questons lke the followng: Is the law of nerta (Newton s frst law) a law of moton (of free bodes) or s t a statement of exstence (of nertal reference frames)? Are the frst two of Newton s laws ndependent of each other? It appears that the frst law s redundant, beng no more than a specal case of the second law! Is the second law a true law or a defnton (of force)? Is the thrd law more fundamental than conservaton of momentum, or s t the other way around? Does the parallelogram rule for composton of forces follow trvally from Newton s laws, or s an adonal, ndependent prncple requred? And, fnally, what s the mnmum number of ndependent laws needed n order to buld a complete theoretcal bass for mechancs? In ths artcle we descrbe an axomatc approach to ntroductory mechancs that s both rgorous and pedagogcal. It purports to clarfy ssues lke the ones mentoned above, at an early stage of the learnng process, thus adng the student to acqure a deep understandng of the basc deas of the theory. It s not the purpose of ths artcle, of course, to present an outlne of a complete course of mechancs! Rather, we wll focus on the most fundamental concepts and prncples, those that are taught at the early chapters of dynamcs (we wll not be concerned wth knematcs, snce ths subject confnes tself to a descrpton of moton rather than nvestgatng the physcal laws governng ths moton). 1 See Note at the end of the artcle. 2 papachrstou@snd.edu.gr
2 2 C. J. Papachrstou The axomatc bass of our approach conssts of two fundamental postulates, presented n Secton 3. The frst postulate (P1) embodes both the exstence of nertal reference frames and the conservaton of momentum, whle the second one (P2) expresses a superposton prncple for nteractons. The law of nerta s deduced from P1. In Sec. 4, the concept of force on a partcle subject to nteractons s defned (as n Newton s second law) and P2 s used to show that a composte nteracton of a partcle wth others s represented by a vector sum of forces. Then, P1 and P2 are used to derve the acton-reacton law. Fnally, a generalzaton to systems of partcles subject to external nteractons s made. For completeness of presentaton, certan dervatve concepts such as angular momentum, work, knetc energy, etc., are dscussed n Sec. 5. To make the artcle self-contaned, proofs of all theorems are ncluded. 2. A crtcal look at Newton s theory There have been several attempts to reexamne Newton s laws even snce Newton s tme. Probably the most mportant revson of Newton s deas and the one on whch modern mechancs teachng s based s that due to Ernst Mach ( ) (for a beautful dscusson of Mach s deas, see the classc artcle by H. A. Smon [1]). Our approach dffers n several aspects from those of Mach and Smon, although all these approaches share common characterstcs n sprt. (For a hstorcal overvew of the varous vewponts regardng the theoretcal bass of classcal mechancs, see, e.g., the frst chapter of [2].) The queston of the ndependence of Newton s laws has troubled many generatons of physcsts. In partcular, stll on ths day some authors assert that the frst law (the law of nerta) s but a specal case of the second law. The argument goes as follows: Accordng to the second law, the acceleraton of a partcle s proportonal to the total force actng on t. Now, n the case of a free partcle the total force on t s zero. Thus, a free partcle must not be acceleratng,.e., ts velocty must be constant. But, ths s precsely what the law of nerta says! Where s the error n ths lne of reasonng? Answer: The error rests n regardng the acceleraton as an absolute quantty ndependent of the observer that measures t. As we well know, ths s not the case. In partcular, the only observer enttled to conclude that a non-acceleratng object s subject to no net force s an nertal observer, one who uses an nertal frame of reference for hs/her measurements. It s precsely the law of nerta that defnes nertal frames and guarantees ther exstence. So, wthout the frst law, the second law becomes ndetermnate, f not altogether wrong, snce t would appear to be vald relatve to any observer regardless of hs/her state of moton. It may be sad that the frst law defnes the terran wthn whch the second law acqures a meanng. Applyng the latter law wthout takng the former one nto account would be lke tryng to play soccer wthout possessng a soccer feld! The completeness of Newton s laws s another ssue. Let us see a sgnfcant example: As s well known, the prncple of conservaton of momentum s a drect consequence of Newton s laws. Ths prncple dctates that the total momentum of a system of partcles s constant n tme, relatve to an nertal frame of reference, when the
3 Foundatons of Newtonan Dynamcs 3 total external force on the system vanshes (n partcular, ths s true for an solated system of partcles,.e., a system subject to no external forces). But, when provng ths prncple we take t for granted that the total force on each partcle s the vector sum of all forces (both nternal and external) actng on t. Ths s not somethng that follows trvally from Newton s laws, however! In fact, t was Danel Bernoull who frst stated ths prncple of superposton after Newton s death. Ths means that classcal Newtonan mechancs s bult upon a total of four rather than just three basc laws. The queston now s: can we somehow compactfy the axomatc bass of Newtonan mechancs n order for t to consst of a smaller number of ndependent prncples? At ths pont t s worth takng a closer look at the prncple of conservaton of momentum mentoned above. In partcular, we note the followng: For an solated system consstng of a sngle partcle, conservaton of momentum reduces to the law of nerta (the momentum, thus also the velocty, of a free partcle s constant relatve to an nertal frame of reference). For an solated system of two partcles, conservaton of momentum takes us back to the acton-reacton law (Newton s thrd law). Thus, startng wth four fundamental laws (the three laws of Newton plus the law of superposton) we derved a new prncple (conservaton of momentum) that yelds, as specal cases, two of the laws we started wth. The dea s then that, by takng ths prncple as our fundamental physcal law, the number of ndependent laws necessary for buldng the theory would be reduced. How about Newton s second law? We take the vew, adopted by several authors ncludng Mach hmself (see, e.g., [1,3-7]) that ths law should be nterpreted as the defnton of force n terms of the rate of change of momentum. We thus end up wth a theory bult upon two fundamental prncples,.e., the conservaton of momentum and the prncple of superposton. In the followng sectons these deas are presented n more detal. 3. The fundamental postulates and ther consequences We begn wth some basc defntons. Defnton 1. A frame of reference (or reference frame) s a system of coordnates (or axes) used by an observer to measure physcal quanttes such as the poston, the velocty, the acceleraton, etc., of any partcle n space. The poston of the observer hm/herself s assumed fxed relatve to hs/her own frame of reference. Defnton 2. An solated system of partcles s a system of partcles subject only to ther mutual nteractons,.e., subject to no external nteractons. Any system of partcles subject to external nteractons that somehow cancel one another n order to make the system s moton dentcal to that of an solated system wll also be consdered solated. In partcular, an solated system consstng of a sngle partcle s called a free partcle. Our frst fundamental postulate of mechancs s stated as follows:
4 4 C. J. Papachrstou Postulate 1. A class of frames of reference (nertal frames) exsts such that, for any solated system of partcles, a vector equaton of the followng form s vald: m v = constant n tme (1) where v s the velocty of the partcle ndexed by ( = 1,2, ) and where m s a constant quantty assocated wth ths partcle, whch quantty s ndependent of the number or the nature of nteractons the partcle s subject to. We call m the mass and p m v = the momentum of the th partcle. Also, we call P = m v = p (2) the total momentum of the system relatve to the consdered reference frame. Postulate 1, then, expresses the prncple of conservaton of momentum: the total momentum of an solated system of partcles, relatve to an nertal reference frame, s constant n tme. (The same s true, n partcular, for a free partcle.) Corollary 1. A free partcle moves wth constant velocty (.e., wth no acceleraton) relatve to an nertal reference frame. Corollary 2. Any two free partcles move wth constant veloctes relatve to each other (ther relatve velocty s constant and ther relatve acceleraton s zero). Corollary 3. The poston of a free partcle may defne the orgn of an nertal frame of reference. We note that Corollares 1 and 2 consttute alternate expressons of the law of nerta (Newton s frst law). By nertal observer we mean an ntellgent free partcle,.e., one that can perform measurements of physcal quanttes such as velocty or acceleraton. By conventon, the observer s assumed to be located at the orgn of hs/her own nertal frame of reference. Corollary 4. Inertal observers move wth constant veloctes (.e., they do not accelerate) relatve to one another. Consder now an solated system of two partcles of masses m1 and m 2. Assume that the partcles are allowed to nteract for some tme nterval t. By conservaton of momentum relatve to an nertal frame of reference, we have: ( p + p ) = 0 p = p m v = m v We note that the changes n the veloctes of the two partcles wthn the (arbtrary) tme nterval t must be n opposte drectons, a fact that s verfed expermentally. Moreover, these changes are ndependent of the partcular nertal frame used to measure the veloctes (although, of course, the veloctes themselves are frame-
5 Foundatons of Newtonan Dynamcs 5 dependent!). Ths latter statement s a consequence of the constancy of the relatve velocty of any two nertal observers (the student s nvted to explan ths n detal). Now, takng magntudes n the above vector equaton, we have: v m 1 2 = = constant (3) v m 2 1 regardless of the knd of nteracton or the tme t (whch also s an expermentally verfed fact). These demonstrate, n practce, the valy of the frst postulate. Equaton (3) allows us to specfy the mass of a partcle numercally, relatve to the mass of some other partcle (whch partcle may arbtrarly be assgned a unt mass), by lettng the two partcles nteract for some tme. As argued above, the result wll be ndependent of the specfc nertal frame used by the observer who makes the measurements. That s, n the classcal theory, mass s a frame-ndependent quantty. So far we have examned the case of solated systems and, n partcular, free partcles. Consder now a partcle subject to nteractons wth the rest of the world. Then, n general (unless these nteractons somehow cancel one another), the partcle s momentum wll not reman constant relatve to an nertal reference frame,.e., wll be a functon of tme. Our second postulate, whch expresses the superposton prncple for nteractons, asserts that external nteractons act on a partcle ndependently of one another and ther effects are supermposed. 1 2 Postulate 2. If a partcle of mass m s subject to nteractons wth partcles m, m,, then, at each nstant t, the rate of change of ths partcle s momentum relatve to an nertal reference frame s equal to where ( d p ) / d p d p = s the rate of change of the partcle s momentum due solely to the nteracton of ths partcle wth the partcle m (.e., the rate of change of p f the partcle m nteracted only wth m ). (4) 4. The concept of force and the Thrd Law We now defne the concept of force, n a manner smlar to Newton s second law: Defnton 3. Consder a partcle of mass m that s subject to nteractons. Let p( t) be the partcle s momentum as a functon of tme, as measured relatve to an nertal reference frame. The vector quantty d p F = (5) s called the total force actng on the partcle at tme t.
6 6 C. J. Papachrstou Takng nto account that, for a sngle partcle, p = m v wth fxed m, we may rewrte Eq. (5) n the equvalent form, dv F = ma = m (6) where a s the partcle s acceleraton at tme t. Gven that both the mass and the acceleraton (prove ths!) are ndependent of the nertal frame used to measure them, we conclude that the total force on a partcle s a frame-ndependent quantty. Corollary 5. Consder a partcle of mass m subject to nteractons wth partcles m1, m2,. Let F be the total force on m at tme t, and let F be the force on m due solely to ts nteracton wth m. Then, by the superposton prncple for nteractons (Postulate 2) as expressed by Eq. (4), we have: F = F Theorem 1. Consder two partcles 1 and 2. Let F 12 be the force on partcle 1 due to ts nteracton wth partcle 2 at tme t, and let F 21 be the force on partcle 2 due to ts nteracton wth partcle 1 at the same nstant. Then, F = F Proof. By the ndependence of nteractons, as expressed by the superposton prncple, the forces F 12 and F 21 are ndependent of the presence or not of other partcles n nteracton wth partcles 1 and 2. Thus, wthout loss of generalty, we may assume that the system of the two partcles s solated. Then, by conservaton of momentum and by usng Eq. (5), d d p1 d p 2 ( p + p ) = 0 = F = F Equaton (8) expresses the acton-reacton law (Newton s thrd law). Theorem 2. The rate of change of the total momentum P( t) of a system of partcles, relatve to an nertal frame of reference, equals the total external force actng on the system at tme t. Proof. Consder a system of partcles of masses m ( = 1, 2, ). Let F be the total external force on m (due to ts nteractons wth partcles not belongng to the sys- m j (by con- tem), and let venton, F j = 0 F j be the nternal force on m due to ts nteracton wth when =j). Then, by Eq. (5) and by takng nto account Eq. (7),. (7) (8)
7 Foundatons of Newtonan Dynamcs 7 d p F. = F + j j By usng Eq. (2) for the total momentum, we have: But, dp d p F. = = F + j = = + = 0, 2 1 F j Fj ( F j Fj ) j j j j where the acton-reacton law (8) has been taken nto account. So, fnally, dp = F = F ext (9) where F ext represents the total external force on the system. 5. Dervatve concepts and theorems Havng presented the most fundamental concepts of mechancs, we now turn to some useful dervatve concepts and related theorems, such as those of angular momentum and ts relaton to torque, work and ts relaton to knetc energy, and conservatve force felds and ther assocaton wth mechancal-energy conservaton. Defnton 4. Let O be the orgn of an nertal reference frame, and let r be the poston vector of a partcle of mass m, relatve to O. The vector quantty L = r p = m r v ( ) (10) (where p = mv s the partcle s momentum n the consdered frame) s called the angular momentum of the partcle relatve to O. Theorem 3. The rate of change of the angular momentum of a partcle, relatve to O, s gven by dl r F T = where F s the total force on the partcle at tme t and where T s the torque of ths force relatve to O, at ths nstant. Proof. Equaton (11) s easly proven by dfferentatng Eq. (10) wth respect to tme and by usng Eq. (5). (11)
8 8 C. J. Papachrstou Corollary 6. If the torque of the total force on a partcle, relatve to some pont O, vanshes, then the angular momentum of the partcle relatve to O s constant n tme (prncple of conservaton of angular momentum). Under approprate conons, the above conservaton prncple can be extended to the more general case of a system of partcles (see, e.g., [2-8]). Defnton 5. Consder a partcle of mass m n a force feld F( r ), where r s the partcle s poston vector relatve to the orgn O of an nertal reference frame. Let C be a curve representng the trajectory of the partcle from pont A to pont B n ths feld. Then, the lne ntegral B W = F( r ) d r AB A (12) represents the work done by the force feld on m along the path C. (Note: Ths defnton s vald ndependently of whether or not adonal forces, not related to the feld, are actng on the partcle;.e., regardless of whether or not F( r ) represents the total force on m.) Theorem 4. Let F( r ) represent the total force on a partcle of mass m n a force feld. Then, the work done on the partcle along a path C from A to B s equal to B W = F( r ) d r = E E = E AB A k, B k, A k (13) where E k p = m v = (14) 2 2m s the knetc energy of the partcle. Proof. By usng Eq. (6), we have: dv F d r = m d r = mv dv = m d( v v) = m d( v ) = mvdv, 2 2 from whch Eq. (13) follows mmedately. Defnton 6. A force feld F( r ) s sad to be conservatve f a scalar functon Ep ( r ) (potental energy) exsts, such that the work on a partcle along any path from A to B can be wrtten as B W = F( r ) d r = E E = E AB A p, A p, B p (15)
9 Foundatons of Newtonan Dynamcs 9 Theorem 5. If the total force F( r ) actng on a partcle m s conservatve, wth an assocated potental energy E ( r ), then the quantty p 1 2 E = Ek + Ep = m v + Ep( r ) 2 (16) (total mechancal energy of the partcle) remans constant along any path traced by the partcle (conservaton of mechancal energy). Proof. By combnng Eq. (13) (whch s generally vald for any knd of force) wth Eq. (15) (whch s vald for conservatve force felds) we fnd: E = E ( E + E ) = 0 E + E = const. k p k p k p Theorems 4 and 5 are readly extended to the case of a system of partcles (see, e.g., [2-8]). 6. Some conceptual problems After establshng our axomatc bass and demonstratng that the standard Newtonan laws are consstent wth t, the development of the rest of mechancs follows famlar paths. Thus, as we saw n the prevous secton, we can defne concepts such as angular momentum, work, knetc and total mechancal energes, etc., and we can state dervatve theorems such as conservaton of angular momentum, conservaton of mechancal energy, etc. Also, rgd bodes and contnuous meda can be treated n the usual way [2-8] as systems contanng an arbtrarly large number of partcles. Despte the more economcal axomatc bass of Newtonan mechancs suggested here, however, certan problems nherent n the classcal theory reman. Let us pont out a few: 1. The problem of nertal frames An nertal frame of reference s only a theoretcal abstracton: such a frame cannot exst n realty. As follows from the dscusson n Sec. 3, the orgn (say, O) of an nertal frame concdes wth the poston of a hypothetcal free partcle and, moreover, any real free partcle moves wth constant velocty relatve to O. However, no such thng as an absolutely free partcle may exst n the world. In the frst place, every materal partcle s subject to the nfntely long-range gravtatonal nteracton wth the rest of the world. Furthermore, n order for a supposedly nertal observer to measure the velocty of a free partcle and verfy that ths partcle s not acceleratng relatve to hm/her, the observer must somehow nteract wth the partcle. Thus, no matter how weak ths nteracton may be, the partcle cannot be consdered free n the course of the observaton. 2. The problem of smultanety In Sec. 4 we used our two postulates, together wth the defnton of force, to derve the acton-reacton law. Implct n our arguments was the requrement that acton
10 10 C. J. Papachrstou must be smultaneous wth reacton. As s well known, ths hypothess, whch suggests nstantaneous acton at a dstance, gnores the fnte speed of propagaton of the feld assocated wth the nteracton and volates causalty. 3. A dmensonless observer As we have used ths concept, an observer s an ntellgent free partcle capable of makng measurements of physcal quanttes such as velocty or acceleraton. Such an observer may use any convenent (preferably rectangular) set of axes (x, y, z) for hs/her measurements. Dfferent systems of axes used by ths observer have dfferent orentatons n space. By conventon, the observer s located at the orgn O of the chosen system of axes. As we know, nertal observers do not accelerate relatve to one another. Thus, the relatve velocty of the orgns (say, O and O ) of two dfferent nertal frames of reference s constant n tme. But, what f the axes of these frames are n relatve rotaton (although the orgns O and O move unformly relatve to each other, or even concde)? How can we tell whch observer (f any) s an nertal one? The answer s that, relatve to the system of axes of an nertal frame, a free partcle does not accelerate. In partcular, relatve to a rotatng frame, a free partcle wll appear to possess at least a centrpetal acceleraton. Such a frame, therefore, cannot be nertal. As mentoned prevously, an object wth fnte dmensons (e.g., a rgd body) can be treated as an arbtrarly large system of partcles. No adonal postulates are thus needed n order to study the dynamcs of such an object. Ths allows us to regard momentum and ts conservaton as more fundamental than angular momentum and ts conservaton, respectvely. In ths regard, our approach dffers sgnfcantly from, e.g., that of Smon [1] who, n hs own treatment, places the aforementoned two conservaton laws on an equal footng from the outset. 7. Summary Newtonan mechancs s the frst subject n Physcs an undergraduate student s exposed to. It contnues to be mportant even at the ntermedate and advanced levels, despte the predomnant role played there by the more general formulatons of Lagrangan and Hamltonan dynamcs. It s ths author s experence as a teacher that, despte ts apparent smplcty, Newtonan mechancs contans certan conceptual subtletes that may leave the deepthnkng student wth some degree of confuson. The average student, of course, s happy wth the dea that the whole theory s bult upon three rather smple laws attrbuted to Newton s genus. In the mnd of the more demandng student, however, puzzlng questons often arse, such as, e.g., how many ndependent laws we really need to fully formulate the theory, or, whch ones should be regarded as truly fundamental laws of Nature, as opposed to others that can be derved as theorems. Ths artcle suggested an axomatc approach to ntroductory mechancs, based on two fundamental, emprcally verfable laws; namely, the prncple of conservaton of momentum and the prncple of superposton for nteractons. We showed that all standard deas of mechancs (ncludng, of course, Newton s laws) naturally follow from these basc prncples. To make our formulaton as economcal as possble, we expressed the frst prncple n terms of a system of partcles and treated the sngle-
11 Foundatons of Newtonan Dynamcs 11 partcle stuaton as a specal case. To make the artcle self-contaned for the beneft of the student, explct proofs of all theorems were gven. By no means do we assert, of course, that ths partcular approach s unque or pedagogcally superor to other establshed methods that adopt dfferent vewponts regardng the axomatc bass of classcal mechancs. Moreover, as noted n Sec. 6, ths approach s not devod of the usual theoretcal problems nherent n Newtonan mechancs (see also [9,10]). In any case, t looks lke classcal mechancs remans a subject open to dscusson and re-nterpretaton, and more can always be sad about thngs that are usually taken for granted by most students (ths s not exclusvely ther fault, of course!). Happly, some of my own students do not fall nto ths category. I apprecate the hard tme they enjoy gvng me n class! References 1. H. A. Smon, The Axoms of Newtonan Mechancs, The Phlosophcal Magazne, Ser. 7, 38 (1947) N. C. Rana, P. S. Joag, Classcal Mechancs (Tata McGraw-Hll, 1991). 3. K. R. Symon, Mechancs, 3rd eon (Addson-Wesley, 1971). 4. J. B. Maron, S. T. Thornton, Classcal Dynamcs of Partcles and Systems, 4th eon (Saunders College, 1995). 5. M. Alonso, E. J. Fnn, Fundamental Unversty Physcs, Volume I: Mechancs (Addson-Wesley, 1967). 6. V. I. Arnold, Mathematcal Methods of Classcal Mechancs, 2nd ed. (Sprnger-Verlag, 1989). 7. C. J. Papachrstou, Introducton to Mechancs of Partcles and Systems (Hellenc Naval Academy Publcatons, 2017) H. Goldsten, Classcal Mechancs, 2nd eon (Addson-Wesley, 1980). 9. C.-E. Khar, Newton s Laws of Moton Revsted: Some Epstemologcal and Ddactc Problems, Lat. Am. J. Phys. Educ. 5 (2011) A. E. Chubykalo, A. Espnoza, B. P. Kosyakov, The Inertal Property of Approxmately Inertal Frames of Reference, Eur. J. Phys. 32 (2011) Note: Ths artcle s a revsed verson of the followng publshed artcle: C. J. Papachrstou, Foundatons of Newtonan Dynamcs: An Axomatc Approach for the Thnkng Student, Nausvos Chora Vol. 4 (2012) ; 3 Avalable on the Internet; please search ttle.
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