The universal Lagrangian for one particle in a potential

Transcription

1 The unversal Lagrangan for one partcle n a potental James Evans a) Department of Physcs, Unversty of Puget Soun, Tacoma, Washngton Receve 28 May 2002; accepte 6 November 2002 In a system consstng of a sngle partcle n a potental, the classcal acton ts the number of phase waves that pass through the movng partcle, as the partcle moves from ts ntal to ts fnal poston Thus the Lagrangan can be cast nto the form L p(v g p ), where v g an v p are the group an phase veloctes an p s the momentum 2003 Amercan Assocaton of Physcs Teachers DOI: / I INTRODUCTION Hamlton s prncpal functon now often calle the acton, t S 2L q,v t, 1 plays an mportant role n both classcal an quantum mechancs In classcal mechancs, Hamlton s prncple, t 2Lt 0, 2 leas to the Euler Lagrange equatons of moton In quantum mechancs, S frequently appears as a phase, for example n Wentzel Kramers Brlloun calculatons an, more funamentally, n Feynman s sum over hstores However, t s not so clear just what S represents For example, n a classcal mechancal system, why shoul S be the tme ntegral of the fference between the knetc an potental energes? Moreover, the Lagrangans for fferent systems seem to have nothng n common wth one another The meanng of the acton can be mae clearer f we connect the classcal partcle wth ts unerlyng e Brogle phase waves In all that follows the reaer wll be aske to vsualze the classcal partcle an ts phase waves smultaneously We shall see that, n a sngle-partcle system, S has a smple physcal nterpretaton Apart from a multplcatve constant, S s the number of phase waves that pass through the movng partcle between tmes t 1 an t 2 Therefore, all sngle-partcle Lagrangans o share a common form In Sec II, we shall see how ths theorem can be proven Secton III wll llustrate ts utlty n several applcatons In Sec IV we shall see how to calculate the equatons of moton rectly from a new form of the Lagrangan Fnally, n Sec V, we wll see that ths treatment can be extene to other kns of waves II AN ALTERNATIVE FORM FOR THE SINGLE- PARTICLE ACTION The Lagrangan L(q,v ) s a functon of the coornates q an ther veloctes v q /t By a reversal of the usual Legenre transformaton, L may be calculate from the Hamltonan H: L p v g H p v g H, 3 where v g s the partcle velocty an the nex labels ts components p s the momentum conjugate to q Because the partcle velocty s equal to the group velocty of the e Brogle waves, we may also regar v g as the group velocty: the subscrpt g wll serve as a remner Now, for a partcle wth a well-efne energy, H p /k pv p, where h/2, h s Planck s constant, s the frequency, k s the wave number, an v p s the phase velocty of the e Brogle waves assocate wth the partcle Usng ths substtuton for H, we obtan L p v g pv p If the momentum an partcle velocty are parallel, Eq 5 smplfes to L p g p Thus the acton becomes S t 2pg p t Now, (v g p )t s the relatve splacement of the partcle an ts phase wave Moreover, p h/, where s the partcle s e Brogle wavelength So we see that the acton s equal to h tmes the number of phase waves that pass through the partcle, as the partcle moves between the ntal an the fnal poston The term acton therefore seems rather apt: the acton expresses a sort of nteracton between the partcle an ts phase wave Most treatments of the relaton of the classcal acton to the quantum mechancal phase wave epen on the rather auntng apparatus of the Hamlton Jacob equaton, 1 whch reflects the route followe hstorcally by Schrönger Equaton 6 certanly oes not replace these approaches, but t oes prove another perspectve The acton of Eq 1 s a Lorentz nvarant a fact that usually s establshe by conseratons from relatvty But the Lorentz nvarance of the acton becomes obvous when we see that the acton s mae up of countable enttes or events A certan number of phase waves pass through the partcle as the partcle moves from ts ntal confguaton to ts fnal one The number of these events cannot epen upon the frame of reference of the observer Fnally, let us note that Eq 5, oreq 6 f the potental s sotropc, proves a unversal form of the snglepartcle Lagrangan Am J Phys 71 5, May Amercan Assocaton of Physcs Teachers 457

2 III EXAMPLES A Nonrelatvstc partcle n a scalar potental As our frst example, conser a nonrelatvstc partcle wth mass m, momentum p, an potental energy U(x) that vares wth poston x The Lagrangan s L p2 2m U p p 2m U p The group velocty s v g p/m, an the phase velocty s v p k H p p 2m U p 9 Thus the expresson n parentheses n Eq 8 s equal to (v g p ) an the Lagrangan assumes the form of Eq 6 2 B Relatvstc partcle n a scalar potental As a secon example of the utlty of Eq 6 for the sngle-partcle Lagrangan, let us see how t can be use to euce the form of the relatvstc Lagrangan for a partcle n a scalar potental In many textbooks, ths Lagrangan s often just wrtten own wth the remark that t can be verfe by showng that ts Euler Lagrange equatons are the correct equatons of moton 3 We begn from the expressons for the relatvstc momentum p an energy H: p mv g, 10 H mc 2 U, 11 where (1 2 g /c 2 ) 1/2 an c s the vacuum spee of lght Then, because v p H/p, Eq 6 yels L mc g /c 2 1/2 U 12 Ths ervaton may be the smplest possble way to justfy the form of the relatvstc Lagrangan C Relatvstc partcle n an electromagnetc fel If the vector potental A s ntrouce, the canoncal momentum p nee not be parallel to the group an partcle velocty v g In ths case we must use the Lagrangan n the form of Eq 5 Nevertheless, the acton t contnues to be the number of phase waves that pass through the movng partcle To emonstrate ths, we frst recall that Eq 5 gves the correct Lagrangan Let a partcle of mass m an charge e move n an electromagnetc fel characterze by the vector potental A an scalar potental The canoncal momentum p an energy H are p mv g e c A, 8 13 H mc 2 e 14 If we substtute Eqs 13 an 14 nto Eq 3, we have L mv g e c A v g mc 2 e 15 Fg 1 Phase waves an group velocty n the presence of a vector potental The recton ab of the wave vector nee not conce wth the recton ac of the partcle velocty the usual form of the Lagrangan for ths system There s nothng new here: H an L are relate by the usual prescrpton To complete the emonstraton, we must show that, even f the momentum s not n the recton of the group velocty, the acton S stll represents the number of phase waves that pass through the partcle as the partcle moves between ts ntal an fnal postons At the outset we recall that the phase velocty v p s not a vector 4 However, for the purpose of vsualzaton, v p may be regare as recte along k, that s, n the recton of p So the Lagrangan coul also be wrtten as L p g p kˆ, 17 where kˆ s a unt vector n the recton of k Let enote the angle between p an v g In Fg 1, the x axs represents the recton of the partcle an group velocty v g, whle ab s the recton of k or p 5 The wave fronts of the phase waves are at rght angles to k Let phase crest 0 pass through the partcle at pont a at tme 0 In the short tme nterval t, the phase crest avances a stance v p t an occupes poston b Crest 0 now crosses the x axs that s, the partcle trajectory at pont Thus the noe of ntersecton between the partcle trajectory an the phase wave has move a stance v N t Ths s one reason why v p s not a vector: ts x component s longer than the vector tself From Fg 1 we see that v N p /cos 18 The stance between two successve crests of the phase wave, measure along the partcle trajectory, s /cos Now, n tme t the partcle moves a stance v g t, avancng from a to c The number of phase crests that pass through the partcle between tme 0 an tme t s therefore c /cos N g t /cos p g cos t 19a p h p g cos t pv p p v g t/h 19b mc 2 1 g 2 /c 2 e e c v g A, 16 Apart from the factor of h an the overall mnus sgn nvolve n the efnton of the classcal acton, the rght-han 458 Am J Phys, Vol 71, No 5, May 2003 James Evans 458

3 se of Eq 19b s precsely the nfntesmal verson of Eq 1 So, even wth the vector potental nclue, the acton s equal to h tmes the number of phase waves that pass through the partcle D Relatvstc partcle n a strong gravtatonal fel We have seen that for a partcle movng at relatvstc spees n ether a scalar or a vector potental, the classcal sngle-partcle acton s rgorously gven by the number of phase waves that pass through the movng partcle Ths theorem contnues to hol even n the curve space of general relatvty As an example, conser a statc, sotropc gravtatonal fel represente by the metrc s 2 2 x c 2 t 2 n 2 x x 2, 20 where an n are functons of the spatal coornates x (r,, ) or(x,y,z) Many metrcs of physcal nterest can be put nto ths form, nclung the Schwarzschl metrc The orbts of massve partcles are obtane by requrng that they be geoescs: s 0 21 If we nsert the assume form of the lne element s, we can express the geoesc conton n the form of Hamlton s prncple, Eq 2, where the Lagrangan s L x,v g mc g n 2 /c 2 1/2 22 The factor of the rest mass m has been nclue for mensonal convenence The canoncal momenta are p m n g n 2 /c 2 1/2 v g 23 The Hamltonan s H mc g n 2 /c 2 1/2, 24 or, f expresse n terms of the momenta, H mc 2 2 p 2 /n 2 m 2 c 2 1/2 25 If we square both ses of Eq 25 an substtute H an p k, we obtan m 2 c 4 2 c 2 2 k 2 /n 2 26 By fferentatng both ses of Eq 26 wth respect to k, we obtan the sperson relaton v p v g c 2 /n 2, 27 where v p /k s the phase velocty an v g /k s the group velocty of the e Brogle waves Now we have all the necessary peces n han Usng Eq 23, we may wrte Eq 22 n the form L p v g c2 28 n 2 v g Then, wth the use of Eq 27, L may be put nto the form gven by Eq 6 In ths example, we have begun wth a Lagrangan that s val for relatvstc spees an arbtrarly strong gravtatonal fels Thus, n spte of ts smple form, for the class of metrcs uner conseraton, Eq 6 s an exact general-relatvstc relaton Incentally, Eq 6 also gves nsght nto why the geoescs of lght rays are null: for lght n a gravtatonal fel, the phase an group veloctes are equal, so the acton vanshes IV CLASSICAL EQUATIONS OF MOTION It mght be thought that havng the Lagrangan n the form of Eq 5 or 6 woul be of lttle utlty for calculatng equatons of moton, because we woul nee to re-express the momenta p n terms of the coornates q an veloctes v g However, ths s not the case For we can operate rectly wth the Lagrangan n ths form f we regar the p as coornates, along wth the q The generalze veloctes are then the ṗ an the v g where, of course, v g q ) 7 We shall regar the phase velocty v p as a functon of the q an p But we coul equally well conser v p to be a functon of q an v g ) Then for the sngle-partcle problem we have sx Euler Lagrange equatons: t g, 29 q an t ṗ, p 30 where 1,2,3 Let us conser the q-equaton 29 frst, an apply t to a Cartesan coornate, x If we operate on the Lagrangan of Eq 5 we fn: g p, 31 p p 32 x x Thus, the Euler Lagrange q equaton becomes p t p p 33 x Let us now conser the p equaton, 30 Because the Lagrangan oes not actually epen upon the generalze velocty ṗ, 0 34 ṗ Now, p g pv p 35 p Then, because p/ p p /p, Eq 30 becomes v g p p p p p v p 36 Equatons 33 an 36 are equvalent to Hamlton s canoncal equatons of moton Equaton 33 correspons to Hamlton s ynamcal equaton p /t H/ x, an Eq 36 correspons to Hamlton s knematc equaton ẋ H/ p Ths corresponence may be easly verfe for the partcular example of Sec III A 459 Am J Phys, Vol 71, No 5, May 2003 James Evans 459

4 Fnally, t wll be helpful to express the Lagrangan n an alternatve form If we multply Eq 36 by p an sum over, we obtan p p p p p v g v p p p 2 37 But because p 2 p 2, the rght-se of Eq 37 s the Lagrangan tself Thus L p p p p V GENERALIZATION TO OTHER WAVES 38 Untl now, we have been exclusvely concerne wth partcle mechancs In the applcaton of our theorem, we have smultaneously pcture a classcal partcle an ts unerlyng quantum-mechancal phase waves But the expresson of the acton as the number of phase waves that pass through the movng partcle as t moves from ts ntal to ts fnal poston s broaer than mechancs It apples n the geometrcal optcs lmt to any lnear wave system That s, f t s possble to follow the moton of the center of a wave group, then the equatons of moton of the group center can be euce from a Lagrangan that takes the form of Eq 5 Ths may be emonstrate very smply Conser a general wave sturbance The Cartesan components of the group velocty are v g 39 k Because kv p, we have v g k p k k p k, 40 whch has the form of Eq 36 If we multply by k an sum over, we obtan k v g k p k k p k 41 Snce, by analogy to Eq 38, k k p / k may be nterprete as the Lagrangan, we fn L k v g kv p, 42 n complete conformty wth Eq 5 In operatng on ths Lagrangan to obtan the equatons of moton, one shoul regar v p as a functon of the x an k 8 Agan, L tself s a functon of the x, ẋ ( g ), k, an k, although of course the latter o not actually appear We have euce the form of the Lagrangan from the knematc equaton It s, of course, necessary to show that the Lagrangan also leas to the correct ynamcal equaton The q equaton for the Lagrangan of Eq 42 s k t k v p 43 A smple way to see that Eq 43 s the correct ynamcal equaton s to examne the famlar specal case of lght waves n the geometrcal optcs lmt We take the phase velocty to be an sotropc functon of the coornates alone, so that we may wrte v p c/n, 44 where n(x) s the nex of refracton, assume to be nepenent of t, so that, for a gven frequency component, remans constant along the ray Because k ponts along the ray, we may wrte k k x s, 45 where x s a recte element of the ray, an s s the length of ths element of stance taken along the ray Thus x/s s a unt vector tangent to the ray Fnally, we note that k n/c 46 If we substtute Eqs nto Eq 43, an treat as a constant along the ray, we fn n s x n, 47 s whch s the stanar equaton for the shape of a ray n a meum of varable nex of refracton In ths example, n whch n oes not epen upon the k, the wave group follows the rays, that s, travels n the recton of k Ths may be seen from Eq 40 by puttng p / k 0 In the more general case n whch v p epens upon the k as well as the x, we may frst solve Eq 43 for the shapes of the rays an then apply Eq 40 to etermne the path of the wave group VI CONCLUDING REMARKS For a we range of stuatons, the classcal acton assocate wth a one-partcle system s the number of phase waves that pass through the partcle as the partcle moves from ts ntal to ts fnal poston The applcatons scusse not specfcally aress the queston of whether ths nterpretaton contnues to hol even n tme-epenent potentals But, n fact, t oes Ths we mght surmse from Sec III C, n whch we not have to assume that the vector potental A was constant n tme As long as we are able to entfy the quantum-mechancal phase wave wth the surface of constant acton n classcal Hamlton Jacob theory, the phase spee s rgorously equal to H/p, whether or not the Hamltonan epens explctly upon tme, 9 an ths s the key fact that unerles the theorem Moreover, the theorem hols also for nonsotropc potentals Secton III D proves a convncng proof There we chose to work wth a partcle n a generalrelatvstc gravtatonal fel by castng the metrc nto sotropc form However, the valty of the theorem cannot epen upon ths convenence for the followng reason Events are nepenent of conventons Thus, f a certan number of phase waves pass through the movng partcle as the partcle moves from one space tme poston to another, ths number must be nepenent of whether or not we choose to work n sotropc coornates 10 The Lagrangans of Eqs 8, 12, 16, an 22 lttle resemble one another, yet they o all have somethng n common, the unversal form of Eq 5 The nterpretaton of the sngle-partcle Lagrangan as p(v g p ) can often be useful when one nees to wrte own, or check, a Lagrangan Moreover, t proves a reay way to vsualze the acton Most reaers have spent some tme at the waterfront watchng boat wakes The next tme you get a chance to o ths, 460 Am J Phys, Vol 71, No 5, May 2003 James Evans 460

5 count the crests of the phase waves as they pass through the wave group The number that you get s the acton ACKNOWLEDGMENTS I am grateful to Paul Alsng, Georges Lochak, Dav Grffths, Anrew Rex, an Alan Thornke for comments that helpe me sharpen the argument I woul also lke to acknowlege the vtal nput of Pp McCasln a Electronc mal: jcevans@upseu 1 Goo treatments are foun n Ewn C Kemble, The Funamental Prncples of Quantum Mechancs McGraw Hll, New York, 1937, pp 35 51; Wolfgang Yourgrau an Stanley Manelstam, Varatonal Prncples n Dynamcs an Quantum Theory Dover, New York, 1968, pp 45 64, As wth all treatments of phase waves for the nonrelatvstc Lagrangan, ths s somewhat fortutous, because the true frequency of the e Brogle waves ffers from H/ by the enormous contrbuton mc 2 / ue to the rest mass However, when the rest mass energy s nclue, the result s only an effectve constant ncrement of mc 2 to the potental energy U, wth no consequences for the valty of the theorem 3 For example, Jerry B Maron an Stephen T Thornton, Classcal Dynamcs of Partcles an Systems Harcourt Brace Jovanovch, San Dego, 1988, 3r e, p 539; Herbert Golsten, Classcal Mechancs Ason Wesley, Reang, MA, 1980, 2n e, p 321 In contrast, a very clear an complete ervaton s gven by Wolfgang Rnler, Introucton to Specal Relatvty Clarenon, Oxfor, 1991, pp Max Born an Eml Wolf, Prncples of Optcs Pergamon, Oxfor, 1980, 6th e, p 18 5 The fact that k ponts n the recton of p, even when a vector potental A s present, s, of course, ue to Lous e Brogle See Ones et Mouvements Gauther-Vllars, Pars, 1926, pp For a more etale justfcaton, see J Evans, P M Alsng, S Gorgett, an K K Nan, Matter waves n a gravtatonal fel: An nex of refracton for massve partcles n general relatvty, Am J Phys 69, We are, of course, free to choose the coornates as we please Choosng the p as coornates s one stanar way, among several, of passng from Euler Lagrange to Hamltonan equatons of moton For a etale scusson, nclung the fact that ths proceure has no consequences for the unerlyng varatonal prncple, see Golsten Ref 3, pp Ths s usually a convenent way to regar v p For example, for gravty surface waves on water, the phase velocty epens upon the wave number k an the epth of the water, v p (g/k)tanh(k) 1/2 See for example, Wllam C Elmore an Mark A Heal, Physcs of Waves McGraw Hll, New York, 1969, p For an especally clear an smple proof, see T T Taylor, Mechancs: Classcal an Quantum Pergamon, Oxfor, 1976, pp An explct calculaton, showng that the theorem hols n an arbtrary metrc, has been mae by Paul M Alsng personal communcaton