Principle of Least Action

Transcription

1 The Based on par of Chaper 19, Volume II of The Feynman Lecures on Physics Addison-Wesley, 1964: pages 19-1 hru 19-3 & 19-8 hru Edwin F. Taylor July. The Acion Sofware The se of exercises on Acion will combine hand calculaions wih use of a compuer program called ACTION ha allows more rapid and graphical analysis. An Appendix o his uni has a se of inroducory exercises for he Acion program (for he case of zero poenial -- no graviaional field) o help you ge used o is various feaures. You will probably find i helpful o go hrough hese appendix exercises (wih he prefix Z) before beginning he exercises in he body of his uni (wih he prefix Q). I. Inroducion Almos all of he nonrelaivisic classical mechanics of single paricle moion can be derived from he. In paricular, he Principle of Leas Acion predics paricle moion in he presence of a poenial, such as ha due o a graviaional field. And he lends iself o he ransiion from nonrelaivisic classical mechanics o nonrelaivisic quanum mechanics. In his exercise we apply he classical o a much simplified analysis of he moion of a one-kilogram sone moving verically upward in a graviaional field. x (x, ) x =x 1 (x 1, 1 ) (x, ) 1 Figure 1. Two possible alernaive worldlines beween he evens (,) and (x, ) ha occur a differen heighs x in a graviaional field. 31

2 Figure 1 shows a diagram of wo alernaive pahs of a sone in a graviaional field ha acs verically downward. There are several imporan hings o say abou hese moions: 1. The moion in space is one-dimensional; he sone moves verically upward. Feynman uses x as he verical space coordinae. Mos of us would ordinarily use he variable y for his posiion.. The diagrams in Figure 1 are spaceime diagrams, wih ime ploed along he horizonal axis. The pah of a sone ploed on a spaceime diagram is called a worldline. A worldline is NOT a pah in space, bu raher a plo of locaion as a funcion of ime. The worldline ells much more abou moion han a picure showing he rajecory in space alone. In paricular, he slope of he worldline ells us he velociy of he sone a every poin in is moion. On each sraigh segmen of he worldlines of Figure 1, he sone is moving wih consan velociy and herefore wih consan kineic energy. 3. A poin on a spaceime diagram gives us no only he locaion of he sone in space bu also he ime a which i reaches ha locaion. Such a spaceime poin is called an even. 4. Descripions of he wo moions represened in Figure 1 are as follows: Case A: The sone sars a he origin and rises wih consan velociy o heigh x. Case B: The sone sars a he origin and rises wih consan velociy greaer han ha of Case A, arriving a heigh x 1 = x sooner, where i sops a ha heigh for an addiional ime. 5. Neiher of hese moions is he naural moion of a sone as i flies in a graviaional field. Bu i is no hard o calculae he Acion for hese unnaural pahs, so we use hem o inform ourselves. 6. Typically, energy is no conserved along candidae worldlines shown in Figure 1. In paricular, a sone rising verically wih consan velociy has a consan kineic energy bu an increasing poenial energy. Therefore is oal energy is increasing during ha porion of he worldline. Only for he correc, naural pah followed by a rising sone is energy conserved. The Acion program will help us find he correc "naural" pah ha exhibis conservaion of energy. 3

3 II. The Acion for Case A We find an expression for he Acion in Case A. x (x, ) Figure : Direc worldline saring a origin. The formal definiion of he acion is given on page 19-3 of he Feynman reading: Acion = S = KE PE d = KEd PEd (1) where 1 and are he imes of he iniial and final evens. For Case A, ime 1 = : Acion = S = KE PE d = KEd PEd () These inegrals, like all inegrals, are an indicaor ha somehing is being summed. For he firs inegral on he righ side of equaion (), he sum adds up a lo of small conribuions KE d. Muliply he kineic energy KE during each shor inerval of ime by ha inerval of ime d and add up all of hese lile producs. The resul is an approximaion of he inegral: KE d KE d + KE d + KE d +... (3) a a b b c c Case A shows a sraigh segmen of worldline wih a consan slope, which means a consan velociy along ha segmen. This means also a consan kineic energy along his sraigh segmen. So along his segmen of worldline he consan kineic energy, call i KE, can be facored ou of he bracke on he righ side of equaion (). Wha is lef is jus he sum of all he lile ime elemens beween = and =. This sum is jus equal o : 33

4 KE d KE d + d + d +... KE (sraigh worldline) (4) = a b c This gives an exac expression for he firs inegral on he righ side of equaion () for he case of a sraigh worldline. The poenial energy conribuion o he Acion, he second inegral on he righ side of equaion (), is almos as simple, bu no quie. The poenial energy has he form: PE = mgx (5) where m is he mass of he sone, g is he acceleraion of graviy, and x is he heigh of he sone above he ground. The PE inegral in equaion () corresponds o he summaion: PE d mg x d + x d + x d +... (6) a a b b c c No one can sop us from making all incremens of ime d equal: PE d mg x d + x d + x d +... mg x x x... d (7) = ( ) a b c a b c Now suppose he ime inerval is divided ino N equal segmens of size d. Then we can wrie d = N (8) Subsiue his ino equaion (7) and perform a sleigh of hand by placing N under he sum of x-erms: PE d mg x a + x b + x c +... N (9) The expression in he square brackes has a simple inerpreaion. The numeraor is he sum of he values of x a N equally-spaced locaions along he worldline segmen. Dividing by N gives a resul ha approximaes he average value of x over his segmen. Bu his segmen is a sraigh line, so he average value of x is jus is value a he midpoin of he segmen: xa + xb + xc +... x N = (1) 34

5 Wih his subsiuion, equaion (9) becomes: PE d mg x = PEavg (11) where PE avg is he average of he poenial energy over he worldline of Case A. Again, equaion (11) is exac for a sraigh worldline. So for his sraigh segmen of worldline from (,) o (x, ), he Acion has he value: Acion = S= ( KE PE) d = KE PEavg (Case A) (1) Or, expressed in erms of he endpoins of he worldline: 1 Acion = S = m x mg x (Case A) (13) Here are he firs exercises: Q1. Using a hand calculaor, find he numerical value of S (for a mass m = 1 kg) for Case A, given he specific values: x = 5 meers = 7.5 seconds (14) Q. Check your answer o Q1 by seing up he same sraigh worldline using he ACTION sofware. (You may wan o do he inroducory exercises in he Appendix before rying his quesion.) NOTE #1: You mus se up he GRAVITY poenial using he POTENTIAL menu a he op of he screen ha appears when you press he NEW CASE buon. NOTE #: Pay aenion o he scales on he horizonal and verical axis, which are labeled in a funny way. NOTE #3: To help you correcly place even-dos, ry ou various seings in he GRID menu. NOTE #4: Under he ENDPOINTS menu choose Graviy Case o ener he above endpoins direcly. Or choose Digial Enry o do he same digially. NOTE #5: The values of Acion derived as answers o he Q-quesions in his secion are NEGATIVE numbers. The Acion can be a negaive number, 35

6 and he LEAST acion can mean he MOST NEGATIVE number. (Go figure!) Example: S = 134 has a lower value han S = 9. NOTE #6: Do no worry if he ACTION sofware does no yield a number exacly equal o your calculaed value. The ACTION sofware is accurae o only a few percen. III. Generalizaion: Acion for a Sraigh Segmen of Worldline Now we are going o make a grea simplificaion by approximaing every worldline as a series of sraigh segmens. This makes i easy for a compuer o calculae he Acion for each sraigh segmen, hen add up he Acion for all segmens o yield he oal Acion. One such segmen is shown in Figure 3: x x 1 x (x 1, 1 ) (x, ) 1 Figure 3. General sraigh segmen of a worldline The derivaion of he Acion for his segmen is a simple exension of Case A. The kineic energy depends only on he slope of he sraigh segmen of worldline, so where he segmen is on he x diagram does no maer as far as he KE conribuion o he Acion is concerned. The resul for Case A can simply be applied o he general case. The only difference is ha he ime lapse over he segmen becomes ( 1 ), as shown in equaion (15) below. As before, he poenial energy erm is almos as simple bu no quie. We can choose he zero of classical poenial energy anywhere. In he case of a uniform graviaional field, we can choose he zero of poenial energy o be a he ground, a he second floor, or a he op of a able on he second floor. Differen choices simply add or subrac a consan poenial, wihou changing any predicions abou he resuling moion. Changing he origin of x from Figure o Figure 3 increases he poenial a all poins of he segmen of he worldline. Bu he average PE is sill calculaed using he average value of he heigh, which his ime is given by he expression: (x 1 + x )/. These simple ranslaions change equaions (1) and (13) o equaions (15) and (16): 36

7 ( avg ) Acion = S = KE PE d = KE PE 1 1 (sraigh segmen) (15) or ( + ) Acion = S = m x x 1 1 ( ) mg x x 1 1 ( 1 ) (sraigh segmen) (16) 1 The form of equaion (16) makes i fairly clear where each erm comes from. However, for compuaional purposes, you may wan o simplify his algebraically a bi: 1 Acion = S = m x x 1 1 gx ( + x1) ( 1 ) (sraigh segmen) (17) The sofware program ACTION uses equaion (17) wih m = 1 kilogram o compue he acion along each segmen of he worldline ha you consruc beween an iniial even and a final even. IV. CASE B: More Time a Aliude Now i is your urn. Case A is a very unrealisic. Take he nex sep oward realiy. Find an expression for he oal Acion along he slighly more realisic wo-segmen pah shown for Case B in Figure 1, repeaed as Figure 4. x =x 1 (x 1, 1 ) (x, ) 1 Figure 4. Two-segmen worldline joining same iniial and final evens as Case A in Figure 3. On he way o your soluion, you may answer he following quesions in sequence or do i your own way and wrie up your resuls. Q3. Adap he resul for Case A o he shorer ime 1 hus finding he value of he Acion for he firs segmen, he upward par of he 37

8 worldline in Case B. Wha is his algebraic expression for he Acion along his firs, rising segmen? Do NOT urn in his complicaed expression. Insead, use a hand calculaor o find he value along his firs segmen using he values given in equaion (14) plus he value: 1 = 5 seconds (18) Turn in your numerical resul. Q4. Check your numerical resul of quesion Q3 using he ACTION program o find he value of S along he firs, rising segmen for Case B. (In his case here is only a lef-hand poin a (,) and a righ-hand poin a...?) Repor your numerical value from he program. Commen: You will find a large variaion in he resul, depending on exac placemen of he dos a each end. This is because kineic energy and poenial energy are nearly equal along his segmen, so (KE PE) varies widely as KE varies a lile bi. You can minimize his variaion by choosing Digial Enry from he Endpoins menu. Q5. From he general resul for a sraigh segmen, equaion (17), use a hand calculaor o find a numerical value for he Acion along he second, horizonal segmen of worldline in Figure 4. Q6. Check your numerical resul of quesion Q5 using he ACTION program o find he value of S along he second, horizonal segmen for Case B. (In his case here is only a lef-hand poin a (x 1, 1 ) and a righ-hand poin a...?) Repor your numerical value from he program. Q7. Now sar again, and use he ACTION program o draw he complee wo-segmen worldline shown in Figure 4 for he poins described in equaions (14) and (18). (Hin: Choose he endpoins firs.) Read off he TOTAL acion for his worldline and compare his oal wih he numbers derived for he individual segmens in quesions Q3 hrough Q6. Q8. Now click on CHANGE TO MOVE DOTS and move he middle even up and down o deermine he minimum value of he Acion for his wo-segmen worldline. Repor he value of his minimum Acion and compare i o he value of he Acion for he one-segmen worldline analyzed in Secion II. 38

9 V. An Even More General Case This is geing edious. Compuers were made o do edious work. The sofware program ACTION helps you o compue he Acion beween wo evens along a worldline wih as many segmens as you wan. Secion III above derives he algebraic expression ha he compuer uses for each segmen of he worldline. Q9. Se up he Acion program wih he GRAVITY poenial and iniial and final poins he same as for Cases A and B [iniial poin a (,), final poin a 5 meers and 7.5 seconds]. Add TWO inermediae poins equally spaced in ime a =.5 seconds and 5 seconds. Click on CHANGE TO MOVE DOTS, hen drag hese inermediae poins UP and DOWN (no lef o righ) o find he minimum value of he Acion. Repor his numerical value. Q1. Now click on CHANGE TO ADD DOTS and click on a bunch more inermediae dos along his worldline. Now click on CHANGE TO MOVE DOTS and use he HUNT MINIMUM menu a he op of he screen o Hun Verical for he worldline of minimum Acion. Repor he value of his minimum Acion. Q11. Wih many dos in place in Q1, is energy (approximaely) conserved along his worldline? Drag one of he dos a long way up or down. Wha does his do o he value of he oal energy for he segmens on eiher side of he moved do? Commen: Equal values of oal energy along each segmen is wha we mean by he conservaion of energy. You have demonsraed ha conservaion of energy is a naural consequence of he Principle of Leas Acion. AN ASIDE: The is very powerful. I ells us how a paricle moves in a poenial and leads o he conservaion of energy. In conras, he conservaion of energy does no ell us how a paricle moves. A candidae moion of a paricle may saisfy he conservaion of energy and sill be incorrec. Consider he following. The picher hrows a ball oward he baer. How does he ball move? One (incorrec) possibiliy is ha he ball says a he same heigh above he ground and moves wih consan speed along a sraigh line from picher o baer. For such a candidae moion, kineic energy KE is consan and poenial energy PE is consan; herefore oal energy E = KE + PE is consan energy is conserved. Bu his is NOT he way a piched ball moves. The acual moion is prediced by varying he worldline o find he worldline of minimum Acion. For his worldline, oo, energy is conserved. Bu he argumen does no work he oher way: conservaion of energy does no necessarily predic he correc moion. Leas Acion does. 39

10 APPENDIX: INTRODUCTION TO THE ACTION PROGRAM The following exercises are inended o inroduce you o he ACTION sofware. For hese exercises here is no graviaional field. The paricle moves in an inerial frame (as in a space laboraory orbiing Earh). NOTE #1: Mos of he menus described below appear a he op of he screen only when you press he NEW CASE buon. NOTE #: Use he MACRO defaul under he DEFAULT. This is he one for classical Acion. NOTE #3: The defaul direcion for he ime axis is horizonal, he same as in Feynman's lecure on he subjec. If you need o change his, use he TIME AXIS menu. Use he ACTION sofware o read off he answers o he following quesions for a one-kilogram mass. Place he endpoins a (space =, ime = ) and (space = 1 meers, ime = 1 seconds). NOTE: The space and ime axes are labeled in a funny way. Z1. Wha is he value of he Acion S along he sraigh worldline beween hese wo evens? (See display a he boom of he screen.) Z. Wha is he KE of he paricle along his worldline? (See display a he righ.) Z3. Wha is he value of he Acion S along a wo-segmen worldline ha begins a poin (space =, ime = ) moves o poin (space = 75 meers, ime = 5 seconds) and ends a poin (space = 1 meers, ime = 1 seconds)? HINT: Place he endpoins firs. Z4. Is energy (approximaely) conserved along his wo-segmen worldline? Now click on he buon CHANGE TO MOVE DOTS and drag he inermediae poin up and down o find he minimum Acion S min (boom of he screen). Click on buons o allow you o drag lef-righ as well as up-down. NOTE: Wrien a he boom of he screen is boh he value of he Acion S and also he minimum value S min recorded for pas posiions of he inermediae poin or poins. This will help you find he minimum value as you drag he poin(s) in various direcions. Z5. Wha is he value of his minimum Acion S min? Z6. Is energy (approximaely) conserved along his wo-segmen worldline of minimum Acion? 4

11 Click on CHANGE TO ADD DOTS and click on a bunch more inermediae dos along his worldline. Now click on CHANGE TO MOVE DOTS and use he HUNT MINIMUM menu a he op of he screen o search auomaically for he worldline of minimum Acion. Z7. Wha is he value of he minimum Acion for his many-segmen line? Z8. Is energy (approximaely) conserved along his worldline of minimum Acion? RECESS! Play wih he Menus a he op of he screen, he ones ha appear when you sar a NEW CASE. In paricular, learn how o change he poenial and how o place he endpoins digially. (Noe ha digial placemen of endpoins may include he exremes, such as and 1 -- and ha he program will accep second-decimal-place accuracy in endpoins, such as 1.34.) Copyrigh Edwin F. Taylor 41

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