Figure 1: Double pendulum system

Transcription

1 Lecture 1 Introuction MATH-GA Mechnics The purpose of this lecture is to motivte vritionl formultions of clssicl mechnics n to provie brief reminer of the essentil ies n results of the clculus of vrition tht we will use in the next few lectures. 1 Clssicl Mechnics of Discrete Systems 1.1 Description of the physicl system Clssicl mechnics of iscrete systems The subject of the next few lectures will be the mthemticl formultion n the stuy of the physicl behvior of iscrete systems s escribe by the brnch of physics known s clssicl mechnics. Discrete mens tht the system is me of finite number of elementry prts, n this number will often be quite smll. Discrete is in opposition here to brnch of clssicl mechnics known s continuum mechnics, which is often pplie to the stuy of solis n itself is very rich n interesting fiel of pplie mthemtics. Clssicl mens tht we will not consier quntum effects Configurtion spce The elementry prts forming the iscrete systems re often clle point msses. A point mss is point prticle, i.e. single point in spce, with mss but no internl structure. We will then view extene boies s compose of lrge number of these elementry prts with specific sptil reltionships mong them. This is obviously n ieliztion, which will nevertheless be stisfying for ll the situtions we will cover in this course. Consier for exmple the ouble penulum system shown in Figure 1. If the msses of ech bob re much lrger thn the msses of the rigi ros holing them, we cn just view the system s me of two point msses, with msses m 1 n m, loclize t the center of grvity of ech bob, whose motion in spce n with respect to one nother is constrine by the fct tht the bobs remin ttche to the ros t ll times. Figure 1: Double penulum system Specifying the position of ll the constituent prticles of system specifies the configurtion of the system. The existence of constrints mong prts of the system mens tht the constituent prticles cnnot ssume ll possible positions. The set of ll configurtions of the system tht cn be ssume is clle the configurtion spce of the system. The imension of the configurtion spce is the smllest number of prmeters tht hve to be given to completely specify configurtion. This number is commonly clle the number of egrees of freeom of the system. Consier the motion of point mss in free spce subject to some force F. Three quntities re require to specify its position t ny given moment in time. The number of egrees of freeom of tht system is 3. Consier the spring mss system shown in Figure. The system is usully consiere in the ielize frmework in which the motion of the bll with mss m is constrine to only be in the verticl irection. In other wors, it is unerstoo tht the initil conitions n the rigiity of the spring re such tht the motion is long the z xis. In tht cse, the system hs one egree of freeom. 1

2 Figure : Spring mss system The prmeters tht re use to escribe the configurtion of system re clle the generlize coorintes. For complete escription of system, one nees t lest s mny generlize coorintes s there re egrees of freeom in the system. Depening on how one chooses to escribe the system, one my however use more generlize coorintes thn there re egrees of freeom. If we go bck to the ouble penulum exmple shown in Figure 1, one my t first think tht convenient wy to stuy the system is to tke the Crtesin coorintes of ech of the two msses s generlize coorintes, n to keep in min, while setting up the equtions escribing the system, tht there re constrints on the system tht limit the possible configurtions, since the msses m 1 n m re not llowe to move inepenently from ech other. With such n pproch, the number of generlize coorintes woul be 4, n there woul be two constrints, nmely the fct tht the lengths l 1 n l of the rigi ros re fixe. After further thought, however, it ppers tht since the system hs two egrees of freeom, it is more convenient to tke the ngles θ 1 n θ s generlize coorintes. Sets of coorintes with the sme imension s the configurtion spce re generlly esier to work with becuse we o not hve to el with explicit constrints mong the coorintes. In this clss, we will lern how to el with both situtions in n effective mnner. Now tht we hve seen how to escribe system t given moment in time, we re rey to lern how to etermine the motion of the system s function of time. In clssicl mechnics, there re t lest two equivlent wys of specifying the physiclly relevnt pth of the system in configurtion spce: either through ifferentil equtions known s Newton s lws, or through integrl equtions, ssocite with vritionl formultion of clssicl mechnics often referre to s Lgrngin mechnics. Both pproches hve seprte merits, tht we will iscuss in etil. Since you re more likely to hve been expose to Newtonin mechnics t this stge of your cemic creer, we will strt with review of the Newtonin pproch. 1. Newtonin pproch to clssicl mechnics 1..1 Newton s lw Consier N point msses m i,i=1...n whose positions t ny moment in time re given by the position vectors r i (t) = (x i (t), y i (t), z i (t)) in Crtesin coorintes, n re subject to the forces F i tht my epen on time, the positions of the point msses r i (t) n their velocities ṙ i (t). Newton s lw sys tht the motion of the point msses s function of time is given by the vector eqution m i r i (t) = F i (t, r 1 (t),..., r N (t), ṙ 1 (t),..., ṙ N (t)) (1) r i cn be recognize s the ccelertion of the point mss i, so Newton s lw is often memorize s F=m. Eqution (1) is secon orer orinry ifferentil eqution, so when viewe s n initil vlue problem strting t time t 0, both r(t 0 ) n ṙ(t 0 ) re require to fully specify the motion t lter times. The problem is mthemticlly well-pose when specifie s follows: m i r i (t) = F i (t, r 1 (t),..., r N (t), ṙ 1 (t)... ṙ N (t)) for t t 0, i = 1... N r i (t 0 ) = r i0, i = 1... N () ṙ i (t 0 ) = v i0, i = 1... N Note the very remrkble fct lthough by now quite intuitive for most humn beings tht if the functionl form of the forces is known for ll times, then the motion of the point msses cn be compute for

3 ll times t t 0. In fct, it cn lso be reconstructe for times t t 0. The system is entirely eterministic. Tht oes not lwys men tht the motion cn be compute numericlly with stisfying ccurcy for ll times. This is becuse the ynmics cn be chotic, n very smll chnges in the initil conitions cn le to significnt ifferences t lter times. This is the cse for the motion of objects (plnets, stellites, sterois) in the solr system, or for the ynmics of riven penulum, sitution we will stuy in etil lter in the course. Let us now look t few stnr pplictions of Newton s lw, to refresh our memory n gin fmilirity with systems we will regulrly go bck to in the course of the semester. 1.. Exmples Object in free fll uner the effect of grvity: Newton s pple The pple flling from the tree onto Newton s he cn be ielize s point mss with mss m n if the initil conitions re such tht the pple oes not hve ny horizontl velocity initilly, the system hs one egree of freeom, escribe by the verticl coorinte z. The pple is subject to the force of grvity, which physicists hve shown to be equl to F g = mge z if the z-xis is pointing upwr, with g 9.8m.s the grvity constnt. Newton s eqution for the z coorinte is m z = mg z = g (3) This is the well-known result tht in free fll, the mgnitue of the ccelertion is equl to the grvity constnt. If we re given initil conitions z(0) = z 0 n ż(0) = v 0, Eqution (3) is esily integrte, to give the motion for ny lter time t z(t) = z 0 + v 0 t g t Conservtion of energy Multiplying Eqution (3) by ż, we hve ż z + gż = 0 (ż ) t + gz = 0 (4) We conclue tht the quntity H(z, ż) = ż + gz (5) is conserve uring the motion of the pple. The function H(z, ż) is clle the Hmiltonin, n represents the energy of the system. We will go bck to Hmiltonins very soon in this course. For the moment, it suffices to sy tht the first term in the Hmiltonin ż is clle the kinetic energy, n gz is the potentil energy. Physicists usully efine m ż n mgz s the kinetic n potentil energy respectively. Phse spce As we hve seen previously, the position n velocity of point mss re sufficient to unequivoclly specify its motion t lter times, provie we know the force cting on it. In clssicl mechnics, the pir position-velocity is clle the stte of the system. A very efficient wy to vizulize the ynmics of system is to look t the physiclly llowe trjectories of the stte of the system in the spce of ll imginble sttes. This spce is usully clle phse spce for historicl resons, n the plot of the llowble trjectories is often clle phse portrit. Since for Newton s pple ż + gz is constnt quntity equl to the totl energy of the pple, the phse portrit for Newton s pple is obtine by plotting the contours in the (z, ż) phse spce of the Hmiltonin H(z, ż) for severl vlues of the totl energy. It is cler from the form of the Hmiltonin tht these contours re prbols, s shown in Figure 3. 3

4 z/t z Figure 3: Phse portrit for Newton s pple (g = 9.8 ws use for this figure) The simple penulum The simple penulum is stnr system in clssicl mechnics, me of bob of mss m ttche to rigi ro of length l n negligible mss s compre to the mss of the bob, which is ttche to frictionless pivot (see Figure 4). One coul use the x n y coorintes of the position of the bob to escribe the ynmics, but it is not the most intuitive strtegy. The reson is tht the length l of the ro is fixe, so we hve the constrint x + y = l In other wors, the system hs only one egree of freeom, n it is more convenient to use the coorinte θ to escribe the ynmics of the penulum. Let us therefore consier the nturl polr coorinte system (r, θ) ssocite with the geometry, n fix r = l to tke the constrint into ccount. In polr coointes, the position of n object is r = re r where e r is the unit vector in the r irection. Since e r epens on θ, the velocity of n object in polr coorintes is ṙ = ṙe r + r θe θ where e θ is the unit vector in the θ irection, which lso epens on θ. The ccelertion is r = ( r r θ )e r + (r θ + ṙ θ)e θ In our cse, r = cst = l, so the ccelertion tkes the simple form Figure 4: Schemtic of the simple penulum r = l θ e r + l θe θ 4

5 There re two forces cting on the bob: the grvity force n the tension force of the ro on the bob. The tension force is entirely in the ril irection. The component of the grvity force in the θ irection is F θ = mgsinθ Therefore, Newton s lw in the e θ irection is ml θ = mgsinθ g θ + sinθ = 0 (6) l In the smll ngle limit, θ 0, sinθ θ, so Eqution (6) becomes the ifferentil eqution for simple hrmonic oscilltor θ + ω θ = 0 (7) where is the ngulr frequency of the motion. The solution of Eqution (7) is strightforwr: ω = g l θ(t) = θ 0 cos(ωt + φ) where θ 0 is the mximum mplitue of the motion, n φ the phse. Conservtion of energy Just s in the cse of Newton s pple, we cn erive reltion for the conservtion of energy for the simple penulum by multiplying Eqution (6) by θ n integrting ( ) θ t g l cos θ = 0 H(θ, θ) = θ g cos θ = Cst = E (8) l where H(θ, θ) is the Hmiltonin for the simple penulum. The first term in the Hmiltonin correspons to the kinetic energy of the penulum; the secon term is the potentil energy. Phse spce The vlue of phse portrits becomes truly pprent in the cse of the simple penulum. It is quite esy, using computer, to plot the contours of the Hmiltonin for the penulum. This is wht we o in Figure 5. We only show the portrit for θ [ π π] since the whole portrit is esily prouce using the π perioicity of the equtions. We recognize the key fetures of the simple penulum, which woul not be obtine so esily by looking t the ifferentil eqution (6): close contours, corresponing to energies below the threshol for the full turn roun the pivot; by tking the smll ngle limit for the Hmiltonin, we cn see tht these contours re ellipses the open contours corresponing to energies llowing the bob to fully turn roun the pivot the yellow curve is the seprtrix, corresponing to E th = g/l. Chrge prticles in uniform mgnetic fiel We finish this section with the stuy of the motion of chrge prticles in uniform mgnetic fiel. The reson we look t this prticulr exmple, which is not often covere in clssicl mechnics courses, is tht this sitution les to more generl form for the Lgrngin n for conjugte moment thn the forms usully presente in elementry presenttions of Lgrngin mechnics. We will therefore iscuss this sitution in the following lectures, n this is goo plce to introuce it. Consier spce equippe with the Crtesin coorintes (x, y, z) n fille with uniform mgnetic fiel in the z-irection: B = B 0 e z. Physicists hve iscovere tht chrge prticle with chrge q n velocity ṙ is subject to the Lorentz force F L = qṙ B (9) 5

6 10 5 θ/t θ Figure 5: Phse portrit for the simple penulum (l = 1 n g = 9.8 ws use for this figure) In the following, we will consier tht the mss of the prticle is smll enough tht grvity cn be ignore. This is very goo pproximtion for electrons, protons n nuclei insie toms for exmple. The system hs three egrees of freeom, n given the geometry of the problem, the Crtesin coorintes (x, y, z) re pproprite. Conservtion of energy We invert the orer of the presenttion n first erive the eqution for the conservtion of energy. The reson for this is tht we will mke explicit use of the conservtion of kinetic energy when eriving the equtions for the motion of the prticle. Newton s lw for prticle of mss m subject to the Lorentz force is Dotting this eqution with ṙ, we immeitely fin (mṙ ) = 0 t m r = qṙ B (10) t The Lorentz force oes not o ny work on moving prticles, so the kinetic energy is conserve. Motion of chrge prticles in uniform mgnetic fiel Defining ω c = qb 0 /m, clle the cyclotron frequency, Newton s equtions cn be written s Eqution (13) is reily solve: ẍ = ω c ẏ (11) ÿ = ω c ẋ (1) z = 0 (13) ż(t) = ż(0) = v z0 (14) The motion in the irection of the mgnetic fiel is unffecte by the mgnetic force. The motion perpeniculr to the mgnetic fiel is more interesting. Tking time erivtive of Eqution (11) n using (1), we fin... x + ωc ẋ = 0 (15) 6

7 Figure 6: Helicl trjectories of electrons immerse in uniform mgnetic fiel The generl solution of Eqution (15) is ẋ(t) = Acos(ω c t + ϕ) (16) where the mplitue A n the phse ϕ re etermine from the initil conitions. Plugging (16) into (11), we then hve ẏ(t) = Asin(ω c t + ϕ) (17) Now we sw erlier tht the Lorentz force conserves the kinetic energy of the prticle. The prllel kinetic energy is constnt, s we sw in Eqution (14). This mens tht the perpeniculr kinetic energy, efine s W = 1 mv = 1 m ( vx + vy ) is lso constnt: W = cst = 1 mv 0 (18) For ny time t, we thus hve so tht the velocities re given for ll times t by ẋ + ẏ = A = v 0 ẋ = v 0 cos(ω c t + ϕ) ẏ = v 0 sin(ω c t + ϕ) (19) These equtions clerly escribe circulr motion. Let us cll (x gc, y gc ) the center of tht circle, clle the gyrocenter. Integrting Equtions (19) we obtin the prticle s trjectory for the perpeniculr motion: x(t) = x gc + v 0 ω c sin(ω c t + ϕ) The rius of the circulr motion is n is clle the Lrmor rius or gyrorius. y(t) = y gc + v 0 ω c cos(ω c t + ϕ) (0) ρ L = v 0 ω c (1) ω c > 0 for positively chrge prticles, while ω c < 0 negtively chrge prticles. This mens tht positively chrge prticles n negtively chrge prticles rotte roun the mgnetic fiel in opposite irections. If the mgnetic fiel is pointing towr you, positively chrge prticles rotte clockwise n negtively chrge prticles rotte counter-clockwise. The trjectory in the prllel irection is foun by integrting Eqution (14): z(t) = z gc + v z0 t where z gc, the z-coorinte of the guiing center position is lso the z-coorinte z 0 of the initil position of the prticle. Summrizing, the totl trjectory is given by x(t) = x gc + v 0 ω c sin(ω c t + ϕ) () y(t) = y gc + v 0 cos(ω c t + ϕ) ω c (3) z(t) = z gc + v z0 t (4) Clerly, these re the prmetric equtions of helicl motion (see Figure ). 7

8 1.3 Motivtion for vritionl formultion of clssicl mechnics From the exmples bove, it seems like quite lot cn be one with the Newtonin pproch to clssicl mechnics. In the cse of the simple penulum, we even mnge to incorporte the constrint tht the ro be rigi in eriving the equtions of motion. Why then look t other formultions of clssicl mechnics? There re severl resons, tht will become more n more pprent s the course unfols. We will mention few here: In the Newtonin pproch, ech point mss is trete iniviully. It is inherently prticle-byprticle pproch. In contrst, s we will see, the vritionl formultion consiers the vrious forms of energy in the system. These ifferent energies o not epen on the wy the system is escribe The equtions of motion re erive in the sme wy regrless of the choice of the coorinte system In the vritionl formultion, constrints cn be expresse in n esier mnner, n even built into the coorintes The vritionl formultion provies nturl venue for reveling symmetries n conserve quntities in the system The vritionl formultion cn le to significntly simpler erivtions of the motion within the frmework of symptotic theories The vritionl formultion shres very close similrities with other fiels of physics, such s quntum mechnics Clculus of Vritions Before we go into the etils of the vritionl formultions of clssicl mechnics, we will hve brief review of funmentl result of the clculus of vrition: the Euler-Lgrnge eqution. The presenttion follows in lrge prt tht of Oliver Bühler in A Brief Introuction to Clssicl, Sttisticl, n Quntum Mechnics..1 Vrition of functionls Consier function q of one vrible efine on the intervl [, b], n smooth enough to llow us to tke ll the erivtives we nee to tke. We efine the following integrl S[q] = b L(q(x), q (x), x)x (5) x where L is sufficiently smooth in ll its three vrible slots to llow prtil erivtives up to secon orer to exist. S is sclr function tht epens on the function q. We sy tht S is functionl of q, n the epenence of S on q is usully enote with squre brcket. In the vritionl formultion of clssicl mechnics, we will extremize functionl clle the ction. Let us therefore look t the chnge of S s the function q is subject to smll vrition. More precisely, we wnt to see how S chnges s we let q q + δq where δq is smll in the sense tht sup{ δq(x), x [, b]} 1 n sup{ x, x [, b]} 1 The vrition of S cn be pproximte s follows S[q + δq] S[q] = = b b L(q + δq(x), q x + δq x, x)x b [ 1 L(q, q x, x)δq + L(q, q x, x)δq x L(q(x), q x, x)x ] x + o(δq, δq x ) 8

9 where the symbol i inictes prtil erivtive with respect to whichever symbol ppers in the i-th slot of L. The secon term insie the squre brcket cn be integrte by prt, so the integrl in the eqution bove, clle the first vrition of S n written δs becomes δs =. Extremls b [ 1 L(q, q x, x) ( L(q, q )] [ x x, x) δqx + L(q, q ] b x, x)δq The functionl S is si to be extremize, n the function q is n extreml of S if δs mesure roun q vnishes for ll δq. Looking t (6), this cn only be the cse if q solves the following secon-orer orinry ifferentil eqution: [ L(q, q ] x x, x) (6) = 1 L(q, q, x) (7) x Inee, if tht were not true, then we coul choose function δq such tht δq() = δq(b) = 0 n such tht the integrl is not zero. For such function, we woul hve δs 0, contricting the fct tht S is extremize. Eqution (7) is clle the Euler-Lgrnge eqution, iscovere jointly by Euler n Lgrnge in the 18th century, s they were working on severl problems in clssicl mechnics. To see why the Euler-Lgrnge eqution le to secon-orer ODE in generl, one cn use the chin rule to rewrite the left-hn sie of Eqution (7): [ L(q, q ] x x, x) = 1 L(q, q q, x) x x + L(q, q x, x) q x + 3 L(q, q x, x) There re mny possible bounry conitions on q, epening on the question one is sking. Often, q() n q(b) re fixe, s will be the most common cse for us. These bounry conitions re clle rigi bounry conitions, or essentil bounry conitions. If q(b) is not fixe, δs = 0 n Eqution (6) imply tht L(q(b), q (b), b) = 0 x This bounry conition is clle nturl bounry conition..3 Illustrtion: shortest pth between two points Wht we hve just lerne cn be use to etermine the shortest pth between two points A n B with coorintes (x A, y A ) n (x B, y B ) in two-imensionl Euclien spce. The length functionl is, ssuming tht x B > x A B B xb ( ) y S = s = x + y = 1 + x x We cn ientify the integrn with L(y, y x 1, x) = + the following Euler-Lgrnge eqution A A x A x ( L) = 0 L y = y = cons 1 + y ( y x) so tht the function y tht minimizes S stisfies where y y/x n the bounry conitions re y(x A ) = y A n y(x B ) = y B. The eqution bove implies tht y = const, so the extreml y is stright line, s one woul expect. Note tht in principle, we shoul o more work to prove tht the extreml bove correspons to minimum of S, n not mximum. This is not lwys so esy; fortuntely, it is cler tht in our sitution, we inee foun the minimum. 9

10 .4 Euler-Lgrnge equtions for functionls of severl functions In the course of the semester, we will encounter systems tht hve more thn one egree of freeom. To be rey for such situtions, we nee to consier functionls S tht epen on N functions q n. If the integrn hs the following epenence, L(q 1,..., q N, q1 x,..., q N x, x), you cn repet the steps we erive bove for the integrn tht only epene on one function, n convince yourself tht the Euler-Lgrnge equtions etermining the N functions extremizing the functionl re x ( N+iL) = i L i = 1,..., N (8) This represents system of N couple secon-orer ODEs for the functions q i. Illustrtion: shortest pth between two points To illustrte the generliztion bove, let s reconsier the question of the shortest pth between the points A(x A, y A ) n B(x B, y B ), viewing ll possible pths γ(t) between these two points s prmetrize by the prmeter t: The length functionl is γ(t) : {x(t), y(t)}, t [0, 1], (x(0), y(0)) = (x A, y A ), (x(1), y(1)) = (x B, y B ) S = We cn ientify the functionl B A s = B A 1 (x ) x + y = + t L(x, y, x t, y t, t) = 0 (x ) + t ( ) y t ( ) y t (9) t so the Euler-Lgrnge equtions re t ( ẋ 3L) = 0 ẋ = const +ẏ t ( 4L) = 0 ẏ ẋ = const (30) +ẏ where we hve use the nottion ẋ = x/t n ẏ = y/t. The couple equtions (30) implies tht there exists constnt C such tht ẏ = Cẋ.γ thus is stright line, s expecte. 10