1.2 Single Particle Kinematics

Transcription

1 1.. SINGLE PARTICLE KINEMATICS 3 multpole moments of the sun forces due to other planets effects of correctons to Newtonan gravty due to general relatvty frcton due to the solar wnd and gas n the solar system Learnng how to estmate or ncorporate such effects s not trval. Secondly, classcal mechancs s not a dead feld of research n fact, n the last few decades there has been a great deal of nterest n dynamcal systems. Attenton has shfted from calculaton of the trajectory over fxed ntervals of tme to questons of the long-term stablty of the moton. New ways of lookng at dynamcal behavor have emerged, such as chaos and fractal systems. Thrdly, the fundamental concepts of classcal mechancs provde the conceptual framework of quantum mechancs. For example, although the Hamltonan and Lagrangan were developed as sophstcated technques for performng classcal mechancs calculatons, they provde the basc dynamcal objects of quantum mechancs and quantum feld theory respectvely. One vew of classcal mechancs s as a steepest path approxmaton to the path ntegral whch descrbes quantum mechancs. Ths ntegral over paths s of a classcal quantty dependng on the acton of the moton. So classcal mechancs s worth learnng well, and we mght as well jump rght n. 1. Sngle Partcle Knematcs We start wth the smplest knd of system, a sngle unconstraned partcle, free to move n three dmensonal space, under the nfluence of a force F Moton n confguraton space The moton of the partcle s descrbed by a functon whch gves ts poston as a functon of tme. These postons are ponts n Eucldean space. Eucldean space s smlar to a vector space, except that there s no specal pont whch s fxed as the orgn. It does have a metrc, that s, a noton of dstance between any two ponts, D(A, B). It also has the concept of a dsplacement A B from one pont B n the Eucldean space to another, 4 CHAPTER 1. PARTICLE KINEMATICS A. These dsplacements do form a vector space, and for a three-dmensonal Eucldean space, the vectors form a three-dmensonal real vector space R 3, whch can be gven an orthonormal bass such that the dstance between A 3=1 and B s gven by D(A, B) = [(A B) ]. Because the mathematcs of vector spaces s so useful, we often convert our Eucldean space to a vector space by choosng a partcular pont as the orgn. Each partcle s poston s then equated to the dsplacement of that poston from the orgn, so that t s descrbed by a poston vector r relatve to ths orgn. But the orgn has no physcal sgnfcance unless t has been choosen n some physcally meanngful way. In general the multplcaton of a poston vector by a scalar s as meanngless physcally as sayng that 4nd street s three tmes 14th street. The cartesan components of the vector r, wth respect to some fxed though arbtrary coordnate system, are called the coordnates, cartesan coordnates n ths case. We shall fnd that we often (even usually) prefer to change to other sets of coordnates, such as polar or sphercal coordnates, but for the tme beng we stck to cartesan coordnates. The moton of the partcle s the functon r(t) of tme. Certanly one of the central questons of classcal mechancs s to determne, gven the physcal propertes of a system and some ntal condtons, what the subsequent moton s. The requred physcal propertes s a specfcaton of the force, F. The begnnngs of modern classcal mechancs was the realzaton early n the 17th century that the physcs, or dynamcs, enters nto the moton (or knematcs) through the force and ts effect on the acceleraton, and not through any drect effect of dynamcs on the poston or velocty of the partcle. Most lkely the force wll depend on the poston of the partcle, say for a partcle n the gravtatonal feld of a fxed (heavy) source at the orgn, for whch F ( r) = GMm r. (1.1) r 3 But the force mght also depend explctly on tme. For example, for the moton of a spaceshp near the Earth, we mght assume that the force s gven by sum of the Newtonan gravtatonal forces of the Sun, Moon and Earth. Each of these forces depends on the postons of the correspondng heavenly body, whch vares wth tme. The assumpton here s that the moton of these bodes s ndependent of the poston of the lght spaceshp. We assume someone else has already performed the nontrval problem of fndng the postons of these bodes as functons of tme. Gven that, we

2 1.. SINGLE PARTICLE KINEMATICS 5 can wrte down the force the spaceshp feels at tme t f t happens to be at poston r, r R F ( r, t) = GmM S (t) S r R S (t) GmM r R E (t) 3 E r R E (t) 3 r R GmM M (t) M r R M (t). 3 So there s an explct dependence on t Fnally, the force mght depend on the velocty of the partcle, as for example for the Lorentz force on a charged partcle n electrc and magnetc felds F ( r, v, t) =q E( r, t)+q v B( r, t). (1.) However the force s determned, t determnes the moton of the partcle through the second order dfferental equaton known as Newton s Second Law F ( r, v, t) =m a = m d r dt. As ths s a second order dfferental equaton, the soluton depends n general on two arbtrary (3-vector) parameters, whch we mght choose to be the ntal poston and velocty, r(0) and v(0). For a gven physcal stuaton and a gven set of ntal condtons for the partcle, Newton s laws determne the moton r(t), whch s a curve n confguraton space parameterzed by tme t, known as the trajectory n confguraton space. If we consder the curve tself, ndependent of how t depends on tme, ths s called the orbt of the partcle. For example, the orbt of a planet, n the approxmaton that t feels only the feld of a fxed sun, s an ellpse. That word does not mply any nformaton about the tme dependence or parameterzaton of the curve. 1.. Conserved Quanttes Whle we tend to thnk of Newtonan mechancs as centered on Newton s Second Law n the form F = m a, he actually started wth the observaton that n the absence of a force, there was unform moton. We would now say that under these crcumstances the momentum p(t) s conserved, d p/dt = 6 CHAPTER 1. PARTICLE KINEMATICS 0. In hs second law, Newton stated the effect of a force as producng a rate of change of momentum, whch we would wrte as F = d p/dt, rather than as producng an acceleraton F = m a. In focusng on the concept of momentum, Newton emphaszed one of the fundamental quanttes of physcs, useful beyond Newtonan mechancs, n both relatvty and quantum mechancs 1. Only after usng the classcal relaton of momentum to velocty, p = m v, and the assumpton that m s constant, do we fnd the famlar F = m a. One of the prncpal tools n understandng the moton of many systems s solatng those quanttes whch do not change wth tme. A conserved quantty s a functon of the postons and momenta, and perhaps explctly of tme as well, Q( r, p, t), whch remans unchanged when evaluated along the actual moton, dq( r(t), p(t),t)/dt = 0. A functon dependng on the postons, momenta, and tme s sad to be a functon on extended phase space. When tme s not ncluded, the space s called phase space. In ths language, a conserved quantty s a functon on extended phase space wth a vanshng total tme dervatve along any path whch descrbes the moton of the system. A sngle partcle wth no forces actng on t provdes a very smple example. As Newton tells us, p = d p/dt = F = 0, so the momentum s conserved. There are three more conserved quanttes Q( r, p, t) := r(t) t p(t)/m, whch have a tme rate of change dq/dt = r p/m t p/m = 0. These sx ndependent conserved quanttes are as many as one could have for a system wth a sx dmensonal phase space, and they completely solve for the moton. Of course ths was a very smple system to solve. We now consder a partcle under the nfluence of a force. Energy Consder a partcle under the nfluence of an external force F. In general, the momentum wll not be conserved, although f any cartesan component of the force vanshes along the moton, that component of the momentum 1 The relatonshp of momentum to velocty s changed n these extensons, however. Phase space s dscussed further n secton 1.4.

3 1.. SINGLE PARTICLE KINEMATICS 7 wll be conserved. Also the knetc energy, defned as T = 1m v, wll not n general be conserved, because dt dt = m v v = F v. As the partcle moves from the pont r to the pont r f the total change n the knetc energy s the work done by the force F, T = rf r F d r. If the force law F ( r, p, t) applcable to the partcle s ndependent of tme and velocty, then the work done wll not depend on how quckly the partcle moved along the path from r to r f. If n addton the work done s ndependent of the path taken between these ponts, so t depends only on the endponts, then the force s called a conservatve force and we assoscate wth t potental energy U( r) =U( r 0 )+ r0 r F ( r ) d r, where r 0 s some arbtrary reference poston and U( r 0 ) s an arbtrarly chosen reference energy, whch has no physcal sgnfcance n ordnary mechancs. U( r) represents the potental the force has for dong work on the partcle f the partcle s at poston r. r f r f 8 CHAPTER 1. PARTICLE KINEMATICS Thus the requrement that the ntegral of F d r vansh around any closed path s equvalent to the requrement that the curl of F vansh everywhere n space. By consderng an nfntesmal path from r to r + r, we see that U( r + ) U( r) = F r, or F (r) = U(r). The value of the concept of potental energy s that t enables fndng a conserved quantty, the total energy, n stutatons n whch all forces are conservatve. Then the total energy E = T + U changes at a rate de dt = dt dt + d r dt U = F v v F =0. The total energy can also be used n systems wth both conservatve and nonconservatve forces, gvng a quantty whose rate of change s determned by the work done only by the nonconservatve forces. One example of ths usefulness s n the dscusson of a slghtly damped harmonc oscllator drven by a perodc force near resonance. Then the ampltude of steady-state moton s determned by a balence between the average power nput by the drvng force and the average power dsspated by frcton, the two nonconservatve forces n the problem, wthout needng to worry about the work done by the sprng. The condton for the path ntegral to be ndependent of the path s that t gves the same results along any two cotermnous paths Γ 1 and Γ, or alternatvely that t gve zero when evaluated along any closed path such as Γ = Γ 1 Γ, the path consstng of followng Γ 1 and then takng Γ backwards to the startng pont. By Stokes Theorem, ths lne ntegral s equvalent to an ntegral over any surface S bounded by Γ, F d r = FdS. Γ S r Γ Γ 1 r Independence of path Γ 1 = Γ s equvalent to vanshng of the path ntegral over closed paths Γ, whch s n turn equvalent to the vanshng of the curl on the surface whose boundary s Γ. Γ Angular momentum Another quantty whch s often useful because t may be conserved s the angular momentum. The defnton requres a reference pont n the Eucldean space, say r 0. Then a partcle at poston r wth momentum p has an angular momentum about r 0 gven by L =( r r 0 ) p. Very often we take the reference pont r 0 to be the same as the pont we have chosen as the orgn n convertng the Eucldan space to a vector space, so r 0 = 0, and d L dt L = r p = d r d p p + r dt dt = 1 m p p + r F =0+ τ = τ. where we have defned the torque about r 0 as τ =( r r 0 ) F n general, and τ = r F when our reference pont r 0 s at the orgn.

4 1.3. SYSTEMS OF PARTICLES 9 10 CHAPTER 1. PARTICLE KINEMATICS We see that f the torque τ(t) vanshes (at all tmes) the angular momentum s conserved. Ths can happen not only f the force s zero, but also f the force always ponts to the reference pont. Ths s the case n a central force problem such as moton of a planet about the sun. Let us defne F E Thrd Law holds, = F E to be the total external force. If Newton s F j = F j, so j F j =0, and 1.3 Systems of Partcles P = F E. (1.3) So far we have talked about a system consstng of only a sngle partcle, possbly nfluenced by external forces. Consder now a system of n partcles wth postons r, =1,...,n, n flat space. The confguraton of the system then has 3n coordnates (confguraton space s R 3n ), and the phase space has 6n coordnates { r, p } External and nternal forces Let F be the total force actng on partcle. It s the sum of the forces produced by each of the other partcles and that due to any external force. Let F j be the force partcle j exerts on partcle and let F E be the external force on partcle. Usng Newton s second law on partcle, wehave F = F E + j F j = p = m v, where m s the mass of the th partcle. Here we are assumng forces have dentfable causes, whch s the real meanng of Newton s second law, and that the causes are ether ndvdual partcles or external forces. Thus we are assumng there are no three-body forces whch are not smply the sum of two-body forces that one object exerts on another. Defne the center of mass and total mass R = m r m, M = m. Then f we defne the total momentum we have P = p = m v = d dt d P dt = P = p = F = m r = M d R dt, F E + j F j. Thus the nternal forces cancel n pars n ther effect on the total momentum, whch changes only n response to the total external force. As an obvous but very mportant consequence 3 the total momentum of an solated system s conserved. The total angular momentum s also just a sum over the ndvdual angular momenta, so for a system of pont partcles, L = L = r p. Its rate of change wth tme s dl dt = L = v p + r F =0+ r F E + j r F j. 3 There are stuatons and ways of descrbng them n whch the law of acton and reacton seems not to hold. For example, a current 1 flowng through a wre segment d s 1 contrbutes, accordng to the law of Bot and Savart, a magnetc feld db = µ 0 1 d s 1 r/4π r 3 at a pont r away from the current element. If a current flows through a segment of wre d s at that pont, t feels a force F 1 = µ 0 4π d s (d s 1 r) 1 r 3 due to element 1. On the other hand F 1 s gven by the same expresson wth d s 1 and d s nterchanged and the sgn of r reversed, so F 1 + F 1 = µ 0 1 4π r 3 [d s 1(d s r) d s (d s 1 r)], whch s not generally zero. One should not despar for the valdty of momentum conservaton. The Law of Bot and Savart only holds for tme-ndependent current dstrbutons. Unless the currents form closed loops, there wll be a charge buldup and Coulomb forces need to be consdered. If the loops are closed, the total momentum wll nvolve ntegrals over the two closed loops, for whch F 1 + F 1 can be shown to vansh. More generally, even the sum of the momenta of the current elements s not the whole story, because there s momentum n the electromagnetc feld, whch wll be changng n the tme-dependent stuaton.

5 1.3. SYSTEMS OF PARTICLES 11 The total external torque s naturally defned as τ = r F E, so we mght ask f the last term vanshes due the Thrd Law, whch permts us to rewrte F ( j = 1 Fj ) F j. Then the last term becomes r F j = 1 r F j j 1 r F j j j = 1 r F j 1 r j F j j j = 1 ( r r j ) F j. j Ths s not automatcally zero, but vanshes f one assumes a stronger form of the Thrd Law, namely that the acton and reacton forces between two partcles acts along the lne of separaton of the partcles. If the force law s ndependent of velocty and rotatonally and translatonally symmetrc, there s no other drecton for t to pont. For spnnng partcles and magnetc forces the argument s not so smple n fact electromagnetc forces between movng charged partcles are really only correctly vewed n a context n whch the system ncludes not only the partcles but also the felds themselves. For such a system, n general the total energy, momentum, and angular momentum of the partcles alone wll not be conserved, because the felds can carry all of these quanttes. But properly defnng the energy, momentum, and angular momentum of the electromagnetc felds, and ncludng them n the totals, wll result n quanttes conserved as a result of symmetres of the underlyng physcs. Ths s further dscussed n secton 8.3. Makng the assumpton that the strong form of Newton s Thrd Law holds, we have shown that τ = d L dt. (1.4) The conservaton laws are very useful because they permt algebrac soluton for part of the velocty. Takng a sngle partcle as an example, f E = 1 mv + U( r) s conserved, the speed v(t) s determned at all tmes (as a functon of r) by one arbtrary constant E. Smlarly f L s conserved, 1 CHAPTER 1. PARTICLE KINEMATICS the components of v whch are perpendcular to r are determned n terms of the fxed constant L. Wth both conserved, v s completely determned except for the sgn of the radal component. Examples of the usefulness of conserved quanttes are everywhere, and wll be partcularly clear when we consder the two body central force problem later. But frst we contnue our dscusson of general systems of partcles. As we mentoned earler, the total angular momentum depends on the pont of evaluaton, that s, the orgn of the coordnate system used. We now show that t conssts of two contrbutons, the angular momentum about the center of mass and the angular momentum of a fcttous pont object located at the center of mass. Let r be the poston of the th partcle wth respect to the center of mass, so r = r R. Then L = m r v = ( m r + R ) ( r + ) R = m r r + m r R + R m r + MR R = r p + R P. Here we have noted that m r = 0, and also ts dervatve m v =0. We have defned p = m v, the momentum n the center of mass reference frame. The frst term of the fnal form s the sum of the angular momenta of the partcles about ther center of mass, whle the second term s the angular momentum the system would have f t were collapsed to a pont at the center of mass. Notce we dd not need to assume the center of mass s unaccelerated. What about the total energy? The knetc energy T = 1 m v = 1 ( m v + V ) ( v + V ) = 1 m v + 1 MV, (1.5) where V = R s the velocty of the center of mass. The cross term vanshes once agan, because m v = 0. Thus the knetc energy of the system can also be vewed as the sum of the knetc energes of the consttuents about

6 1.3. SYSTEMS OF PARTICLES 13 the center of mass, plus the knetc energy the system would have f t were collapsed to a partcle at the center of mass. If the forces on the system are due to potentals, the total energy wll be conserved, but ths ncludes not only the potental due to the external forces but also that due to nterpartcle forces, U j ( r, r j ). In general ths contrbuton wll not be zero or even constant wth tme, and the nternal potental energy wll need to be consdered. One excepton to ths s the case of a rgd body Constrants A rgd body s defned as a system of n partcles for whch all the nterpartcle dstances are constraned to fxed constants, r r j = c j, and the nterpartcle potentals are functons only of these nterpartcle dstances. As these dstances do not vary, nether does the nternal potental energy. These nterpartcle forces cannot do work, and the nternal potental energy may be gnored. The rgd body s an example of a constraned system, n whch the general 3n degrees of freedom are restrcted by some forces of constrant whch place condtons on the coordnates r, perhaps n conjuncton wth ther momenta. In such descrptons we do not wsh to consder or specfy the forces themselves, but only ther (approxmate) effect. The forces are assumed to be whatever s necessary to have that effect. It s generally assumed, as n the case wth the rgd body, that the constrant forces do no work under dsplacements allowed by the constrants. We wll consder ths pont n more detal later. If the constrants can be phrased so that they are on the coordnates and tme only, as Φ ( r 1,... r n,t)=0, =1,...,k, they are known as holonomc constrants. These constrants determne hypersurfaces n confguraton space to whch all moton of the system s confned. In general ths hypersurface forms a 3n k dmensonal manfold. We mght descrbe the confguraton pont on ths manfold n terms of 3n k generalzed coordnates, q j,j =1,...,3n k, so that the 3n k varables q j, together wth the k constrant condtons Φ ({ r }) = 0, determne the r = r (q 1,...,q 3n k,t) 14 CHAPTER 1. PARTICLE KINEMATICS The constraned subspace of confguraton space need not be a flat space. Consder, for example, a mass on one end of a rgd z lght rod of length L, the other end of whch s fxed to be at the θ orgn r = 0, though the rod s L completely free to rotate. Clearly the possble values of the cartesan coordnates r of the poston ϕ y of the mass satsfy the constrant r = L, so r les on the surface of a sphere of radus L. We x mght choose as generalzed coordnates the standard sphercal an- Generalzed coordnates (θ, φ) for a partcle constraned to le on a gles θ andφ. Thus the constraned sphere. subspace s two dmensonal but [Note: mathematcs books tend not flat rather t s the surface to nterchange θ and φ from the of a sphere, whch mathematcans choce we use here, whch s what call S. It s natural to reexpress most physcs books use.] the dynamcs n terms of θ and φ. Note that wth ths constraned confguraton space, we see that deas common n Eucldean space are no longer clear. The dsplacement between two ponts A and B, as a three vector, cannot be added to a general pont C, and n two dmensons, a change, for example, of φ s a very dffernent change n confguraton dependng on what θ s. The use of generalzed (non-cartesan) coordnates s not just for constraned systems. The moton of a partcle n a central force feld about the orgn, wth a potental U( r) = U( r ), s far more naturally descrbed n terms of sphercal coordnates r, θ, and φ than n terms of x, y, and z. Before we pursue a dscusson of generalzed coordnates, t must be ponted out that not all constrants are holonomc. The standard example s a dsk of radus R, whch rolls on a fxed horzontal plane. It s constraned to always reman vertcal, and also to roll wthout slppng on the plane. As coordnates we can choose the x and y of the center of the dsk, whch are also the x and y of the contact pont, together wth the angle a fxed lne on the dsk makes wth the downward drecton, φ, andtheangletheaxsofthe dsk makes wth the x axs, θ.

7 1.3. SYSTEMS OF PARTICLES 15 As the dsk rolls through an angle dφ, the pont of contact moves a dstance Rdφ n a drecton dependng on θ, Rdφ sn θ = dx Rdφ cos θ = dy Dvdng by dt, we get two constrants nvolvng the postons and veloctes, Φ 1 := R φ sn θ ẋ =0 Φ := R φ cos θ ẏ =0. The fact that these nvolve veloctes does not automatcally make them nonholonomc. In the smpler one-dmensonal problem n whch the dsk s confned to the yz plane, rollng x z A vertcal dsk free to roll on a plane. A fxed lne on the dsk makes an angle of φ wth respect to the vertcal, and the axs of the dsk makes an angle θ wth the x-axs. The long curved path s the trajectory of the contact pont. The three small paths are alternate trajectores llustratng that x, y, and φ can each be changed wthout any net change n the other coordnates. along x =0(θ = 0), we would have only the coordnates φ and y, wth the rollng constrant R φ ẏ = 0. But ths constrant can be ntegrated, Rφ(t) y(t) =c, for some constant c, so that t becomes a constrant among just the coordnates, and s holomorphc. Ths cannot be done wth the twodmensonal problem. We can see that there s no constrant among the four coordnates themselves because each of them can be changed by a moton whch leaves the others unchanged. Rotatng θ wthout movng the other coordnates s straghtforward. By rollng the dsk along each of the three small paths shown to the rght of the dsk, we can change one of the varables x, y, or φ, respectvely, wth no net change n the other coordnates. Thus all values of the coordnates 4 can be acheved n ths fashon. There are other, less nterestng, nonholonomc constrants gven by nequaltes rather than constrant equatons. A bug sldng down a bowlng 4 Thus the confguraton space s x R, y R, θ [0, π) and φ [0, π), or, f we allow more carefully for the contnuty as θ and φ go through π, the more accurate statement s that confguraton space s R (S 1 ), where S 1 s the crcumference of a crcle, θ [0, π], wth the requrement that θ = 0 s equvalent to θ =π. θ R φ y 16 CHAPTER 1. PARTICLE KINEMATICS ball obeys the constrant r R. Such problems are solved by consderng the constrant wth an equalty ( r = R), but restrctng the regon of valdty of the soluton by an nequalty on the constrant force (N 0), and then supplementng wth the unconstraned problem once the bug leaves the surface. In quantum feld theory, anholonomc constrants whch are functons of the postons and momenta are further subdvded nto frst and second class constrants àladrac, wth the frst class constrants leadng to local gauge nvarance, as n Quantum Electrodynamcs or Yang-Mlls theory. But ths s headng far afeld Generalzed Coordnates for Unconstraned Systems Before we get further nto constraned systems and D Alembert s Prncple, we wll dscuss the formulaton of a conservatve unconstraned system n generalzed coordnates. Thus we wsh to use 3n generalzed coordnates q j, whch, together wth tme, determne all of the 3n cartesan coordnates r : r = r (q 1,..., q 3n,t). Notce that ths s a relatonshp between dfferent descrptons of the same pont n confguraton space, and the functons r ({q},t) are ndependent of the moton of any partcle. We are assumng that the r and the q j are each a complete set of coordnates for the space, so the q s are also functons of the { r } and t: q j = q j ( r 1,..., r n,t). The t dependence permts there to be an explct dependence of ths relaton on tme, as we would have, for example, n relatng a rotatng coordnate system to an nertal cartesan one. Let us change the cartesan coordnate notaton slghtly, wth {x k } the 3n cartesan coordnates of the n 3-vectors r, deemphaszng the dvson of these coordnates nto trplets. A small change n the coordnates of a partcle n confguraton space, whether an actual change over a small tme nterval dt or a vrtual change between where a partcle s and where t mght have been under slghtly altered crcumstances, can be descrbed by a set of δx k or by a set of δq j.if

8 1.3. SYSTEMS OF PARTICLES 17 we are talkng about a vrtual change at the same tme, these are related by the chan rule δx k = j x k δq j, δq j = k x k δx k, (for δt =0). (1.6) For the actual moton through tme, or any varaton where δt s not assumed to be zero, we need the more general form, δx k = j x k δq j + x k t δt, δq j = k δx k + q k δt. (1.7) x k t A vrtual dsplacement, wth δt = 0, s the knd of varaton we need to fnd the forces descrbed by a potental. Thus the force s 18 CHAPTER 1. PARTICLE KINEMATICS forces Q k are gven by the potental just as for ordnary cartesan coordnates and ther forces. Now we examne the knetc energy T = 1 m r = 1 m j ẋ j j where the 3n values m j are not really ndependent, as each partcle has the same mass n all three dmensons n ordnary Newtonan mechancs 5.Now x j ẋ j = lm t 0 t = lm t 0 k x j q k + x j, q k q,t t t q where F k = U({x}) x k Q j := k = j F k x k U({x({q})}) = x k j x k Q j, (1.8) = U({x({q})}) (1.9) where q,t means that t and the q s other than q k are held fxed. The last term s due to the possblty that the coordnates x (q 1,..., q 3n,t)mayvary wth tme even for fxed values of q k. So the chan rule s gvng us ẋ j = dx j dt = k x j q q k k + x j. (1.10) q,t t q s known as the generalzed force. We may thnk of Ũ(q, t) :=U(x(q),t) as a potental n the generalzed coordnates {q}. Note that f the coordnate transformaton s tme-dependent, t s possble that a tme-ndependent potental U(x) wll lead to a tme-dependent potental Ũ(q, t), and a system wth forces descrbed by a tme-dependent potental s not conservatve. The defnton of the generalzed force Q j n the left part of (1.9) holds even f the cartesan force s not descrbed by a potental. The q k do not necessarly have unts of dstance. For example, one q k mght be an angle, as n polar or sphercal coordnates. The correspondng component of the generalzed force wll have the unts of energy and we mght consder t a torque rather than a force Knetc energy n generalzed coordnates We have seen that, under the rght crcumstances, the potental energy can be thought of as a functon of the generalzed coordnates q k, and the generalzed Pluggng ths nto the knetc energy, we see that T = 1 x j x j m j q k q l + x j x j m j q k + 1 m j,k,l q k q l j,k q k t j x j. (1.11) q j t q What s the nterpretaton of these terms? Only the frst term arses f the relaton between x and q s tme ndependent. The second and thrd terms are the sources of the r ( ω r) and ( ω r) terms n the knetc energy when we consder rotatng coordnate systems 6. 5 But n an ansotropc crystal, the effectve mass of a partcle mght n fact be dfferent n dfferent drectons. 6 Ths wll be fully developed n secton 4.

9 1.3. SYSTEMS OF PARTICLES 19 Let s work a smple example: we wll consder a two dmensonal system usng polar coordnates wth θ measured from a drecton rotatng at angular velocty ω. Thus the angle the radus vector to an arbtrary pont (r, θ) makes wth the nertal x 1 -axs s θ + ωt, and the relatons are x 1 = r cos(θ + ωt), x = r sn(θ + ωt), wth nverse relatons r = x 1 + x, θ = sn 1 (x /r) ωt. r θ ωt x 1 Rotatng polar coordnates related to nertal cartesan coordnates. So ẋ 1 =ṙ cos(θ + ωt) θr sn(θ + ωt) ωr sn(θ + ωt), where the last term s from x j / t, and ẋ =ṙ sn(θ + ωt)+ θr cos(θ + ωt)+ωr cos(θ + ωt). In the square, thngs get a bt smpler, ẋ =ṙ + r (ω + θ). We see that the form of the knetc energy n terms of the generalzed coordnates and ther veloctes s much more complcated than t s n cartesan nertal coordnates, where t s coordnate ndependent, and a smple dagonal quadratc form n the veloctes. In generalzed coordnates, t s quadratc but not homogeneous 7 n the veloctes, and wth an arbtrary dependence on the coordnates. In general, even f the coordnate transformaton s tme ndependent, the form of the knetc energy s stll coordnate dependent and, whle a purely quadratc form n the veloctes, t s not necessarly dagonal. In ths tme-ndependent stuaton, we have T = 1 M kl ({q}) q k q l, wth M kl ({q}) = kl j x m j x j q k x j q l, (1.1) where M kl s known as the mass matrx, and s always symmetrc but not necessarly dagonal or coordnate ndependent. The mass matrx s ndependent of the x j / t terms, and we can understand the results we just obtaned for t n our two-dmensonal example 7 It nvolves quadratc and lower order terms n the veloctes, not just quadratc ones. 0 CHAPTER 1. PARTICLE KINEMATICS above, M 11 = m, M 1 = M 1 =0, M = mr, by consderng the case wthout rotaton, ω = 0. We can also derve ths expresson for the knetc energy n nonrotatng polar coordnates by expressng the velocty vector v = ṙê r + r θê θ n terms of unt vectors n the radal and tangental drectons respectvely. The coeffcents of these unt vectors can be understood graphcally wth geometrc arguments. Ths leads more quckly to v =(ṙ) +r ( θ), T = 1 mṙ + 1 mr θ, and the mass matrx follows. Smlar geometrc arguments are usually used to fnd the form of the knetc energy n sphercal coordnates, but the formal approach of (1.1) enables us to fnd the form even n stuatons where the geometry s dffcult to pcture. It s mportant to keep n mnd that when we vew T as a functon of coordnates and veloctes, these are ndependent arguments evaluated at a partcular moment of tme. Thus we can ask ndependently how T vares as we change x or as we change ẋ, each tme holdng the other varable fxed. Thus the knetc energy s not a functon on the 3n-dmensonal confguraton space, but on a larger, 6n-dmensonal space 8 wth a pont specfyng both the coordnates {q } and the veloctes { q }. 1.4 Phase Space If the trajectory of the system n confguraton space, r(t), s known, the velocty as a functon of tme, v(t) s also determned. As the mass of the partcle s smply a physcal constant, the momentum p = m v contans the same nformaton as the velocty. Vewed as functons of tme, ths gves nothng beyond the nformaton n the trajectory. But at any gven tme, r and p provde a complete set of ntal condtons, whle r alone does not. We defne phase space as the set of possble postons and momenta for the system at some nstant. Equvalently, t s the set of possble ntal condtons, or the set of possble motons obeyng the equatons of moton 9. For a sngle partcle n cartesan coordnates, the sx coordnates of phase 8 Ths space s called the tangent bundle to confguraton space. For cartesan coordnates t s almost dentcal to phase space, whch s n general the cotangent bundle to confguraton space. 9 As each ntal condton gves rse to a unque future development of a trajectory, there s an somorphsm between ntal condtons and allowed trajectores.

10 1.4. PHASE SPACE 1 space are the three components of r and the three components of p. Atany nstant of tme, the system s represented by a pont n ths space, called the phase pont, and that pont moves wth tme accordng to the physcal laws of the system. These laws are emboded n the force functon, whch we now consder as a functon of p rather than v, n addton to r and t. We may wrte these equatons as d r = p dt m, d p = F dt ( r, p, t). Note that these are frst order equatons, whch means that the moton of the pont representng the system n phase space s completely determned 10 by where the phase pont s. Ths s to be dstngushed from the trajectory n confguraton space, where n order to know the trajectory you must have not only an ntal pont (poston) but also ts ntal tme dervatve Dynamcal Systems We have spoken of the coordnates of phase space for a sngle partcle as r and p, but from a mathematcal pont of vew these together gve the coordnates of the phase pont n phase space. We mght descrbe these coordnates n terms of a sx dmensonal vector η =(r 1,r,r 3,p 1,p,p 3 ). The physcal laws determne at each pont a velocty functon for the phase pont as t moves through phase space, d η dt = V ( η, t), (1.13) whch gves the velocty at whch the phase pont representng the system moves through phase space. Only half of ths velocty s the ordnary velocty, whle the other half represents the rapdty wth whch the momentum s changng,.e. the force. The path traced by the phase pont as t travels through phase space s called the phase curve. For a system of n partcles n three dmensons, the complete set of ntal condtons requres 3n spatal coordnates and 3n momenta, so phase space s 6n dmensonal. Whle ths certanly makes vsualzaton dffcult, the large 10 We wll assume throughout that the force functon s a well defned contnuous functon of ts arguments. CHAPTER 1. PARTICLE KINEMATICS dmensonalty s no hndrance for formal developments. Also, t s sometmes possble to focus on partcular dmensons, or to make generalzatons of deas famlar n two and three dmensons. For example, n dscussng ntegrable systems (7.1), we wll fnd that the moton of the phase pont s confned to a 3n-dmensonal torus, a generalzaton of one and two dmensonal tor, whch are crcles and the surface of a donut respectvely. Thus for a system composed of a fnte number of partcles, the dynamcs s determned by the frst order ordnary dfferental equaton (1.13), formally a very smple equaton. All of the complcaton of the physcal stuaton s hdden n the large dmensonalty of the dependent varable η and n the functonal dependence of the velocty functon V ( η, t) ont. There are other systems besdes Newtonan mechancs whch are controlled by equaton (1.13), wth a sutable velocty functon. Collectvely these are known as dynamcal systems. For example, ndvduals of an asexual mutually hostle speces mght have a fxed brth rate b and a death rate proportonal to the populaton, so the populaton would obey the logstc equaton 11 dp/dt = bp cp, a dynamcal system wth a one-dmensonal space for ts dependent varable. The populatons of three competng speces could be descrbed by eq. (1.13) wth η n three dmensons. The dmensonalty d of η n (1.13) s called the order of the dynamcal system. Ad th order dfferental equaton n one ndependent varable may always be recast as a frst order dfferental equaton n d varables, so t s one example of a d th order dynamcal system. The space of these dependent varables s called the phase space of the dynamcal system. Newtonan systems always gve rse to an even-order system, because each spatal coordnate s pared wth a momentum. Forn partcles unconstraned n D dmensons, the order of the dynamcal system s d =nd. Even for constraned Newtonan systems, there s always a parng of coordnates and momenta, whch gves a restrctng structure, called the symplectc structure 1, on phase space. If the force functon does not depend explctly on tme, we say the system s autonomous. The velocty functon has no explct dependance on tme, V = V ( η), and s a tme-ndependent vector feld on phase space, whch we can ndcate by arrows just as we mght the electrc feld n ordnary space, or the velocty feld of a flud n moton. Ths gves a vsual ndcaton of 11 Ths s not to be confused wth the smpler logstc map, whch s a recurson relaton wth the same form but wth solutons dsplayng a very dfferent behavor. 1 Ths wll be dscussed n sectons (6.3) and (6.6).

11 1.4. PHASE SPACE 3 the moton of the system s pont. For example, consder a damped harmonc oscllator wth F = kx αp, for whch the velocty functon s ( dx dt, dp ) ( ) p = dt m, kx αp. A plot of ths feld for the undamped (α = 0) and damped oscllators s p x Fgure 1.1: Velocty feld for undamped and damped harmonc oscllators, and one possble phase curve for each system through phase space. shown n Fgure 1.1. The velocty feld s everywhere tangent to any possble path, one of whch s shown for each case. Note that qualtatve features of the moton can be seen from the velocty feld wthout any solvng of the dfferental equatons; t s clear that n the damped case the path of the system must spral n toward the orgn. The paths taken by possble physcal motons through the phase space of an autonomous system have an mportant property. Because the rate and drecton wth whch the phase pont moves away from a gven pont of phase space s completely determned by the velocty functon at that pont, f the system ever returns to a pont t must move away from that pont exactly as t dd the last tme. That s, f the system at tme T returns to a pont n phase space that t was at at tme t = 0, then ts subsequent moton must be just as t was, so η(t + t) = η(t), and the moton s perodc wth perod T. Ths almost mples that the phase curve the object takes through phase space must be nonntersectng 13. In the non-autonomous case, where the velocty feld s tme dependent, t may be preferable to thnk n terms of extended phase space, a 6n An excepton can occur at an unstable equlbrum pont, where the velocty functon vanshes. The moton can just end at such a pont, and several possble phase curves can termnate at that pont. p x 4 CHAPTER 1. PARTICLE KINEMATICS dmensonal space wth coordnates ( η, t). The velocty feld can be extended to ths space by gvng each vector a last component of 1, as dt/dt = 1. Then the moton of the system s relentlessly upwards n ths drecton, though stll complex n the others. For the undamped one-dmensonal harmonc oscllator, the path s a helx n the three dmensonal extended phase space. Most of ths book s devoted to fndng analytc methods for explorng the moton of a system. In several cases we wll be able to fnd exact analytc solutons, but t should be noted that these exactly solvable problems, whle very mportant, cover only a small set of real problems. It s therefore mportant to have methods other than searchng for analytc solutons to deal wth dynamcal systems. Phase space provdes one method for fndng qualtatve nformaton about the solutons. Another approach s numercal. Newton s Law, and more generally the equaton (1.13) for a dynamcal system, s a set of ordnary dfferental equatons for the evoluton of the system s poston n phase space. Thus t s always subject to numercal soluton gven an ntal confguraton, at least up untl such pont that some sngularty n the velocty functon s reached. One prmtve technque whch wll work for all such systems s to choose a small tme nterval of length t, and use d η/dt at the begnnng of each nterval to approxmate η durng ths nterval. Ths gves a new approxmate value for η at the end of ths nterval, whch may then be taken as the begnnng of the next Ths s a very unsophstcated method. The errors made n each step for r and p are typcally O( t). As any calculaton of the evoluton from tme t 0 to t f wll nvolve anumber([t f t 0 ]/ t) of tme steps whch grows nversely to t, the cumulatve error can be expected to be O( t). In prncple therefore we can approach exact results for a fnte tme evoluton by takng smaller and smaller tme steps, but n practse there are other consderatons, such as computer tme and roundoff errors, whch argue strongly n favor of usng more sophstcated numercal technques, wth errors of hgher order n t. Increasngly sophstcated methods can be generated whch gve cumulatve errors of order O(( t) n ), for any n. A very common technque s called fourth-order Runge-Kutta, whch gves an error O(( t) 5 ). These methods can be found n any text on numercal methods.

12 1.4. PHASE SPACE 5 As an example, we show the meat of a calculaton for the damped harmonc oscllator. Ths same technque wll work even wth a very complcated stuaton. One need only add lnes for all the components of the poston and momentum, and change the force law approprately. Ths s not to say that numercal soluton s a good way to solve ths problem. An analytcal soluton, f t can be found, s almost always preferable, because whle (t < tf) { dx = (p/m) * dt; dp = -(k*x+alpha*p)*dt; x = x + dx; p = p + dp; t = t + dt; prnt t, x, p; } Integratng the moton, for a damped harmonc oscllator. It s far more lkely to provde nsght nto the qualtatve features of the moton. Numercal solutons must be done separately for each value of the parameters (k, m, α) and each value of the ntal condtons (x 0 and p 0 ). Numercal solutons have subtle numercal problems n that they are only exact as t 0, and only f the computatons are done exactly. Sometmes uncontrolled approxmate solutons lead to surprsngly large errors. Nonetheless, numercal solutons are often the only way to handle a real problem, and there has been extensve development of technques for effcently and accurately handlng the problem, whch s essentally one of solvng a system of frst order ordnary dfferental equatons Phase Space Flows As we just saw, Newton s equatons for a system of partcles can be cast n the form of a set of frst order ordnary dfferental equatons n tme on phase space, wth the moton n phase space descrbed by the velocty feld. Ths could be more generally dscussed as a d th order dynamcal system, wth a phase pont representng the system n a d-dmensonal phase space, movng 6 CHAPTER 1. PARTICLE KINEMATICS wth tme t along the velocty feld, sweepng out a path n phase space called the phase curve. The phase pont η(t) s also called the state of the system at tme t. Many qualtatve features of the moton can be stated n terms of the phase curve. Fxed Ponts There may be ponts η k, known as fxed ponts, at whch the velocty functon vanshes, V ( η k ) = 0. Ths s a pont of equlbrum for the system, for f the system s at a fxed pont at one moment, η(t 0 )= η k, t remans at that pont. At other ponts, the system does not stay put, but there may be sets of states whch flow nto each other, such as the ellptcal orbt for the undamped harmonc oscllator. These are called nvarant sets of states. In a frst order dynamcal system 15, the fxed ponts dvde the lne nto ntervals whch are nvarant sets. Even though a frst-order system s smaller than any Newtonan system, t s worthwhle dscussng brefly the phase flow there. We have been assumng the velocty functon s a smooth functon genercally ts zeros wll be frst order, and near the fxed pont η 0 we wll have V (η) c(η η 0 ). If the constant c<0, dη/dt wll have the opposte sgn from η η 0, and the system wll flow towards the fxed pont, whch s therefore called stable. On the other hand, f c>0, the dsplacement η η 0 wll grow wth tme, and the fxed pont s unstable. Of course there are other possbltes: f V (η) =cη, the fxed pont η = 0 s stable from the left and unstable from the rght. But ths knd of stuaton s somewhat artfcal, and such a system s structually unstable. What that means s that f the velocty feld s perturbed by a small smooth varaton V (η) V (η) +ɛw(η), for some bounded smooth functon w, the fxed pont at η = 0 s lkely to ether dsappear or splt nto two fxed ponts, whereas the fxed ponts dscussed earler wll smply be shfted by order ɛ n poston and wll retan ther stablty or nstablty. Thus the smple zero n the velocty functon s structurally stable. Note that structual stablty s qute a dfferent noton from stablty of the fxed pont. In ths dscusson of stablty n frst order dynamcal systems, we see that genercally the stable fxed ponts occur where the velocty functon decreases through zero, whle the unstable ponts are where t ncreases through zero. 15 Note that ths s not a one-dmensonal Newtonan system, whch s a two dmensonal η =(x, p) dynamcal system.

13 1.4. PHASE SPACE 7 Thus genercally the fxed ponts wll alternate n stablty, dvdng the phase lne nto open ntervals whch are each nvarant sets of states, wth the ponts n a gven nterval flowng ether to the left or to the rght, but never leavng the open nterval. The state never reaches the stable fxed pont because the tme t = dη/v (η) (1/c) dη/(η η 0 ) dverges. On the other hand, n the case V (η) =cη, a system startng at η 0 at t = 0 has a moton gven by η =(η0 1 ct) 1, whch runs off to nfnty as t 1/η 0 c. Thus the soluton termnates at t =1/η 0 c, and makes no sense thereafter. Ths form of soluton s called termnatng moton. For hgher order dynamcal systems, the d equatons V ( η) = 0 requred for a fxed pont wll genercally determne the d varables η j, so the generc form of the velocty feld near a fxed pont η 0 s V ( η) = j M j (η j η 0j ) wth a nonsngular matrx M. The stablty of the flow wll be determned by ths d-dmensonal square matrx M. Genercally the egenvalue equaton, a d th order polynomal n λ, wll have d dstnct solutons. Because M s a real matrx, the egenvalues must ether be real or come n complex conjugate pars. For the real case, whether the egenvalue s postve or negatve determnes the nstablty or stablty of the flow along the drecton of the egenvector. For a par of complex conjugate egenvalues λ = u + v and λ = u v, wth egenvectors e and e respectvely, we may descrbe the flow n the plane δ η = η η 0 = x( e + e )+y( e e ), so η = M δ η = x(λ e + λ e )+y(λ e λ e ) = (ux vy)( e + e )+(vx + uy)( e e ) so ( ) ( )( ) { ẋ u v x x = Ae =, or ut cos(vt + φ) ẏ v u y y = Ae ut sn(vt + φ). Thus we see that the moton sprals n towards the fxed pont f u s negatve, and sprals away from the fxed pont f u s postve. Stablty n these drectons s determned by the sgn of the real part of the egenvalue. In general, then, stablty n each subspace around the fxed pont η 0 depends on the sgn of the real part of the egenvalue. If all the real parts are negatve, the system wll flow from anywhere n some neghborhood of η 0 towards the fxed pont, so lm t η(t) = η 0 provded we start n that neghborhood. Then η 0 s an attractor and s a strongly stable fxed pont. On the other hand, f some of the egenvalues have postve real parts, there are unstable drectons. Startng from a generc pont n any neghborhood 8 CHAPTER 1. PARTICLE KINEMATICS of η 0, the moton wll eventually flow out along an unstable drecton, and the fxed pont s consdered unstable, although there may be subspaces along whch the flow may be nto η 0. An example s the lne x = y n the hyperbolc fxed pont case shown n Fgure 1.. Some examples of two dmensonal flows n the neghborhood of a generc fxed pont are shown n Fgure 1.. Note that none of these descrbe the fxed pont of the undamped harmonc oscllator of Fgure 1.1. We have dscussed generc stuatons as f the velocty feld were chosen arbtrarly from the set of all smooth vector functons, but n fact Newtonan mechancs mposes constrants on the velocty felds n many stuatons, n partcular f there are conserved quanttes. ẋ = x + y, ẏ = x y. Strongly stable spral pont. λ = 1 ±. ẋ = 3x y, ẏ = x 3y. Strongly stable fxed pont, λ = 1,. ẋ = 3x + y, ẏ = x +3y. Unstable fxed pont, λ =1,. ẋ = x 3y, ẏ = 3x y. Hyperbolc fxed pont, λ =, 1. Fgure 1.: Four generc fxed ponts for a second order dynamcal system. Effect of conserved quanttes on the flow If the system has a conserved quantty Q(q, p) whch s a functon on phase space only, and not of tme, the flow n phase space s consderably changed. Ths s because the equatons Q(q, p) = K gves a set of subsurfaces or contours n phase space, and the system s confned to stay on whchever contour t s on ntally. Unless ths conserved quantty s a trval functon,

14 1.4. PHASE SPACE 9 30 CHAPTER 1. PARTICLE KINEMATICS.e. constant, n the vcnty of a fxed pont, t s not possble for all ponts to flow nto the fxed pont, and thus t s not strongly stable. For the case of a sngle partcle n a potental, the total energy E = p /m + U( r) s conserved, and so the moton of the system s confned to one surface of a gven energy. As p/m s part of the velocty functon, a fxed pont must have p = 0. The vanshng of the other half of the velocty feld gves U( r 0 ) = 0, whch s the condton for a statonary pont of the potental energy, and for the force to vansh. If ths pont s a maxmum or a saddle of U, the moton along a descendng path wll be unstable. If the fxed pont s a mnmum of the potental, the regon E( r, p) <E( r 0, 0) + ɛ, for suffcently small ɛ, gves a neghborhood around η 0 =( r 0, 0) to whch the moton s confned f t starts wthn ths regon. Such a fxed pont s called stable 16, but t s not strongly stable, as the flow does not settle down to η 0. Ths s the stuaton we saw for the undamped harmonc oscllator. For that stuaton F = kx, so the potental energy may be taken to be U(x) = 0 x kx dx = 1 kx, As an example of a conservatve system wth both stable and unstable fxed ponts, consder a partcle n one dmenson wth a cubc potental U(x) = ax bx 3, as shown n Fg There s a stable equlbrum at x s = 0 and an unstable one at x u = a/3b. Each has an assocated fxed pont n phase space, an ellptc fxed pont η s = (x s, 0) and a hyperbolc fxed pont η u =(x u, 0). The velocty feld n phase space and several possble orbts are shown. Near the stable equlbrum, the trajectores are approxmately ellpses, as they were for the harmonc oscllator, but for larger energes they begn to feel the asymmetry of the potental, and the orbts become egg-shaped U U(x) p x Fgure 1.3. Moton n a cubc potental. x and so the total energy E = p /m + 1 kx s conserved. The curves of constant E n phase space are ellpses, and each moton orbts the approprate ellpse, as shown n Fg. 1.1 for the undamped oscllator. Ths contrasts to the case of the damped oscllator, for whch there s no conserved energy, and for whch the orgn s a strongly stable fxed pont. 16 A fxed pont s stable f t s n arbtrarty small neghborhoods, each wth the property that f the system s n that neghborhood at one tme, t remans n t at all later tmes. If the system has total energy precsely U(x u ), the contour lne crosses tself. Ths contour actually conssts of three separate orbts. One starts at t at x = x u, completes one trp though the potental well, and returns as t + to x = x u. The other two are orbts whch go from x = x u to x =, one ncomng and one outgong. For E>U(x u ), all the orbts start and end at x = +. Note that genercally the orbts deform contnuously as the energy vares, but at E = U(x u ) ths s not the case the character of the orbt changes as E passes through U(x u ). An orbt wth ths crtcal value of the energy s called a separatrx, as t separates regons n phase space where the orbts have dfferent qualtatve characterstcs. Qute generally hyperbolc fxed ponts are at the ends of separatrces. In our case the contour E = U(x u ) conssts of four nvarant sets of states, one of whch s the pont η u tself, and the other three are the orbts whch are