. The kinetic energy of this system is T = T i. m i. Now let s consider how the kinetic energy of the system changes in time. Assuming each.
|
|
- Charles Walters
- 5 years ago
- Views:
Transcription
1 Chapter 2 Systems of Partcles 2. Work-Energy Theorem Consder a system of many partcles, wth postons r and veloctes ṙ. The knetc energy of ths system s T = T = 2 mṙ2. 2. Now let s consder how the knetc energy of the system changes n tme. Assumng each m s tme-ndependent, we have dt dt = m ṙ r. 2.2 Here, we ve used the relaton d A 2 = 2A da dt dt. 2.3 We now nvoke Newton s 2nd Law, m r = F, to wrte eqn. 2.2 as T = F ṙ. We ntegrate ths equaton from tme t A to t B : T B t B T A = dt t A t B dt dt = dt F ṙ W A B, 2.4 t A where W A B s the total work done on partcle durng ts moton from state A to state B, Clearly the total knetc energy s T = T and the total work done on all partcles s W A B = W A B. Eqn. 2.4 s known as the work-energy theorem. It says that In the evoluton of a mechancal system, the change n total knetc energy s equal to the total work done: T B T A = W A B.
2 2 CHAPTER 2. SYSTEMS OF PARTICLES Fgure 2.: Two paths jonng ponts A and B. 2.2 Conservatve and Nonconservatve Forces For the sake of smplcty, consder a sngle partcle wth knetc energy T = 2 mṙ2. The work done on the partcle durng ts mechancal evoluton s W A B = t B t A dt F v, 2.5 where v = ṙ. Ths s the most general expresson for the work done. If the force F depends only on the partcle s poston r, we may wrte dr = v dt, and then Consder now the force W A B = r B r A dr F r. 2.6 F r = K y ˆx + K 2 xŷ, 2.7 where K,2 are constants. Let s evaluate the work done along each of the two paths n fg. 2.: x B y B W I = K dx y A + K 2 dy x B = K y A x B x A + K 2 x B y B y A 2.8 x A y A x B y B W II = K dx y B + K 2 dy x A = K y B x B x A + K 2 x A y B y A. 2.9 x A y A
3 2.2. CONSERVATIVE AND NONCONSERVATIVE FORCES 3 Note that n general W I W II. Thus, f we start at pont A, the knetc energy at pont B wll depend on the path taken, snce the work done s path-dependent. The dfference between the work done along the two paths s W I W II = K 2 K x B x A y B y A. 2.0 Thus, we see that f K = K 2, the work s the same for the two paths. In fact, f K = K 2, the work would be path-ndependent, and would depend only on the endponts. Ths s true for any path, and not just pecewse lnear paths of the type depcted n fg. 2.. The reason for ths s Stokes theorem: dl F = ds ˆn F. 2. C C Here, C s a connected regon n three-dmensonal space, C s mathematcal notaton for the boundary of C, whch s a closed path, ds s the scalar dfferental area element, ˆn s the unt normal to that dfferental area element, and F s the curl of F : ˆx ŷ ẑ F = det x y z F x F y F z Fz = y F y ˆx + z Fx z F z Fy ŷ + x x F x ẑ. 2.2 y For the force under consderaton, F r = K y ˆx + K 2 xŷ, the curl s F = K 2 K ẑ, 2.3 whch s a constant. The RHS of eqn. 2. s then smply proportonal to the area enclosed by C. When we compute the work dfference n eqn. 2.0, we evaluate the ntegral C dl F along the path γ II γ I, whch s to say path I followed by the nverse of path II. In ths case, ˆn = ẑ and the ntegral of ˆn F over the rectangle C s gven by the RHS of eqn When F = 0 everywhere n space, we can always wrte F = U, where Ur s the potental energy. Such forces are called conservatve forces because the total energy of the system, E = T + U, s then conserved durng ts moton. We can see ths by evaluatng the work done, r B W A B = dr F r r A r B = dr U r A = Ur A Ur B. 2.4 If C s multply connected, then C s a set of closed paths. For example, f C s an annulus, C s two crcles, correspondng to the nner and outer boundares of the annulus.
4 4 CHAPTER 2. SYSTEMS OF PARTICLES The work-energy theorem then gves T B T A = Ur A Ur B, 2.5 whch says E B = T B + Ur B = T A + Ur A = E A. 2.6 Thus, the total energy E = T + U s conserved Example : ntegratng F = U If F = 0, we can compute Ur by ntegratng, vz. Ur = U0 r 0 dr F r. 2.7 The ntegral does not depend on the path chosen connectng 0 and r. For example, we can take x,0,0 x,y,0 x,y,z Ux,y,z = U0,0,0 dx F x x,0,0 dy F y x,y,0 dz F z x,y,z ,0,0 x,0,0 z,y,0 The constant U0,0,0 s arbtrary and mpossble to determne from F alone. As an example, consder the force F r = ky ˆx kxŷ 4bz 3 ẑ, 2.9 where k and b are constants. We have F x = Fz y F y = z F y = Fx z F z = x F z = Fy x F x = 0, 2.22 y so F = 0 and F must be expressble as F = U. Integratng usng eqn. 2.8, we have x,0,0 x,y,0 x,y,z Ux,y,z = U0,0,0 + dx k 0 + dy kxy + dz 4bz ,0,0 x,0,0 z,y,0 = U0,0,0 + kxy + bz
5 2.3. CONSERVATIVE FORCES IN MANY PARTICLE SYSTEMS 5 Another approach s to ntegrate the partal dfferental equaton U = F. Ths s n fact three equatons, and we shall need all of them to obtan the correct answer. We start wth the ˆx-component, = ky x Integratng, we obtan Ux,y,z = kxy + fy,z, 2.26 where fy, z s at ths pont an arbtrary functon of y and z. The mportant thng s that t has no x-dependence, so f/ x = 0. Next, we have y Fnally, the z-component ntegrates to yeld We now equate the frst two expressons: = kx = Ux,y,z = kxy + gx,z z = 4bz3 = Ux,y,z = bz 4 + hx,y kxy + fy,z = kxy + gx,z Subtractng kxy from each sde, we obtan the equaton fy, z = gx, z. Snce the LHS s ndependent of x and the RHS s ndependent of y, we must have fy,z = gx,z = qz, 2.30 where qz s some unknown functon of z. But now we nvoke the fnal equaton, to obtan bz 4 + hx,y = kxy + qz. 2.3 The only possble soluton s hx,y = C + kxy and qz = C + bz 4, where C s a constant. Therefore, Ux,y,z = C + kxy + bz Note that t would be very wrong to ntegrate / x = ky and obtan Ux,y,z = kxy + C, where C s a constant. As we ve seen, the constant of ntegraton we obtan upon ntegratng ths frst order PDE s n fact a functon of y and z. The fact that fy,z carres no explct x dependence means that f/ x = 0, so by constructon U = kxy + fy, z s a soluton to the PDE / x = ky, for any arbtrary functon fy,z. 2.3 Conservatve Forces n Many Partcle Systems T = U = 2 m ṙ V r + <j v r r j. 2.34
6 6 CHAPTER 2. SYSTEMS OF PARTICLES Here, V r s the external or one-body potental, and vr r s the nterpartcle potental, whch we assume to be central, dependng only on the dstance between any par of partcles. The equatons of moton are wth m r = F ext + F nt, 2.35 F ext = V r 2.36 r F nt = v r r j F nt j r j j Here, F nt j s the force exerted on partcle by partcle j: F nt j = v r r j = r r j r r r j v r r j Note that F nt j = F nt j, otherwse known as Newton s Thrd Law. It s convenent to abbrevate r j r r j, n whch case we may wrte the nterpartcle force as F nt j = ˆr j v r j Lnear and Angular Momentum Consder now the total momentum of the system, P = p. Its rate of change s dp dt = ṗ = F ext + F nt j +F nt j =0 {}}{ j F nt j = F ext tot, 2.40 snce the sum over all nternal forces cancels as a result of Newton s Thrd Law. We wrte P = m ṙ = MṘ 2.4 M = m total mass 2.42 R = m r center-of-mass m Next, consder the total angular momentum, L = r p = m r ṙ. 2.44
7 2.4. LINEAR AND ANGULAR MOMENTUM 7 The rate of change of L s then dl dt = = { m ṙ ṙ + m r r } r F ext + j r F nt j = r F ext + 2 j r j F nt j =0 { }}{ r r j F nt j = N ext tot Fnally, t s useful to establsh the result T = 2 m ṙ 2 = 2 MṘ2 + 2 m ṙ Ṙ 2, 2.46 whch says that the knetc energy may be wrtten as a sum of two terms, those beng the knetc energy of the center-of-mass moton, and the knetc energy of the partcles relatve to the center-of-mass. Recall the work-energy theorem for conservatve systems, fnal fnal fnal 0 = de = dt + du ntal ntal ntal = T B T A dr F, 2.47 whch s to say T = T B T A = dr F = U In other words, the total energy E = T + U s conserved: E = 2 m ṙ2 + V r + <j v r r j Note that for contnuous systems, we replace sums by ntegrals over a mass dstrbuton, vz. m φ r d 3 r ρrφr, 2.50 where ρr s the mass densty, and φr s any functon.
8 8 CHAPTER 2. SYSTEMS OF PARTICLES 2.5 Scalng of Solutons for Homogeneous Potentals 2.5. Euler s theorem for homogeneous functons In certan cases of nterest, the potental s a homogeneous functon of the coordnates. Ths means U λr,...,λr N = λ k U r,...,r N. 2.5 Here, k s the degree of homogenety of U. Famlar examples nclude gravty, and the harmonc oscllator, U r,...,r N = G U q,...,q n = 2 <j m m j r r j ; k =, 2.52 σ,σ V σσ q σ q σ ; k = The sum of two homogeneous functons s tself homogeneous only f the component functons themselves are of the same degree of homogenety. Homogeneous functons obey a specal result known as Euler s Theorem, whch we now prove. Suppose a multvarable functon Hx,...,x n s homogeneous: Hλx,...,λx n = λ k Hx,...,x n Then d H n H λx dλ,...,λx n = x = k H 2.55 x λ= = Scaled equatons of moton Now suppose the we rescale dstances and tmes, defnng r = α r, t = β t Then The force F s gven by dr dt = α β d r d t F = r U r,...,r N, = α r αk U r,..., r N d 2 r dt 2 = α β 2 d 2 r d t = α k F. 2.58
9 2.5. SCALING OF SOLUTIONS FOR HOMOGENEOUS POTENTIALS 9 Thus, Newton s 2nd Law says If we choose β such that We now demand α β 2 m d 2 r d t 2 = αk F α β 2 = αk β = α 2 k, 2.60 then the equaton of moton s nvarant under the rescalng transformaton! Ths means that f rt s a soluton to the equatons of moton, then so s α r α 2 k t. Ths gves us an entre one-parameter famly of solutons, for all real postve α. If rt s perodc wth perod T, the r t;α s perodc wth perod T = α 2 k T. Thus, T L 2 k =. 2.6 T L Here, α = L /L s the rato of length scales. Veloctes, energes and angular momenta scale accordngly: [ v ] = L T [ E ] = ML 2 T 2 [ L ] = ML 2 T v / v = L T L T = α 2 k 2.62 E E = L L = L 2 / T 2 = α k 2.63 L T L 2 / T L T = α+ 2 k As examples, consder: Harmonc Oscllator : Here k = 2 and therefore q σ t q σ t;α = α q σ t Thus, rescalng lengths alone gves another soluton. Kepler Problem : Ths s gravty, for whch k =. Thus, rt rt;α = α r α 3/2 t Thus, r 3 t 2,.e. L 3 T 2 =, 2.67 also known as Kepler s Thrd Law. L T
10 0 CHAPTER 2. SYSTEMS OF PARTICLES 2.6 Appendx I : Curvlnear Orthogonal Coordnates The standard cartesan coordnates are {x,...,x d }, where d s the dmenson of space. Consder a dfferent set of coordnates, {q,...,q d }, whch are related to the orgnal coordnates x µ va the d equatons q µ = q µ x,...,x d In general these are nonlnear equatons. Let ê 0 = ˆx be the Cartesan set of orthonormal unt vectors, and defne ê µ to be the unt vector perpendcular to the surface dq µ = 0. A dfferental change n poston can now be descrbed n both coordnate systems: d d ds = ê 0 dx = ê µ h µ qdq µ, 2.69 = where each h µ q s an as yet unknown functon of all the components q ν. Fndng the coeffcent of dq µ then gves h µ qê µ = d = µ= x q µ ê 0 ê µ = d M µ ê 0, 2.70 where M µ q = x. 2.7 h µ q q µ The dot product of unt vectors n the new coordnate system s then ê µ ê ν = MM t µν = d x x h µ qh ν q q µ q ν The condton that the new bass be orthonormal s then d x x = h 2 q µ q µqδ µν ν Ths gves us the relaton Note that = = = h µ q = d 2 x q µ ds 2 = = d h 2 µ qdq µ µ= For general coordnate systems, whch are not necessarly orthogonal, we have d ds 2 = g µν qdq µ dq ν, 2.76 µ,ν= where g µν q s a real, symmetrc, postve defnte matrx called the metrc tensor.
11 2.6. APPENDIX I : CURVILINEAR ORTHOGONAL COORDINATES Fgure 2.2: Volume element Ω for computng dvergences Example : sphercal coordnates Consder sphercal coordnates ρ,θ,φ: x = ρ sn θ cos φ, y = ρ sn θ snφ, z = ρ cos θ It s now a smple matter to derve the results h 2 ρ =, h 2 θ = ρ2, h 2 φ = ρ2 sn 2 θ Thus, ds = ˆρ dρ + ρ ˆθ dθ + ρ sn θ ˆφ dφ Vector calculus : grad, dv, curl Here we restrct our attenton to d = 3. The gradent U of a functon Uq s defned by du = q dq + q 2 dq 2 + q 3 dq 3 U ds Thus, = ê h q q + ê2 h 2 q q 2 + ê3 h 3 q q For the dvergence, we use the dvergence theorem, and we appeal to fg. 2.2: dv A = ds ˆn A, 2.82 Ω Ω
12 2 CHAPTER 2. SYSTEMS OF PARTICLES where Ω s a regon of three-dmensonal space and Ω s ts closed two-dmensonal boundary. The LHS of ths equaton s The RHS s LHS = A h dq h 2 dq 2 h 3 dq q +dq q RHS = A h 2 h 3 dq 2 dq 3 + A 2 h h 2+dq 2 q 3 dq dq 3 + A 3 h h +dq 3 2 dq dq 2 q q 2 q [ ] 3 = A h q 2 h 3 + A2 h q h 3 + A3 h 2 q h 2 dq dq 2 dq We therefore conclude A = h h 2 h 3 [ ] A h q 2 h 3 + A2 h q h 3 + A3 h 2 q h To obtan the curl A, we use Stokes theorem agan, ds ˆn A = dl A, 2.86 Σ where Σ s a two-dmensonal regon of space and Σ s ts one-dmensonal boundary. Now consder a dfferental surface element satsfyng dq = 0,.e. a rectangle of sde lengths h 2 dq 2 and h 3 dq 3. The LHS of the above equaton s The RHS s Σ LHS = ê Ah 2 dq 2 h 3 dq q 2 +dq 2 q RHS = A 3 h 3 dq 3 A 2 h 3+dq 3 2 dq 2 q 2 q [ 3 ] = A3 h 3 A2 h q 2 q 2 dq 2 dq Therefore A = h3 A 3 h 2 A h 2 h 3 q 2 q 3 Ths s one component of the full result h ê h 2 ê 2 h 3 ê 3 A = det q h h 2 h q 2 q 3 2 h A h 2 A 2 h 3 A 3 The Laplacan of a scalar functon U s gven by 2 U = U { h2 h 3 = h h 2 h 3 q h + h h 3 q q 2 h 2 + h h 2 q 2 q 3 h }. 2.9 q 3
13 2.7. COMMON CURVILINEAR ORTHOGONAL SYSTEMS Common curvlnear orthogonal systems 2.7. Rectangular coordnates In rectangular coordnates x,y,z, we have h x = h y = h z = Thus and the velocty squared s ds = ˆxdx + ŷ dy + ẑ dz 2.93 ṡ 2 = ẋ 2 + ẏ 2 + ż The gradent s The dvergence s The curl s A = U = ˆx x + ŷ y + ẑ z A = A x x + A y y + A z z Az y A y Ax ˆx + z z A z Ay ŷ + x x A x ẑ y The Laplacan s 2 U = 2 U x U y U z Cylndrcal coordnates In cylndrcal coordnates ρ, φ, z, we have ˆρ = ˆx cos φ + ŷ snφ ˆx = ˆρ cos φ ˆφ snφ dˆρ = ˆφ dφ 2.99 ˆφ = ˆx sn φ + ŷ cos φ ŷ = ˆρ sn φ + ˆφ cos φ d ˆφ = ˆρdφ The metrc s gven n terms of h ρ =, h φ = ρ, h z =. 2.0 Thus and the velocty squared s ds = ˆρ dρ + ˆφ ρdφ + ẑ dz 2.02 ṡ 2 = ρ 2 + ρ 2 φ2 + ż
14 4 CHAPTER 2. SYSTEMS OF PARTICLES The gradent s The dvergence s The curl s A = U = ˆρ ρ + ˆφ ρ A = ρ ρa ρ ρ φ + ẑ z ρ A φ φ + A z z A z ρ φ A φ Aρ ˆρ + z z A z ρa φ ˆφ + ρ ρ ρ ρ A ρ ẑ φ The Laplacan s 2 U = ρ ρ + 2 U ρ ρ ρ 2 φ U z Sphercal coordnates In sphercal coordnates r,θ,φ, we have ˆr = ˆxsn θ cos φ + ŷ sn θ sn φ + ẑ sn θ 2.08 ˆθ = ˆxcos θ cos φ + ŷ cos θ sn φ ẑ cos θ 2.09 ˆφ = ˆxsn φ + ŷ cos φ, 2.0 for whch ˆr ˆθ = ˆφ, ˆθ ˆφ = ˆr, ˆφ ˆr = ˆθ. 2. The nverse s ˆx = ˆr sn θ cos φ + ˆθ cos θ cos φ ˆφ sn φ 2.2 ŷ = ˆr sn θ sn φ + ˆθ cos θ sn φ + ˆφ cos φ 2.3 ẑ = ˆr cos θ ˆθ snθ. 2.4 The dfferental relatons are dˆr = ˆθ dθ + sn θ ˆφ dφ 2.5 dˆθ = ˆr dθ + cos θ ˆφ dφ 2.6 d ˆφ = sn θ ˆr + cos θ ˆθ dφ 2.7 The metrc s gven n terms of h r =, h θ = r, h φ = r sn θ. 2.8 Thus ds = ˆr dr + ˆθ r dθ + ˆφr sn θ dφ 2.9
15 2.7. COMMON CURVILINEAR ORTHOGONAL SYSTEMS 5 and the velocty squared s ṡ 2 = ṙ 2 + r 2 θ2 + r 2 sn 2 θ φ The gradent s The dvergence s U = ˆr ρ + ˆθ r θ + ˆφ r sn θ φ. 2.2 The curl s The Laplacan s A = r 2 r 2 A r r A = r sn θ 2 U = r 2 r sn θ Aφ θ + raθ r r r 2 + r + sn θ A θ + A φ r sn θ θ r sn θ φ r 2 sn θ A θ φ A r θ A r sn θ φ ra φ ˆθ r ˆr + r ˆφ snθ + θ θ 2 U r 2 sn 2 θ φ Knetc energy Note the form of the knetc energy of a pont partcle: ds 2 T = 2 m = dt 2 m ẋ 2 + ẏ 2 + ż 2 3D Cartesan 2.25 = 2 m ρ 2 + ρ 2 φ2 2D polar 2.26 = 2 m ρ 2 + ρ 2 φ2 + ż 2 3D cylndrcal 2.27 = 2 m ṙ 2 + r 2 θ2 + r 2 sn 2 θ φ 2 3D polar. 2.28
(δr i ) 2. V i. r i 2,
Cartesan coordnates r, = 1, 2,... D for Eucldean space. Dstance by Pythagoras: (δs 2 = (δr 2. Unt vectors ê, dsplacement r = r ê Felds are functons of poston, or of r or of {r }. Scalar felds Φ( r, Vector
More information12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationTensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q
For orthogonal curvlnear coordnates, eˆ grad a a= ( aˆ ˆ e). h q (98) Expandng the dervatve, we have, eˆ aˆ ˆ e a= ˆ ˆ a h e + q q 1 aˆ ˆ ˆ a e = ee ˆˆ ˆ + e. h q h q Now expandng eˆ / q (some of the detals
More informationCelestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestial-mechanics - J. Hedberg
PHY 454 - celestal-mechancs - J. Hedberg - 207 Celestal Mechancs. Basc Orbts. Why crcles? 2. Tycho Brahe 3. Kepler 4. 3 laws of orbtng bodes 2. Newtonan Mechancs 3. Newton's Laws. Law of Gravtaton 2. The
More informationQuantum Mechanics I Problem set No.1
Quantum Mechancs I Problem set No.1 Septembe0, 2017 1 The Least Acton Prncple The acton reads S = d t L(q, q) (1) accordng to the least (extremal) acton prncple, the varaton of acton s zero 0 = δs = t
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationPoisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationLagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013
Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More information1. Review of Mechanics Newton s Laws
. Revew of Mechancs.. Newton s Laws Moton of partcles. Let the poston of the partcle be gven by r. We can always express ths n Cartesan coordnates: r = xˆx + yŷ + zẑ, () where we wll always use ˆ (crcumflex)
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
More informationPHYS 705: Classical Mechanics. Hamilton-Jacobi Equation
1 PHYS 705: Classcal Mechancs Hamlton-Jacob Equaton Hamlton-Jacob Equaton There s also a very elegant relaton between the Hamltonan Formulaton of Mechancs and Quantum Mechancs. To do that, we need to derve
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More information10. Canonical Transformations Michael Fowler
10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst
More informationcoordinates. Then, the position vectors are described by
Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,
More informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
More informationTHEOREMS OF QUANTUM MECHANICS
THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat many-electron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationFrom Biot-Savart Law to Divergence of B (1)
From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to
More informationCausal Diamonds. M. Aghili, L. Bombelli, B. Pilgrim
Causal Damonds M. Aghl, L. Bombell, B. Plgrm Introducton The correcton to volume of a causal nterval due to curvature of spacetme has been done by Myrhem [] and recently by Gbbons & Solodukhn [] and later
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More information8.592J: Solutions for Assignment 7 Spring 2005
8.59J: Solutons for Assgnment 7 Sprng 5 Problem 1 (a) A flament of length l can be created by addton of a monomer to one of length l 1 (at rate a) or removal of a monomer from a flament of length l + 1
More informationPhysics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2
Physcs 607 Exam 1 Please be well-organzed, and show all sgnfcant steps clearly n all problems. You are graded on your wor, so please do not just wrte down answers wth no explanaton! Do all your wor on
More informationWeek 9 Chapter 10 Section 1-5
Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informationare called the contravariant components of the vector a and the a i are called the covariant components of the vector a.
Non-Cartesan Coordnates The poston of an arbtrary pont P n space may be expressed n terms of the three curvlnear coordnates u 1,u,u 3. If r(u 1,u,u 3 ) s the poston vector of the pont P, at every such
More informationHomework Notes Week 7
Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O
More informationCHAPTER 10 ROTATIONAL MOTION
CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n x-y plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationTHE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George
More information8.022 (E&M) Lecture 4
Topcs: 8.0 (E&M) Lecture 4 More applcatons of vector calculus to electrostatcs: Laplacan: Posson and Laplace equaton url: concept and applcatons to electrostatcs Introducton to conductors Last tme Electrc
More information11. Dynamics in Rotating Frames of Reference
Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons
More informationIn this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
More informationNotes on Analytical Dynamics
Notes on Analytcal Dynamcs Jan Peters & Mchael Mstry October 7, 004 Newtonan Mechancs Basc Asssumptons and Newtons Laws Lonely pontmasses wth postve mass Newtons st: Constant velocty v n an nertal frame
More informationThree views of mechanics
Three vews of mechancs John Hubbard, n L. Gross s course February 1, 211 1 Introducton A mechancal system s manfold wth a Remannan metrc K : T M R called knetc energy and a functon V : M R called potental
More informationMechanics Physics 151
Mechancs Physcs 5 Lecture 0 Canoncal Transformatons (Chapter 9) What We Dd Last Tme Hamlton s Prncple n the Hamltonan formalsm Dervaton was smple δi δ p H(, p, t) = 0 Adonal end-pont constrants δ t ( )
More informationSolutions to Problem Set 6
Solutons to Problem Set 6 Problem 6. (Resdue theory) a) Problem 4.7.7 Boas. n ths problem we wll solve ths ntegral: x sn x x + 4x + 5 dx: To solve ths usng the resdue theorem, we study ths complex ntegral:
More informationPhysics 207: Lecture 20. Today s Agenda Homework for Monday
Physcs 207: Lecture 20 Today s Agenda Homework for Monday Recap: Systems of Partcles Center of mass Velocty and acceleraton of the center of mass Dynamcs of the center of mass Lnear Momentum Example problems
More informationWork is the change in energy of a system (neglecting heat transfer). To examine what could
Work Work s the change n energy o a system (neglectng heat transer). To eamne what could cause work, let s look at the dmensons o energy: L ML E M L F L so T T dmensonally energy s equal to a orce tmes
More informationLAGRANGIAN MECHANICS
LAGRANGIAN MECHANICS Generalzed Coordnates State of system of N partcles (Newtonan vew): PE, KE, Momentum, L calculated from m, r, ṙ Subscrpt covers: 1) partcles N 2) dmensons 2, 3, etc. PE U r = U x 1,
More informationCalculus of Variations Basics
Chapter 1 Calculus of Varatons Bascs 1.1 Varaton of a General Functonal In ths chapter, we derve the general formula for the varaton of a functonal of the form J [y 1,y 2,,y n ] F x,y 1,y 2,,y n,y 1,y
More informationLecture 20: Noether s Theorem
Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external
More informationIntegrals and Invariants of Euler-Lagrange Equations
Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
More information= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.
Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationError Bars in both X and Y
Error Bars n both X and Y Wrong ways to ft a lne : 1. y(x) a x +b (σ x 0). x(y) c y + d (σ y 0) 3. splt dfference between 1 and. Example: Prmordal He abundance: Extrapolate ft lne to [ O / H ] 0. [ He
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More informationLagrangian Field Theory
Lagrangan Feld Theory Adam Lott PHY 391 Aprl 6, 017 1 Introducton Ths paper s a summary of Chapter of Mandl and Shaw s Quantum Feld Theory [1]. The frst thng to do s to fx the notaton. For the most part,
More informationCHAPTER 5: Lie Differentiation and Angular Momentum
CHAPTER 5: Le Dfferentaton and Angular Momentum Jose G. Vargas 1 Le dfferentaton Kähler s theory of angular momentum s a specalzaton of hs approach to Le dfferentaton. We could deal wth the former drectly,
More informationSpin-rotation coupling of the angularly accelerated rigid body
Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationModelli Clamfim Equazioni differenziali 7 ottobre 2013
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 7 ottobre 2013 professor Danele Rtell danele.rtell@unbo.t 1/18? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationLecture Note 3. Eshelby s Inclusion II
ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte
More information10/23/2003 PHY Lecture 14R 1
Announcements. Remember -- Tuesday, Oct. 8 th, 9:30 AM Second exam (coverng Chapters 9-4 of HRW) Brng the followng: a) equaton sheet b) Calculator c) Pencl d) Clear head e) Note: If you have kept up wth
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationModule 1 : The equation of continuity. Lecture 1: Equation of Continuity
1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum
More informationLecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2
P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The non-relatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons
More informationReport on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationStudy Guide For Exam Two
Study Gude For Exam Two Physcs 2210 Albretsen Updated: 08/02/2018 All Other Prevous Study Gudes Modules 01-06 Module 07 Work Work done by a constant force F over a dstance s : Work done by varyng force
More informationChapter 8. Potential Energy and Conservation of Energy
Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and non-conservatve forces Mechancal Energy Conservaton of Mechancal
More informationPES 1120 Spring 2014, Spendier Lecture 6/Page 1
PES 110 Sprng 014, Spender Lecture 6/Page 1 Lecture today: Chapter 1) Electrc feld due to charge dstrbutons -> charged rod -> charged rng We ntroduced the electrc feld, E. I defned t as an nvsble aura
More informationCIVL 8/7117 Chapter 10 - Isoparametric Formulation 42/56
CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 4/56 Newton-Cotes Example Usng the Newton-Cotes method wth = ntervals (n = 3 samplng ponts), evaluate the ntegrals: x x cos dx 3 x x dx 3 x x dx 4.3333333
More information1 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 -Davd Klenfeld - Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys
More informationPhysics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.
Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More information9 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 - Chapter 9R -Davd Klenfeld - Fall 2005 9 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys a set
More informationMechanics Physics 151
Mechancs hyscs 151 Lecture Canoncal Transformatons (Chater 9) What We Dd Last Tme Drect Condtons Q j Q j = = j, Q, j, Q, Necessary and suffcent j j for Canoncal Transf. = = j Q, Q, j Q, Q, Infntesmal CT
More informationPhysics 114 Exam 3 Spring Name:
Physcs 114 Exam 3 Sprng 015 Name: For gradng purposes (do not wrte here): Queston 1. 1... 3. 3. Problem 4. Answer each of the followng questons. Ponts for each queston are ndcated n red. Unless otherwse
More informationPhysics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints
Physcs 106a, Caltech 11 October, 2018 Lecture 4: Constrants, Vrtual Work, etc. Many, f not all, dynamcal problems we want to solve are constraned: not all of the possble 3 coordnates for M partcles (or
More informationMechanics Physics 151
Mechancs Physcs 5 Lecture 7 Specal Relatvty (Chapter 7) What We Dd Last Tme Worked on relatvstc knematcs Essental tool for epermental physcs Basc technques are easy: Defne all 4 vectors Calculate c-o-m
More informationFirst Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.
Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More informationSo far: simple (planar) geometries
Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector
More informationLinear Momentum. Center of Mass.
Lecture 6 Chapter 9 Physcs I 03.3.04 Lnear omentum. Center of ass. Course webste: http://faculty.uml.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcssprng.html
More informationσ τ τ τ σ τ τ τ σ Review Chapter Four States of Stress Part Three Review Review
Chapter Four States of Stress Part Three When makng your choce n lfe, do not neglect to lve. Samuel Johnson Revew When we use matrx notaton to show the stresses on an element The rows represent the axs
More informationWeek 8: Chapter 9. Linear Momentum. Newton Law and Momentum. Linear Momentum, cont. Conservation of Linear Momentum. Conservation of Momentum, 2
Lnear omentum Week 8: Chapter 9 Lnear omentum and Collsons The lnear momentum of a partcle, or an object that can be modeled as a partcle, of mass m movng wth a velocty v s defned to be the product of
More informationImplicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the Runge-Kutta method one employs an teratve mplct technque. Ths s because the structure
More informationChapter 4 The Wave Equation
Chapter 4 The Wave Equaton Another classcal example of a hyperbolc PDE s a wave equaton. The wave equaton s a second-order lnear hyperbolc PDE that descrbes the propagaton of a varety of waves, such as
More informationModelli Clamfim Equazioni differenziali 22 settembre 2016
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 22 settembre 2016 professor Danele Rtell danele.rtell@unbo.t 1/22? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationNMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 48/58 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 48/58 7. Robot Dynamcs 7.5 The Equatons of Moton Gven that we wsh to fnd the path q(t (n jont space) whch mnmzes the energy
More informationTensor Smooth Length for SPH Modelling of High Speed Impact
Tensor Smooth Length for SPH Modellng of Hgh Speed Impact Roman Cherepanov and Alexander Gerasmov Insttute of Appled mathematcs and mechancs, Tomsk State Unversty 634050, Lenna av. 36, Tomsk, Russa RCherepanov82@gmal.com,Ger@npmm.tsu.ru
More informationψ = i c i u i c i a i b i u i = i b 0 0 b 0 0
Quantum Mechancs, Advanced Course FMFN/FYSN7 Solutons Sheet Soluton. Lets denote the two operators by  and ˆB, the set of egenstates by { u }, and the egenvalues as  u = a u and ˆB u = b u. Snce the
More informationDECOUPLING THEORY HW2
8.8 DECOUPLIG THEORY HW2 DOGHAO WAG DATE:OCT. 3 207 Problem We shall start by reformulatng the problem. Denote by δ S n the delta functon that s evenly dstrbuted at the n ) dmensonal unt sphere. As a temporal
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationWorkshop: Approximating energies and wave functions Quantum aspects of physical chemistry
Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department
More information