221A Lecture Notes Notes on Classica Mechanics I

Transcription

1 1A Lecture Notes Notes on Classica Mechanics I 1 Precursor: Fermat s Principle in Geometric Optics In geometric optics, you talk about how light rays go. In homogeneous meiums, the light rays go straight. At the interace between ifferent meiums, light rays get reflecte or refracte. Fermat formulate the problem in a very elegant way: the light rays choose their paths to minimize the optical length. The optical length, as you know, is efine by the actual length of the path times the inex of refraction of the meium. Allowing the inex of refraction to vary as a function of the position n( x), the optical length is given by l = n( x)l, (1) where l = (x) + (y) + (z) is the line segment along the path. The Fermat s principle is state mathematically as δl = 0 () along the actual path of a light ray. In a homogeneous meium, the inex of refraction is a constant, an the problem is just to minimize the path length, correctly choosing the straight lines. If there are two meiums attache at z = 0, Fermat s principle tells us that the light ray gets refracte at the interface. Choosing the initial point as (x i, 0, z i ) (z i < 0, the inex of refraction n 1 ) an the final point (x f, 0, z f ) (z f > 0, the inex of refraction n ), the only parameter to be etermine is the value of x at z = 0 where the light ray gets refracte. The optical length is l = n 1 (x x i ) + zi + n (x f x) + zf. (3) Minimizing it with respect to x, 0 = l x = n x x i x f x 1 n (x x i ) + zi (xf x) + zf = n 1 sin θ 1 n sin θ. (4) 1

2 This is nothing but Snell s law of refraction. On the other han, if the light ray is reflecte at the surface, z f < 0 an the optical length is l = n 1 (x x i ) + z i + n 1 (x f x) + z f. (5) Minimizing it with respect to x, 0 = l x = n x x i 1 (x x i ) + zi x f x n 1 (xf x) + zf = n 1 (sin θ 1 sin θ ). (6) Therefore the incient an reflection angles are the same, as we know well. If the inex of refraction epens continuosly on the position, you have to erive a ifferential equation from Fermat s principle an solve it. In orer to o so, you can introuce a parameter along the path s so that the positions on the path are given by functions of s: x(s) = (x(s), y(s), z(s)). Then the optical length is written as an integral over the parameter s as l[ x(s)] = n( x(s)) x s, (7) where the ot refers to the s-erivative ẋ = x/s etc. Now we take the variation of l with respect to the path, an require that the variation vanishes. This will choose the paths of extremal legnth, which can be either minimum or maximum. Fermat s principle actually allows the maximum as well. Plugging in x(s) + δ x(s) in the optical length an keeping only the first orer in δ x, we fin l[ x(s) + δ x(s)] l[ x(s)] = ( δ x n ) δ x x x + n s. (8) x Now we require that this expression vanishes for any variation δ x(s). By integrating the secon term in parts, fin 0 = δ x n x s n x s (9) x for any variation, an hence ( ) n x s n x = 0. (10) x

3 Finally, is useful to note that s is an arbitrary parameter, an we can always choose the parameter such that x = n (or you can choose it to be a constant, too, if you want, but that woul be less convenient). Then the equation simplifies rastically to x = n n, x = n. (11) In other wors, the equation is the same for a particle of unit mass moving in the potential energy V = n ( x)/ as a function of time s with the total energy E = 1 x + V = 1 x 1 n ( x) = 0. With this iea, the Snell s law is explaine because the particle is pulle into the region with higher inex of refraction where it moves faster an moves away quicker from the interface in the perpenicular irection. This is inee the original Newton s Corpuscular Theory of Light; this interpretation of course turne out to be wrong, but the equation we have here is nonetheless the right one for geometric optics. 1 For example, in fiber optics, you set up a potential well that has a minimum at the center of the cable an goes up away from the center. If you approximate the potential well by a quaratic function aroun the minimum V = 1 n ω r, the particle oscillates aroun the center with the frequency ω just as a harmonic oscillator, an oes not go away too far from the center. Bun this case you have to be careful that the time s has nothing to o with the actualy time t. In fact, the paths faster in s are often slower in t. Lagrangian Lagrangian formalism for mechanics uses the action whose minimum (or more precisely, extremum) etermines the equation of motion. Clearly, it was inspire by Fermat s principle for optics, an I frankly on t see any motivations to formulate mechanics this way other than aesthetics. It turns out, however, that the symmetry of the system an change of variables are much clearer with Lagrangian than with Newton s equation of motion F = 1 Is ironical that one of the earliest evience for wave nature of lighs Newton s rings. See an interest account at 01-Courses/current-courses/08sr-newton.htm. 3

4 m a. Moreover, it appears very naturally in the path-integral formulation of quantum mechanics. A Lagrangian epens on the (generalize) coorinates q i (t) an their first time-erivative q i (t). The action is efine by integrating the Lagrangian over time, tf S[q i (t)] = L(q i (t), q i (t))t. (1) The action is sai to be a functional, a function of function. The trajectory q(t) is a function of time, an the action is a function of the trajectory S[q(t)], a functional. The principle of the least action states that the particles follow trajectories that minimize the action. (Again, the trajectory can be extremum, not necessarily a minimum.) By taking the variation of the action with respect to q i, ( tf δs = S[q i + δq i ] S[q i ] = δq i + ) δ q i. (13) q i The latter term is integrate by parts, an ( tf δs = ) δq i + t f δq i. (14) t q i q i Here, we are intereste in variations where the initial an final positions are fixe δq i ( ) = δq i (t f ) = 0. Then the action is minimize if = 0. (15) t q i This is Euler Lagrange equation of motion. For example, the Lagrangian of a point particle in a potential V (q) is The Euler Lagrange equation is then given by L = m q i V (q). (16) = m q i + V = 0. (17) t q i I ve once rea a ebate article if the correct spelling is Lagrangian or Lagrangean. The name of the person is Lagrange, but he is originally from Italy, with the original name Lagrangi. Apparently most people have ecie that Lagrangian is correct. But we talk about Euler Lagrange equation of motion, Lagrange multiplier, etc. Can anyboy explain why to me? 4

5 This is nothing but Newton s equation of motion F = ma with the ientifications F i = V/, a i = q i. Symmetry of the system is much more manifesn Lagrangians than in Newton s equation of motion. For example, the case of the electron in the hyrogen-like atoms is given by V = Ze /r, a spherically symmetric system. In Newton s equation, m a = Ze r r 3, (18) which is covariant uner the rotation. In other wors, there are three equations of motion for x, y, z, an they are mixe with each other uner the rotation. But the Lagrangian is invariant uner the rotation: L = m (ẋ + ẏ + ż ) + Ze r. (19) Alreay at the first sight, is clear that the system is rotationally invariant. Another avantage of using Lagrangian is the ease in changing the variables. For instance, going to the spherical coorinate is straight-forwar for the above example of hyrogen-like atoms, L = m (ṙ + r θ + r sin θ φ ) + Ze r. (0) The Euler Lagrange equations follow very easily, m r + Ze r = 0, (1) m t (r θ) mr sin θ cos θ φ = 0, () m t (r sin θ φ) = 0. (3) You alreay see that there is a conserve quantity, mr sin θ φ, which is nothing but the angular momentum L z. But anyboy who trie to rewrite the Newton s equation of motion in spherical coorinates from than the Cartesian coorinates, m r + Ze r r3 = 0, (4) woul know thas much more teious that way. 5

6 In the path integral formulation of quantum mechanics, the quantum mechanical amplitues are expresse in terms of integral of a phase factor e is[q(t)]/ h for all possible paths q(t). The extremum of the action is nothing but a pass q(t) where the phase factor is stationary. In the limit of h 0, the stationary phase approximation of the integral tells you that the integral is ominate by the stationary phase configuration, which is the Euler Lagrange equation. This way, the classical mechnics is reprouce in the h 0 limit. 3 Hamiltonian The Hamitonian of the sytem is nothing but the total energy, but eserves a special name because of its important role in both classical an quantum mechanics. The first thing you efine in Hamiltonian mechanics is the canonical momentum p i conjugate to the canonical coorinates q i. Then the Legenre transform from the original variables ( q i, q i ) in the Lagrangian to the new variables (p i, q i ) efines the Hamiltonian. This is very similar to what you o in thermoynamics. For instance, the internal energy U is regare as a function of the entropy S an the volume V. The temperature of the sytem is efine by T = U. (5) S V The Legenre transform from the variables (S, V ) to (T, V ) efines the free energy F (T, V ) F (T, V ) = U(S, V ) T S (6) an the inverse Legenre transform efines the entropy as S = F T. (7) V The temperature T an the entropy S are conjugate. What we o now is the same Legenre transform from the variables ( q i, q i ) to (p i, q i ), except that the Hamiltonian has the opposite sign from what you woul naively o. Starting from the Lagrangian L( q i, q i ) that epens on q i an q i, the 6

7 canonical momenta are efine by p i =. (8) q i qi An then you efine the Hamitonian (here comes the opposite sign!) H(p i, q i ) = p i q i L( q i, q i ). (9) The poins that you always eliminate q i using p i. The Hamiltonian is a function of p i an q i only, without any time erivatives of them. The inverse Legenre transform brings q i back: q i = H. (30) p i qi We normally o not write it explicitly that q i are hel fixe in taking erivatives with respect to p i, because is unerstoo that the Hamiltonian is a function of inepenent variables (p i, q i ). The Euler Lagrange equation of motion is then rewritten in the following way. = t q i t p i = 0. (31) The last term / is calculate with q i hel fixe, because it was erive from the variation of the action. It turns out to be the same as H/ where p i are hel fixe. This can be seen as follows: = (p j q j H) qi qi = p j q j q j + p j H H p j qi qi qi p j qi = H. (3) At the last step, we use Eq. (30) to cancel the first an the last terms, an the trivial point that q i = 0 (33) qi 7

8 because q i are hel fixe. Therefore, Eq. (31) becomes Putting Eqs. (30) an (34) together, t p i + H = 0. (34) t p i = H, (35) t q i = H. (36) p i These are calle Hamilton equations of motion. One of the mosmportant properties of Hamilton equations of motion is that they are first orer ifferential equations, while the Euler Lagrange equation of motion is secon orer. This simple point makes the mathematical structure much more tractable. In stuies of chaos, non-linear ynamics an beam ynamics, Hamilton equations of motion are inispensable. Another important property of the Hamiltonian is thas conserve; is nothing but the total energy! The fact that the Hamitonian is conserve H/t = 0 can be checke explicitly using Euler Lagrange equation of motion, H t = t (p i q i L) = ( ) q i L t q i = q i + q i q i q i = 0, (37) t q i q i q i where the first an thir terms cancel because of the Euler Lagrange equation, while the secon an fourth cancel trivially. Note that we use the fact the only time epenence of L comes through the time-epenence of q i an q i, not explicitly / t q, q = 0. Namely, we assume the invariance of the Lagrangian uner time translation. Much more importans the fact that the conservation of the Hamiltonian can also be shown as a consequence of the invariance of the system uner the time translation. Uner the time translation t t + δt, the action changes as S = tf Lt tf +δt +δt Lt = S + L(t f )δt L( )δt. (38) 8

9 an hence δs = L(t f )δt L( )δt. On the other han, it can also be worke out as tf δs = δ Lt tf Comparing above two equations, we fin or = δlt ( tf = q i δt + ) q i δt t q i (( ) tf = q i + ) q i δtt t q i q i = t f q i δt. (39) q i L(t f ) L( ) = t f q i (40) q i (p i q i L)(t f ) = (p i q i L)( ). (41) Therefore, systems invariant uner the time translation lea to the conserve Hamiltonian. A corollary of this statemens that systems nonvariant uner the time translation o not conserve energy. For example, ever-expaning Universe oes not have conserve energy. The energy can ecrease (aiabatic expansion of thermal gas) or can increase (the mysterious negative-pressure Dark Energy foun recently). In quantum mechanics, conserve quantities associate with an invariance of the system actually generate the invariance. I ll make this point clear in later sections. 4 Poisson Bracket Poisson Bracket between two physical quantities A(p i, q i ) an B(p i, q i ) is efine as {A, B} = A B A B. (4) p i p i 9

10 Why o we efine such a thing? Because is useful. First of all, the Hamilton equations of motion Eq. (36) can be written as t p i = {p i, H}, (43) t q i = {q i, H}. (44) In fact, it can easily be shown that the time erivative for any physical quantity can be written as A = {A, H}. (45) t In this sense, the Hamiltonian is sai to generate the time translation, because Poisson brackets with the Hamiltonian pushes the time forwar. In quantum mechanics, this poins much more manifest, because Schröinger equation is nothing but the Hamiltonian pushing the time forwar: i h ψ = H ψ. (46) t To mathematicians, Poisson brackets efine the symplectic structure on the manifol (phase space): a close two-form ω = n i=1 p i q i which gives the phase-space volume element V = ω n. The time evolution of the system is escribe by the Hamiltonian vector t = (ṗ i, q i ) = ( H, H ) = H ω ji, p i p i p i x i x i where x i = (q i, p i ) is the phase space coorinate. Similarly, the total momentum is conserve because of the invariance uner the spatial translation. For example, the total momentum is nothing but P x = i p i,x for one-imensional problems. Then is easy to see that i q i,x A = {A, P x }. (47) Inee, the total momentum generates the overall translation for all positions q i. The connection is also very interesting for angular momenta. Starting with the orbital angular momenta L = q p for a point particle, we can calcualte the Poisson brackets among them. {L x, L y } = {q y p z q z p y, q z p x q x p z } = q y p x + q x p y = L z, (48) {L y, L z } = {q z p x q x p z, q x p y q y p x } = q z p y + q y p z = L x, (49) {L z, L x } = {q x p y q y p x, q y p z q z p y } = q x p z + q z p x = L y. (50) 10

11 This is exactly the same as the commutation relation among the spin operators in quantum mechanics [S x, S y ] = i hs z, [S y, S z ] = i hs x, [S z, S x ] = i hs y. (51) It turns out that the angular momentum is conserve because of the rotational invariance of the system, an the angular momentum generate the rotation. Because rotation aroun x, y, z axes are nonepenent an are relate in a particular way, the Poisson brackets or commutation relation among angular momenta cannot be anything else. Poisson brackets are promote to commutation relations among operators in quantum mechanics. I m truly amaze that the classical mechanis before the awn of quantum mechanics came up with its formulation that so closely resembles the commutation relation among operators! 11