真实之路 宇宙法则的完全指南 (彭罗斯)20拉格朗日与哈.pdf

20 Lagrangians and Hamiltonians 20 1The magical Lagrangian formalism In the centuries following Newton s introduction of his dynamical laws an extremely impressive body of theoretical work was built up from these Newtonian foundations Euler Laplace Lagrange Legendre Gauss Liouville Ostrogradski Poisson Jacobi Hamilton and others came forth with reformulating ideas that led to a profound unifying overview I shall give a brief introduction here to this dynamical overview although I am afraid that my account of it will provide only a very inadequate impression of the magnitude of the achievement It should also be remarked that just the existence of such a mathematically elegant unifying picture appears to be telling us something deep about the mathematical underpinnings of our physical universe even at the level of the laws that were revealed in 17th century Newtonian mechanics Not many suggested laws for a physical universe could lead to mathematical structures of such imposing splendour What elegant unifying picture is it that resulted from Newton s mechanics It occurs basically in two diVerent but closely related forms each having its characteristic virtues Let us refer to the Wrst as the Lagrangian picture and the second as the Hamiltonian one There is the usual diYculty with names here Apparently both pictures were known to Lagrange signiWcantly before Hamilton and the Lagrangian one was at least partially anticipated by Euler Let us consider that we have a Newtonian system consisting of a Wnite number of individual particles and perhaps some rigid bodies each considered as an indivisible entity There will be a conWguration space C of some large number N of dimensions each of whose points represents a single spatial arrange ment of all these particles and bodies see 12 1 As time evolves the single point of C that represents the entire system will move about in C according to some law which encapsulates the Newtonian behaviour of the system see Fig 20 1 It is a remarkable and computationally very valuable fact that this law can be obtained by a direct mathematical procedure from a single function In the Lagrangian picture at least in its 471 Q C Fig 20 1ConWguration space Each point Q of the N dimensional manifold C represents an entire possible conWguration of say a family of Newtonian point particles and rigid bodies As the system evolves in time Q describes some curve in C simplest and most usual form1 this function called the Lagrangian function is deWned on the tangent bundle T C of the conWguration space C Fig 20 2a see 15 5 In the Hamiltonian picture the func tion called the Hamiltonian function is deWned on the cotangent bundle T C see 15 5 called the phase space Fig 20 2b We note that T C each of whose points stands for a point Q of C together with a tangent vector at Q and T C each of whose points stands for a point Q of C together with a cotangent vector at Q are both 2N dimensional manifolds In this section we investigate the Lagrangian picture leaving the Hamil tonian one to the next Coordinates for Lagrange s T C would serve to determine the positions of all the Newtonian bodies including appropriate angles to specify the spatial orientations of the rigid bodies etc and also their velocities including corresponding angular velocities of rigid bodies etc The position coordinates q1 qN usually termed generalized co ordinates label the diVerent points q of the conWguration space C perhaps just given patchwise see 12 2 Any adequate system of coordinates will do They need not be Cartesian or of any other standard kind This is the beauty of the Lagrangian and also Hamiltonian approach The choice of coordinatesisgovernedmerelybyconvenience Thisisjustthesamerolefor coordinates used in Chapters 8 10 12 14 and 15 etc when general manifolds of various kinds were considered Corresponding to the chosen set of generalized coordinates are the generalized velocities q q1 q qN where the dot means the rate of change d dt with respect to time q q 1 dq1 dt q qN dqN dt 20 1CHAPTER 20 472 T C C T C C qq a b Fig 20 2 a In the standard Lagrangian picture the Lagrangian L is a smooth function on the tangent bundle T C of conWguration space C b In the Hamilto nian picture the Hamiltonian H is a smooth function on the cotangent bundle T C called phase space The Lagrangian L would be written as a function of all of these 2 L L q1 qN q q1 q qN Each q qrhas to be treated as an independent variable independent of qr in particular in this expression This is one of the initially baZing features of Lagrangians but it works 3 The normal physical interpretation of the actual value of the function L would be the diVerence L K V between the kinetic energy K of the system and the potential energy V due to the external or internal forces expressed in these coordinates see 18 6 The equations of motion of the system encoding its entire Newtonian behaviour are given by what are called the Euler Lagrange equations which are astonishing in their extra ordinary scope and essential simplicity d dt L q qr L qr r 1 N Rememberthateach q qristobetreatedasanindependentvariable sothatthe expression L q qr whichmeans diVerentiateLformallywithrespectto q qr holdingall the othervariables Wxed actuallymakes sense Lagrangians and Hamiltonians 20 1 473 a b C Fig 20 3Hamilton s principle The Euler Lagrange equations tell us that the motion of Q through C is such as to make the action the integral of L along a curve taken between two Wxed points a b in C stationary under variations of the curve These equations express a remarkable fact sometimes referred to as Hamilton s principle or the principle of stationary action The meaning of this is perhaps clearest if we think in terms of the motion of the point Q in C where we recall that C represents the space of possible spatial conWgura tions of the entire system i e all the locations of all its parts The point Q whose position at any time is labelled by the qr moves along some curve in C at a certain rate this rate together with the tangent direction to the curve being determined by the values of the q qr The Euler Lagrange equations basically tell us that the motion of Q through C is such as to minimize the action this action being the integral of L along the curve taken between two Wxed end points a and b in the conWguration space C see Fig 20 3 More correctly this may not be actually a minimum but the term stationary would be appropriate The situation is basically similar to that which happens in ordinary calculus see 6 2 where the occurrence of a minimum of a smooth real valued function f x requires df dx 0 but where sometimes df dx 0 occurs when the function f is not a minimum it might be a maximum or possibly a point of inXexion or in higher dimensions what is called a saddle point Fig 20 4b All places where df x dx 0 are called stationary See Figs 6 4 and 20 4 Recall the basically similar characterization of a geodesic in pseudo Riemannian space given in 14 8 17 9 and 18 3 as a minimum length path in the positive deWnite case locally and sometimes as a maximum length time like path in the Lorentzian case although merely of stationary length in the general case Thus Q s trajectory can be thought of as some sort of geodesic in the space C 20 1CHAPTER 20 474 f y x a b c Fig 20 4Stationary values of a smooth real valued function f of several vari ables Illustrated is the case of a function f x y of two variables This is stationary where its graph a 2 dimensional surface is horizontal f x 0 f y This occurs a where F has a minimum but also in other situations such as b at a saddle point and c at a maximum In the case of Hamilton s principle Fig 20 3 or a geodesic connecting two points a b the Lagrangian L takes the place of f but the speciWcation of a path requires inWnitely many parameters rather than just x and y Again L may not be a minimum though a stationary point of some kind It is helpful to consider a simple example of a Lagrangian such as that for a single Newtonian particle of mass m moving in some Wxed external Weld given by a potential V which depends on position V V x y z t The meaning of V is that it deWnes the potential energy of the particle due to this external Weld For the case of the gravitational Weld of the Earth near the Earth s surface thought of as a constant downward pull we can take V mgz where z is the height above the ground and g is the downward acceleration due to gravity The three components of velocity are x x y y z z so using the expression 1 2mv 2 for kinetic energy see 18 6 we Wnd the Lagrangian L 1 2m x x 2 y y2 z z2 mgz The Euler Lagrange equation for z now gives us d m z z dt mg from which Galileo s constancy of acceleration in the direction of the Earth follows 20 1 20 2The more symmetrical Hamiltonian picture In the Hamiltonian picture we still use generalized coordinates but now the generalized position coordinates q1 qNare taken together with 20 1 Fill in the full details completing the argument to obtain Galileo s parabolic motion for free fall under gravity Lagrangians and Hamiltonians 20 2 475 what are called their corresponding generalized momentum coordinates p1 pN rather than the velocities For a single free particle the momentum is just the particle s velocity multiplied by its mass But in general the expression for generalized momentum need not be exactly this We can always get it from the Lagrangian however by use of the deWning formula pr L qr In any case these parameters prserve to provide coordinates for the cotangent spaces to C so that a covector can be written as padqa where we recall the summation convention of 12 7 which we adopt here although it is legitimate to read this also as an abstract index expression as in 12 8 This of course is a 1 form and its exterior derivative 12 6 S dpa dqa is a 2 form satisfying dS 0 20 2 which assigns a natural symplectic structure to the phase space T C see 14 8 Much of the strength of the Hamiltonian picture lies in the fact that phase spaces are symplectic manifolds and this symplectic structure is independent of the particular Hamiltonian that is chosen to provide the dynamics Classical physics is thereby intimately connected with the beautiful and surprising geometry of symplectic manifolds that we shall be coming to in 20 4 As a preliminary to understanding the role that this geometry plays let us see the form of Hamilton s dynamical equations These describe the time evolution of a system as a trajectory within the phase space T C of a point P representing the entire Newtonian system This evolution is completely governed by the Hamiltonian function H H p1 pN q1 qN which in the case of the time independent Lagrangians and Hamiltonians that we are concerned with here describes the total energy of the system in terms of the generalized momenta and positions We can actually obtain it from the Lagrangian by means of the expression summation convention or abstract indices H q qr L q qr L 20 2 Why 20 2CHAPTER 20 476 which then has to be rewritten by eliminating all the generalized velocities in favour of the generalized momenta not an easy task in general In terms of these momentum and position coordinates Hamilton s evolution equations are beautifully symmetrical dpr dt H qr dqr dt H pr These equations describe the velocity of a point P in T C This velocity is deWned for every P so we have a vector Weld on T C deWned by the Hamiltonian H In terms of the partial diVerentiation operator notation for a vector Weld given in 12 3 this is 20 3 H pr qr H qr pr This provides a Xow on T C which describes the Newtonian behaviour of the system Fig 20 5 In the particular example of a particle falling in a constant gravitational Weld as given above 20 1 in Lagrangian form the Hamiltonian is Constant H Hamiltonian vector field H Fig 20 5The Hamiltonian Xow H representing the Newtonian time evolution of the system see 20 4 is a vector Weld on phase space T C For the hypersurfaces of Wxed H values Wxed energy taking H to be time independ ent the trajectories remain within the Wxed H hypersurface in accordance with energy conservation 20 3 Explain this Lagrangians and Hamiltonians 20 2 477 H p2 x p 2 y p 2 z 2m mgz p2 2m mgz where px py and pzare the ordinary spatial momentum components in the directions of the Cartesian x y and z axes respectively This can be written downdirectlyfromknowledgeofwhatthetotalenergyoftheparticleought tobewhenexpressedintermsofpositionandmomentumcomponents orelse wecanobtainitfromtheLagrangian asgivenbytheaboveprocedures 20 4 At this point I should confess to a notational awkwardness that I see no way around so I had better come clean We saw in 18 7 that the spatial momentum components p1 p2 p3 in standard Minkowski coordinates for Xat spacetime with my preferred signature are the negatives of the normal momentum components Thus we have in the above example px p1 py p2 and pz p3 In the general discussion of Hamilto nians it is natural to use the downstairs versions of the momenta pa yet this is inconsistent with the pa i e p1 p2 p3 that are natural in relativity with signature The way that I am dealing with this notational problem in this book is simply to give the general formalism using the combination of qaand pawith the usual sign conventions connecting ps to qs whilst being non speciWc about the particular interpretation that each q or p might happen to have so the reader can sort out his her own choices of signs When I am using the combination of xaand pa on the other hand then I really do mean the notation consistent with that of 18 7 so that p1 p2 p3are the ordinary momentum components equal to p1 p2 p3in a standard Minkowski frame of ordinary spatial momentum This has the implication that when written in terms of the xs rather than the qs my Hamiltonian equations appear with the opposite sign dpr dt H xr dxr dt H pr Any reader who is not too concerned with the full details of the formalisms that I shall be presenting is recommended simply to ignore this issue completely Most experts would do the same until the moment comes when they have to write articles or books on the topic 20 3Small oscillations Before moving on in the next section to the remarkable geometry that the Hamiltonian description of things leads us to it will be illuminating Wrst 20 4 Do this explicitly Use Hamilton s equations to obtain the Newtonian equations of motion for a particle falling in a constant gravitational Weld 20 3CHAPTER 20 478 to consider the important topic of vibrations of a physical system about a state of equilibrium The topic has considerable relevance in a number of diVerent areas and it has particular signiWcance for us later in the context of quantum mechanics 22 11 The theory of vibrations can be conveni ently described either in the Lagrangian or the Hamiltonian formalism each of which is very well suited to its treatment I shall give my descrip tions explicitly here in the Hamiltonian formalism primarily because this more directly leads us into the quantum mechanical version of vibrations which we shall catch a good glimpse of in 22 11 The Lagrangian theory of vibrations which is very similar to the Hamiltonian one is left to the reader see Exercise 20 10 A simple example of a vibrating system occurs with an ordinary pendu lum swinging under gravity When the oscillations are small then the motion of the bob backwards and forwards describes a sine wave as a function of time see Fig 20 6 This is the kind of behaviour encountered with the individual Fourier components studied in 9 1 The period of vibration for such small oscillations is actually independent of the ampli tude of the oscillation i e of the distance through which the bob swings a famous early observation of Galileo s in 1583 This type of motion is referred to as simple harmonic motion We shall be seeing in this section how ubiquitous this motion is A general physical structure supposing that frictional eVects can be disre garded can wobble about its equilibrium state only in very speciWc ways We shall Wnd that every small scale wobble can be broken down into particular modes of vibration called normal modes in which the whole Time Fig 20 6A pendulum swinging under gravity For small oscillations the motion of the bob approximates simple harmonic motion the displacement of the bob mapped out as a function of time giving a sine wave Lagrangians and Hamiltonians 20 3 479 structure partakes of a simple harmonic motion with a very speciWc frequency called a normal frequency Let us Wrst see how simple harmonic motion is described analytically Let q denote the horizontal distance of our pendulum bob out from the lowest point or else the outward displacement from equilibrium of what ever other vibrating quantity we might be considering Then the equation of motion for small displacements q is d2q dt2 o2q where the positive constant quantity o 2p is the frequency of the oscilla tion This tells us that the inward acceleration d2q dt2is proportional with factor o2 to the outward displacement We see from 6 5 that q cosot and q sinot both satisfy this equation and so also does the general linear combination q acosot bsinot where a and b are constants 20 5 For a pendulum of length h swinging under gravity in one plane we Wnd an equation of motion that closely approximates the one given above when q is small with o2 g h but for larger values of q deviations from this equation arise 20 6 Suppose that we have a general Hamiltonian system which is in equi librium when the qs take some particular values qa qa 0 It will be conveni ent to choose the origin of our generalized coordinates to represent our equilibrium state i e we choose qa 0 0 Equilibrium refers to a con Wguration where if there is initially no motion then the system will remain stationary We may be interested in whether or not the equilibrium is stable this being the situation where if a small disturbance is made to the system in the equil