PHYS3001Classical MechanicsRobert L. DewarDepartment of Theoretical PhysicsResearch School of Physical Sciences & EngineeringThe Australian National UniversityCanberra ACT 0200Australiarobert.dewar@anu.edu.auVersion 1.51May 20, 2001. c ( R.L. Dewar 19982001.iiContents1 Generalized Kinematics 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Generalized coordinates . . . . . . . . . . . . . . . . . . . . . 21.3 Example: The ideal uid . . . . . . . . . . . . . . . . . . . . . 51.4 Variational Calculus . . . . . . . . . . . . . . . . . . . . . . . 61.4.1 Example: Geodesics . . . . . . . . . . . . . . . . . . . 91.4.2 Trial function method . . . . . . . . . . . . . . . . . . 101.5 Constrained variation: Lagrange multipliers . . . . . . . . . . 101.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6.1 Rigid rod . . . . . . . . . . . . . . . . . . . . . . . . . 131.6.2 Ecliptic . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6.3 Curvature of geodesics . . . . . . . . . . . . . . . . . . 132 Lagrangian Mechanics 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Generalized Newtons 2nd Law . . . . . . . . . . . . . . . . . 162.2.1 Generalized force . . . . . . . . . . . . . . . . . . . . . 162.2.2 Generalized equation of motion . . . . . . . . . . . . . 202.2.3 Example: Motion in Cartesian coordinates . . . . . . . 212.3 Lagranges equations (scalar potential case) . . . . . . . . . . 212.3.1 Hamiltons Principle . . . . . . . . . . . . . . . . . . . 232.4 Lagrangians for some Physical Systems . . . . . . . . . . . . . 242.4.1 Example 1: 1-D motionthe pendulum . . . . . . . . 242.4.2 Example 2: 2-D motion in a central potential . . . . . 252.4.3 Example 3: 2-D motion with time-varying constraint . 262.4.4 Example 4: Atwoods machine . . . . . . . . . . . . . . 272.4.5 Example 5: Particle in e.m. eld . . . . . . . . . . . . 282.4.6 Example 6: Particle in ideal uid . . . . . . . . . . . . 292.5 Averaged Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 302.5.1 Example: Harmonic oscillator . . . . . . . . . . . . . . 312.6 Transformations of the Lagrangian . . . . . . . . . . . . . . . 32iiiiv CONTENTS2.6.1 Point transformations . . . . . . . . . . . . . . . . . . . 322.6.2 Gauge transformations . . . . . . . . . . . . . . . . . . 342.7 Symmetries and Noethers theorem . . . . . . . . . . . . . . . 352.7.1 Time symmetry . . . . . . . . . . . . . . . . . . . . . . 372.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.8.1 Coriolis force and cyclotron motion . . . . . . . . . . . 372.8.2 Anharmonic oscillator . . . . . . . . . . . . . . . . . . 383 Hamiltonian Mechanics 413.1 Introduction: Dynamical systems . . . . . . . . . . . . . . . . 413.2 Mechanics as a dynamical system . . . . . . . . . . . . . . . . 413.2.1 Lagrangian method . . . . . . . . . . . . . . . . . . . . 413.2.2 Hamiltonian method . . . . . . . . . . . . . . . . . . . 433.2.3 Example 1: Scalar potential . . . . . . . . . . . . . . . 453.2.4 Example 2: Physical pendulum . . . . . . . . . . . . . 473.2.5 Example 3: Motion in e.m. potentials . . . . . . . . . . 483.2.6 Example 4: The generalized N-body system . . . . . . 483.3 Time-Dependent and Autonomous Hamiltonian systems . . . 503.4 Hamiltons Principle in phase space . . . . . . . . . . . . . . . 503.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.5.1 Constraints and moving coordinates . . . . . . . . . . . 533.5.2 Anharmonic oscillator phase space . . . . . . . . . . . 533.5.3 2-D motion in a magnetic eld . . . . . . . . . . . . . . 534 Canonical transformations 554.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Generating functions . . . . . . . . . . . . . . . . . . . . . . . 564.2.1 Example 1: Adiabatic Oscillator . . . . . . . . . . . . . 604.2.2 Example 2: Point transformations . . . . . . . . . . . . 624.3 Innitesimal canonical transformations . . . . . . . . . . . . . 634.3.1 Time evolution . . . . . . . . . . . . . . . . . . . . . . 644.4 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4.1 Symmetries and integrals of motion . . . . . . . . . . . 664.4.2 Perturbation theory . . . . . . . . . . . . . . . . . . . . 664.5 Action-Angle Variables . . . . . . . . . . . . . . . . . . . . . . 674.6 Properties of canonical transformations . . . . . . . . . . . . . 694.6.1 Preservation of phase-space volume . . . . . . . . . . . 694.6.2 Transformation of Poisson brackets . . . . . . . . . . . 744.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.7.1 Coriolis yet again . . . . . . . . . . . . . . . . . . . . . 744.7.2 Dierence approximations . . . . . . . . . . . . . . . . 75CONTENTS v5 Answers to Problems 775.1 Chapter 1 Problems . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Chapter 2 Problems . . . . . . . . . . . . . . . . . . . . . . . . 805.3 Chapter 3 Problems . . . . . . . . . . . . . . . . . . . . . . . . 865.4 Chapter 4 Problems . . . . . . . . . . . . . . . . . . . . . . . . 936 References and Index 99vi CONTENTSChapter 1Generalized coordinates andvariational principles1.1 IntroductionIn elementary physics courses you were introduced to the basic ideas of New-tonian mechanics via concrete examples, such as motion of a particle in agravitational potential, the simple harmonic oscillator etc. In this course wewill develop a more abstract viewpoint in which one thinks of the dynamics ofa system described by an arbitrary number of generalized coordinates, but inwhich the dynamics can be nonetheless encapsulated in a single scalar func-tion: the Lagrangian, named after the French mathematician Joseph LouisLagrange (17361813), or the Hamiltonian, named after the Irish mathe-matician Sir William Rowan Hamilton (18051865).This abstract viewpoint is enormously powerful and underpins quantummechanics and modern nonlinear dynamics. It may or may not be more ef-cient than elementary approaches for solving simple problems, but in orderto feel comfortable with the formalism it is very instructive to do some ele-mentary problems using abstract methods. Thus we will be revisiting suchexamples as the harmonic oscillator and the pendulum, but when examplesare set in this course please remember that you are expected to use the ap-proaches covered in the course rather than fall back on the methods youlearnt in First Year.In the following notes the convention will be used of italicizing the rst useor denition of a concept. The index can be used to locate these denitionsand the subsequent occurrences of these words.The present chapter is essentially geometric. It is concerned with the de-scription of possible motions of general systems rather than how to calculate12CHAPTER 1. GENERALIZEDCOORDINATES ANDVARIATIONAL PRINCIPLESphysical motions from knowledge of forces. Thus we call the topic generalizedkinematics.1.2 Generalized coordinatesSuppose we have a system of N particles each moving in 3-space and in-teracting through arbitrary (nite) forces, then the dynamics of the totalsystem is described by the motion of a point q qi[i = 1, 2, . . . , 3N =x1, y1, z1, x2, y2, z2, . . . , xN, yN, zN in a 3N-dimensional generalized cong-uration space. The number n = 3N of generalized coordinates qi is calledthe number of degrees of freedom. No particular metric is assumede.g.we could equally as well use spherical polar coordinates (see Fig. 1.1), qi =r1, 1, 1,r2, 2, 2, . . .,rN, N, N, or a more general curvilinear coordinatesystem.In other systems the generalizedxryzrFigure 1.1: Position vectorr in Cartesian and SphericalPolar coordinates.coordinates need not even be spa-tial coordinatese.g. they couldbe the charges owing in an elec-trical circuit. Thus the convenientvector-like notation q for the arrayof generalized coordinates should notbe confused with the notation r forthe position vector in 3-space. Of-ten the set of generalized coordi-nates is simply denoted q, but inthese notes we use a bold font, dis-tinguishing generalized coordinatearrays from 3-vectors by using a bold slanted font for the former and a boldupright font for the latter.Vectors are entities independent of which coordinates are used to repre-sent them, whereas the set of generalized coordinates changes if we changevariables. For instance, consider the position vector of a particle in Carte-sian coordinates x, y, z and in spherical polar coordinates r, , in Fig. 1.1.The vector r represents a point in physical 3-space and thus does not changewhen we change coordinates,r = xex + yey +zez = rer(, ) . (1.1)However its representation changes, because of the change in the unit basisvectors from ex, ey, ez to er, e and ez. On the other hand, the sets of1.2. GENERALIZED COORDINATES 3generalized coordinates rC x, y, z and rsph r, , z are distinct enti-ties: they are points in two dierent (though related) conguration spacesdescribing the particle.q1q2q3 ... qnq(t)q(t1)q(t2)Figure 1.2: Some possible paths in conguration space, eachparametrized by the time, t.Sometimes the motion is constrained to lie within a submanifold of thefull conguration space.1For instance, we may be interested in the motionof billiard balls constrained to move within a plane, or particles connectedby rigid rods. In such cases, where there exists a set of (functionally inde-pendent) constraint equations, or auxiliary conditionsfj(q) = 0 , j = 1, 2, , m < n , (1.2)the constraints are said to be holonomic.Each holonomic constraint reduces the number of degrees of freedom byone, since it allows us to express one of the original generalized coordinates as1A manifold is a mathematical space which can everywhere be described locally by aCartesian coordinate system, even though extension to a global nonsingular curvilinearcoordinate system may not be possible (as, e.g. on a sphere). A manifold can always beregarded as a surface embedded in a higher dimensional space.4CHAPTER 1. GENERALIZEDCOORDINATES ANDVARIATIONAL PRINCIPLESa function of the others and delete it from the set. For instance, for a particleconstrained to move in a horizontal plane (an idealized billiard table), thevertical position z = const is a trivial function of the horizontal coordinatesx and y and the conguration space becomes two dimensional, q = x, y.Consider the set of all conceivable paths through conguration space (seeFig. 1.2). Each one may be parametrized by the time, t: q = q(t). Bydierentiating eq. (1.2) with respect to time, we nd a set of constraints onthe generalized velocities, qi dqi/ dt, which we write in dierential notationasni=1fj(q)qidqi fj(q)q dq = 0 , (1.3)where f/q f/q1, . . . , f/qn and we use the shorthand dot-productnotationafb ni=1aifbi, (1.4)where ai and bi are arbitrary conguration space variables.The condition for functional independence of the m constraints is thatthere be m nontrivial solutions of eq. (1.3), i.e. that the rank of the matrixfj(q)/qi be its maximal possible value, m.Note that not all such dierential constraints lead to holonomic con-straints. If we are given constraints as a general set of dierential formsni=1(j)i (q) dqi = 0 , (1.5)then we may or may not be able to integrate the constraint equations to theform eq. (1.2). When we can, the forms are said to be complete dierentials.When we cannot, the constraints are said to be nonholonomic.The latter case, where we cannot reduce the number of degrees of freedomby the number of constraints, will not be considered explicitly in these notes.Furthermore, we shall normally assume that any holonomic constraints havebeen used to reduce qi[i = 1, . . . , n to a minimal, unconstrained set. How-ever, we present in Sec. 1.5 an elegant alternative that may be used whenthis reduction is not convenient, or is impossible due to the existence ofnonholonomic constraints.There are situations where there is an innite number of generalizedcoordinates. For instance, consider a scalar eld (such as the instaneousamplitude of a wave), (r, t). Here is a generalized coordinate of thesystem and the position vector r replaces the index i. Since r is a continuousvariable it ranges over an innite number of values.1.3. EXAMPLE: THE IDEAL FLUID 5r0x(r0,t)Figure 1.3: A uid element advected from point r = r0 at time t = 0to r = x(r0, t) at time t.1.3 Example: The ideal uidAs an example of a system with both an innite number of degrees of freedomand holonomic constraints, consider a uid with density eld (r, t), pressureeld p(r, t) and velocity eld v(r, t).Here we are using the Eulerian description, where the uid quantities ,p and v are indexed by the actual position, r, at which they take on theirphysical values at each point in time.However, we can also index these elds by the initial position, r0, ofthe uid particle passing through the point r = x(r0, t) at time t (seeFig. 1.3). This is known as the Lagrangian description. (cf. the Schrodingerand Heisenberg pictures in quantum mechanics.) We shall denote elds inthe Lagrangian description by use of a subscript L: L(r0, t), pL(r0, t) andvL(r0, t) = tx(r0, t).The eld x(r0, t) may be regarded as an innite set of generalized coor-dinates, the specication of which gives the state of the uid at time t. TheJacobian J(r0, t) of the change of coordinates r = x(r0, t) is dened byJ xx0xy0xz0=

xx0yx0zx0xy0yy0zy0xz0yz0zz0

, (1.6)where x0, y0 and z0 are Cartesian components of r0 and x, y and z are thecorresponding components of x(r0, t). This gives the change of volume ofa uid element with initial volume dV0 and nal volume (at time t) dVthroughdV = J(r0, t) dV0 . (1.7)To see this, consider dV0 = dx0 dy0 dz0 to be an innitesimal rectangularbox, as indicated in Fig. 1.3, with sides of length dx0, dy0, dz0. This uid6CHAPTER 1. GENERALIZEDCOORDINATES ANDVARIATIONAL PRINCIPLESelement is transformed by the eect of compression and shear to an innites-imal parallelipiped with sides given by dlx dx0x/x0, dly dy0x/y0,dlz dz0x/z0. The volume of such a parallepiped is dlx dlydlz =J dx0 dy0 dz0 2.Are the elds (r0, t) and p(r0, t) additional generalized coordinates whichneed to be specied at each point in time? In an ideal uid (i.e. one withno dissipation, also called an Euler uid) the answer is no, because massconservation, dV = 0 dV0, allows us to writeL(r0, t) = 0(r0)/J(r0, t) , (1.8)while the ideal equation of state p( dV )= p0( dV0), where is the ratio ofspecic heats, givespL(r0, t) = p0(r0)/J(r0, t) , (1.9)where 0 and p0 are the initial density and pressure elds, respectively. Theseare, by denition, xed in time, so the only time dependence occurs throughthe Jacobian J, which we showed in eq. (1.6) to be completely determinedby the Lagrangian displacement eld x(r0, t). Thus eqs. (1.8) and (1.9) haveallowed us to reduce the number of generalized coordinate elds from 5 to3 (the three components of x)mass conservation and the equation of statehave acted as holonomic constraints.Note: Mass conservation is valid even for nonideal uids (provided theyare not reacting and thus changing from one state to another). However,in a uid with nite dissipation, heat will be generated by the motion andentropy will be increased in each uid element, thus invalidating the useof the adiabatic equation of state. Further, the entropy increase dependson the complete path of the uid through its state space, not just on itsinstantaneous state. Thus the pressure cannot be holonomically constrainedin a nonideal uid.Remark 1.1 A useful model for a hot plasma is the magnetohydrodynamic(MHD) uidan ideal uid with the additional property of being a perfectelectrical conductor. This leads to the magnetic eld B(r, t) being frozen into the plasma, so that BL also obeys a holonomic constraint in the Lagrangianrepresentation, but as it is a vector constraint it is a little too complicated togive here.1.4 Variational CalculusConsider an objective functional I[q], dened on the space of all dierentiablepaths between two points in conguration space, q(t1) and q(t2), as depicted1.4. VARIATIONAL CALCULUS 7in Fig. 1.2I[q]

t2t1dt f(q(t), q(t), t) . (1.10)(As we shall wish to integrate by parts later, we in fact assume the paths tobe slightly smoother than simply dierentiable, so that q is also dened.)We suppose our task is to nd a path that makes I a maximum or min-imum (or at least stationary) with respect to neighbouring paths. Thus wevary the path by an amount q(t): q(t) q(t) + q(t). Then the rstvariation, I, is dened to be the change in I as estimated by linearizing inq:I[q]

t2t1dtq(t)fq + q(t)f q

. (1.11)Our rst task is to evaluate eq. (1.11) in terms of q(t). The crucial stephere is the lemma delta and dot commute. That is q dqdt . (1.12)To prove this, simply go back to denitions: q d(q +q)/ dt dq/ dt =dq/ dt 2.We can now integrate by parts to put I in the formI[q] =qf q

t2t1+

t2t1dt q(t)fq . (1.13)This consists of an endpoint contribution and an integral of the variationalderivative f/q, dened byfq fq ddtf q . (1.14)Remark 1.2 The right-hand side of eq. (1.14) is also sometimes called thefunctional derivative or Frechet derivative of I[q]. When using this termi-nology the notation I/q is used instead of f/q so that we can writeeq. (1.13), for variations qi which vanish in the neighbourhood of the end-points, asI[q]

t2t1dtni=1qi(t)Iqi(t) ,

q, Iq

, (1.15)8CHAPTER 1. GENERALIZEDCOORDINATES ANDVARIATIONAL PRINCIPLESwhich may be taken as the most general dening equation for I/q. Theinner product notation (, ) used above is a kind of innite-dimensional dotproduct where we not only sum over the index i, but integrate over the indext. If we recall that the change in the value of a eld dened on 3-space, e.g.(r), due to an arbitrary innitesimal change r is = r, which maybe regarded as the denition of the gradient , we see that the functionalderivative I/q may be thought of, by analogy, as an innite-dimensionalgradient dened on the function space of paths.A typical variational problem is to make I extremal or stationary underarbitrary variations q(t) holding the endpoints xed. That is, we requireI = 0 functions q(t) such that q(t1) = q(t2) = 0 . (1.16)Note that this condition does not necessarily require I to be a minimumor maximumit can be a kind of saddle point in function space, with someascending and some descending directions. To determine the nature ofa stationary point we would need to expand I to second order in qthesecond variation.In the class of variations in eq.t + qit Figure 1.4: A time-localizedvariation in generalized coor-dinate qi with support in therange t to t + .(1.16), the endpoint contribution ineq. (1.13) vanishes, leaving onlythe contribution of the integral overt. Since q(t) is arbitrary, we can,in particular, consider functions witharbitrarily localized support in t, asindicated in Fig. 1.4. (The supportof a function is just the range overwhich it is nonzero.) As 0,f/q(t) becomes essentially con-stant over the support of q in eq. (1.13)and we can move it outside the in-tegral. Clearly then, I can only bestationary for all such variations if and only if the variational derivative van-ishes for each value of t and each index ifq = 0 . (1.17)These n equations are known as the EulerLagrange equations. Some-times we encounter variational problems where we wish to extremize I under1.4. VARIATIONAL CALCULUS 9variations of the endpoints as well, q(t1) = q(t2) = 0. In such cases wesee from eq. (1.13) that, in addition to eq. (1.17), stationarity implies thenatural boundary conditionsf q = 0 (1.18)at t1 and t2.1.4.1 Example: GeodesicsIn the above development we have used the symbol t to denote the indepen-dent variable because, in applications in dynamics, paths in congurationspace are naturally parametrized by the time. However, in purely geometricapplications t is simply an arbitrary label for the position along a path, andwe shall in this section denote it by to avoid confusion.The distance along a path is given by integrating the lengths dl of in-nitesimal line elements, given a metric tensor gi,j such that( dl)2=ni,j=1dqigi,j dqj . (1.19)In terms of our parameter , we thus have the length l as a functional of theform discussed abovel =

21d ni,j=1 qigi,j(q, ) qj1/2. (1.20)A geodesic is a curve between two points whose length (calculated usingthe given metric) is stationary against innitesimal variations about thatpath. Thus the task of nding geodesics ts within the class of variationalproblems we have discussed, and we can use the EulerLagrange equationsto nd them. Perhaps the best known result on geodesics is the fact that theshortest path between two points in a Euclidean space (one where gi,j = 0for i = j and gi,j = 1 for i = j) is a straight line. Another well-known resultis that the shortest path between two points on the surface of a sphere is agreat circle (see Problem 1.6.3 for a general theorem on geodesics on a curvedsurface).Geodesics are not necessarily purely geometrical objects, but can havephysical interpretations. For instance, suppose we want to nd the shape ofan elastic string stretched over a slippery surface. The string will adjust itsshape to minimize its elastic energy. Since the elastic potential energy is a10CHAPTER 1. GENERALIZEDCOORDINATES ANDVARIATIONAL PRINCIPLESmonotonically increasing function of the length of the string, the string willsettle onto a geodesic on the surface.Geodesics also play an important role in General Relativity, because theworld line of a photon is a geodesic in 4-dimensional space time, with themetric tensor obeying Einsteins equations. If the metric is suciently dis-torted, it can happen that there is not one, but several geodesics betweentwo points, a fact which explains the phenomenon of gravitational lensing(multiple images of a distant galaxy behind a closer massive object).1.4.2 Trial function methodOne advantage of the variational formulation of a problem is that we canuse trial function methods to nd approximate solutions. That is, we canmake a clever guess, q(t) = qK(t, a1, a2, . . . , aK) as to the general form ofthe solution, using some specic function qK (the trial function) involving anite number of parameters ak, k = 1, . . . , K. Then we evaluate the integralin eq. (1.11) (analytically or numerically) and seek a stationary point ofthe resulting function I(a1, a2, . . . , aK) in the K-dimensional space of theparameters ak. Varying the ak the variation in I isI =Kk=1Iakak . (1.21)The condition for a stationary point is thusIak= 0, k = 1, . . . , K , (1.22)that is, that the K-dimensional gradient of I vanish.Since the true solution makes the objective functional I stationary withrespect to small variations, if our guessed trial function solution is close tothe true solution the error in I will be small. (Of course, because qK maynot be a reasonable guess for the solution in all ranges of the parameters,there may be spurious stationary points that must be rejected because theycannot possibly be close to a true solutionsee the answer to Problem 2.8.2in Sec. 5.2.)1.5 Constrained variation: Lagrange multi-pliersAs mentioned in Sec. 1.2 we normally assume that the holonomic constraintshave been used to reduce the dimensionality of the conguration space so1.5. CONSTRAINED VARIATION: LAGRANGE MULTIPLIERS 11that all variations are allowed. However, it may not be possible to do ananalytic elimination explicitly. Or it may be that some variables appear ina symmetric fashion, making it inelegant to eliminate one in favour of theothers.Thus, even in the holonomic case, it is worth seeking a method of han-dling constrained variations: when there are one or more dierential auxiliaryconditions of the form f(j)= 0. In the nonholonomic case it is mandatoryto consider such variations because the auxiliary conditions cannot be inte-grated.We denote the dimension of the conguration space by n. Followingeq. (1.5) we suppose there are m < n auxiliary conditions of the formf(j) (j)(q, t)q = 0 . (1.23)The vectors (j), j = 1, . . . , m may be assumed linearly independent (elsesome of the auxiliary conditions would be redundant) and thus span an m-dimensional subspace, Vm(t), of the full n-dimensional linear vector spaceVn occupied by the unconstrained variations. Thus the equations eq. (1.23)constrain the variations q to lie within an (n m)-dimensional subspace,Vnm(q, t), complementary to Vn.The variational problem we seek to solve is to nd the conditions (thegeneralizations of the EulerLagrange equations) under which the objectivefunctional I[q] is stationary with respect to all variations q in Vm(q, t).Apart from this restriction on the variations, the problem is the same as thatdescribed by eq. (1.16). The generalization of eq. (1.17) isfqq = 0 q Vnm(q, t) . (1.24)If there are no constraints, so that m = 0, then f/q is orthogonal toall vectors in Vn and the only solution is that f/q 0. Thus eq. (1.24)and eq. (1.17) are equivalent in this case. However, if m < n, then f/qcan have a nonvanishing component in the subspace Vm and eq. (1.17) is nolonger valid.An elegant solution to the problem of generalizing eq. (1.17) was found byLagrange. Expressed in our linear vector space language, his idea was thateq. (1.24) can be regarded as the statement that the projection, (f/q)nm,of f/q into Vnm(q, t) is required to vanish.However, we can write (f/q)nm as f/q(f/q)m, where (f/q)mis the projection of f/q into Vm. Now observe that we can write any vectorin Vm as a linear superposition of the (j)since they form a basis spanning12CHAPTER 1. GENERALIZEDCOORDINATES ANDVARIATIONAL PRINCIPLESthis space. Thus we write (f/q)m = j(j), or, equivalently,

fq

m+mj=1j(j)= 0 , (1.25)where the j(q, t) coecients, as yet to be determined, are known as theLagrange multipliers. They can be determined by dotting eq. (1.25) with eachof the m basis vectors (j), thus providing m equations for the m unknowns.Alternatively, we can express this variationally as

fq

m+mj=1j(j)q = 0 q Vm(q, t) . (1.26)Since j(j)is the projection into Vm of f/q, the projection intothe complementary subspace Vnm is found by subtracting (j(j)) fromf/q. That is,

fq

nm= fq +mj=1j(j)= 0 , (1.27)where the Vnm component of the second equality follows by eq. (1.24) andthe Vn component from eq. (1.25). Thus we have n generalized EulerLagrange equations, but they incorporate the m equations for the, so fararbitrary, (j)implicit in eq. (1.25). Thus we really only gain (n m) equa-tions from the variational principle, which is at it should be because we alsoget m kinematic equations from the constraint conditionsif we got morefrom the variational principle the problem would be overdetermined.The variational formulation of the second equality in eq. (1.27) isfq +mj=1j(j)q = 0 q Vn(q, t) . (1.28)That is, by using the Lagrange multipliers we have turned the constrainedvariational problem into an unconstrained one.In the holonomic case, when the auxiliary conditions are of the form ineq. (1.2), we may derive eq. (1.28) by unconstrained variation of the modiedobjective functionalI[q]

t2t1dt (f +mj=1jfj) . (1.29)The auxiliary conditions also follow from this functional if we require that itbe stationary under variation of the j.1.6. PROBLEMS 131.6 Problems1.6.1 Rigid rodTwo particles are connected by a rigid rod so they are constrained to move axed distance apart. Write down a constraint equation of the form eq. (1.2)and nd suitable generalized coordinates for the system incorporating thisholonomic constraint.1.6.2 EclipticSuppose we know that the angular momentum vectors rkmk rk of a systemof particles are all nonzero and parallel to the z-axis in a particular Cartesiancoordinate system. Write down the dierential constraints implied by thisfact, and show that they lead to a set of holonomic constraints. Hence writedown suitable generalized coordinates for the system.1.6.3 Curvature of geodesicsShow that any geodesic r = x() on a two-dimensional manifold S : r =X(, ) embedded in ordinary Euclidean 3-space, where and are arbi-trary curvilinear coordinates on S, is such that the curvature vector () iseverywhere normal to S (or zero).The curvature vector is dened by de

/ dl, where e

() dx/ dl isthe unit tangent vector at each point along the path r = x().Hint: First nd f(, , , ) = l, the integrand of the length functional,l =

f d (which involves nding the metric tensor in , space in terms ofX/ and X/). Then show that, for any path on S,f = e

X(and similarly for the derivative) andf = e

ddX ,and again similarly for the derivative.14CHAPTER 1. GENERALIZEDCOORDINATES ANDVARIATIONAL PRINCIPLESChapter 2Lagrangian Mechanics2.1 IntroductionThe previous chapter dealt with generalized kinematicsthe description ofgiven motions in time and space. In this chapter we deal with one formula-tion (due to Lagrange) of generalized dynamicsthe derivation of dierentialequations (equations of motion) for the time evolution of the generalized co-ordinates. Given appropriate initial conditions, these (in general, nonlinear)equations of motion specify the motion uniquely. Thus, in a sense, the mostimportant task of the physicist is over when the equations of motion havebeen derivedthe rest is just mathematics or numerical analysis (importantthough these are). The goal of generalized dynamics is to nd universal formsof the equations of motion.From elementary mechanics we are all familiar with Newtons SecondLaw, F = ma for a particle of mass m subjected to a force F and undergoingan acceleration a r. If we know the Cartesian components Fi(r, r, t),i = 1, 2, 3, of the force in terms of the Cartesian coordinates x1 = x, x2 = y,x3 = z and their rst time derivatives then the equations of motion are theset of three second-order dierential equations m xiFi = 0.To give a physical framework for developing our generalized dynamicalformalism we consider a set of N Newtonian point masses, which may be con-nected by holonomic constraints so the number n of generalized coordinatesmay be less than 3N. Indeed, in the case of a rigid body N is essentiallyinnite, but the number of generalized coordinates is nite. For example,the generalized coordinates for a rigid body could be the three Cartesiancoordinates of the centre of mass and three angles to specify its orientation(known as the Euler angles), so n = 6 for a rigid body allowed to move freelyin space.1516 CHAPTER 2. LAGRANGIAN MECHANICSWhether the point masses are real particles like electrons, composite par-ticles like nuclei or atoms, or mathematical idealizations like the innitesimalvolume elements in a continuum description, we shall refer to them generi-cally as particles.Having found a very general form of the equations of motion (Lagrangesequations), we then nd a variational principle (Hamiltons Principle) thatgives these equations as its EulerLagrange equations in the case of no fric-tional dissipation. This variational principle forms a basis for generalizingeven beyond Newtonian mechanics (e.g. to dynamics in Special Relativity).2.2 Generalized Newtons 2nd Law2.2.1 Generalized forceLet the (constrained) position of each of the N particles making up the sys-tem be given as a function of the n generalized coordinates q by rk = xk(q, t),k = 1, . . . , N. If there are holonomic constraints acting on the particles, thenumber of generalized coordinates satises the inequality n 3N. Thus, thesystem may be divided into two subsystemsan exterior subsystem de-scribed by the n generalized coordinates and an interior constraint subsys-tem whose (3Nn) coordinates are rigidly related to the q by the geometricconstraints.In a naive Newtonian approach we would have to specify the forces act-ing on each particle (taking into account Newtons Third Law, action andreaction are equal and opposite), derive the 3N equations of motion foreach particle and then eliminate all the interior subsystem coordinates tond the equations of motion of the generalized coordinates only. This isclearly very inecient, and we already know from elementary physics thatit is unnecessarywe do not really need an innite number of equations todescribe the motion of a rigid body. What we seek is a formulation in whichonly the generalized coordinates, and generalized forces conjugate to them,appear explicitly. All the interior coordinates and the forces required to main-tain their constrained relationships to each other (the forces of constraint)should be implicit only.To achieve this it turns out to be fruitful to adopt the viewpoint thatthe total mechanical energy (or, rather, its change due to the performance ofexternal work, W, on the system) is the primitive concept, rather than thevector quantity force. The basic reason is that the energy, a scalar quantity,needs only specication of the coordinates for its full description, whereasthe representation of the force, a vector, depends also on dening a basis2.2. GENERALIZED NEWTONS 2ND LAW 17set on which to resolve it. The choice of basis set is not obvious when weare using generalized coordinates. [Historically, force came to be understoodearlier, but energy also has a long history, see e.g. Ernst Mach The Scienceof Mechanics (Open Court Publishing, La Salle, Illinois, 1960) pp. 309312,QA802.M14 Hancock. With the development of Lagrangian and Hamiltonianmethods, and thermodynamics, energy-based approaches can now be said tobe dominant in physics.]To illustrate the relation between force and work, rst consider justone particle. Recall that the work W done on the particle by a forceF as the particle suers an innesimal displacement r is W = Fr Fxx+Fyy +Fzz. A single displacement r does not give enough informa-tion to determine the three components of F, but if we imagine the thoughtexperiment of displacing the particle in the three independent directions,r = x ex, y ey and z ez, determining the work, Wx, Wy, Wz, done ineach case, then we will have enough equations to deduce the three compo-nents of the force vector, Fx = Wx/x, Fy = Wy/y, Fz = Wz/z. [IfW can be integrated to give a function W(r) (which is not always possi-ble), then we may use standard partial derivative notation: Fx = W/x,Fy = W/y, Fz = W/z.]The displacements x ex, y ey and z ez are historically called virtualdisplacements. They are really simply the same displacements as used inthe mathematical denition of partial derivatives. Note that the virtualdisplacements are done at a xed instant in time as if by some invisiblehands, which perform the work W: this is a thought experimentthedisplacements are ctitious, not dynamical.Suppose we now transform from the Cartesian coordinates x, y, z to anarbitrary curvilinear coordinate system q1, q2, q3(the superscript notationbeing conventional in tensor calculus). Then an arbitrary virtual displace-ment is given by r =iqiei, where the basis vectors ei are in general notorthonormal. The corresponding virtual work is given by W =iqiFi,where Fi eiF. As in the Cartesian case, this can be used to determine thegeneralized forces Fi by determining the virtual work done in three indepen-dent virtual displacements.In tensor calculus the set Fi is known as the covariant representation ofF, and is in general distinct from an alternative resolution, the contravariantrepresentation Fi. (You will meet this terminology again in relativitytheory.) The energy approach shows that the covariant, rather than thecontravariant, components of the force form the natural generalized forcesconjugate to the generalized coordinates qi.Turning now to the N-particle system as a whole, the example abovesuggests we dene the set of n generalized forces, Qi, conjugate to each18 CHAPTER 2. LAGRANGIAN MECHANICSof the n degrees of freedom qi, to be such that the virtual work done on thesystem in displacing it by an arbitrary innitesimal amount q at xed timet is given byW ni=1Qiqi q . (2.1)We now calculate the virtual work in terms of the displacements of the Nparticles assumed to make up the system and the forces Fk acting on them.The virtual work isW =Nk=1Fkrk=ni=1qi

Nk=1Fkxkqi

. (2.2)Comparing eq. (2.1) and eq. (2.2), and noting that they hold for any q, wecan in particular take all but one of the qi to be zero to pick out the ithcomponent, giving the generalized force asQi =Nk=1Fkxkqi. (2.3)When there are holonomic constraints on the system we decompose theforces acting on the particles into what we shall call explicit forces and forcesof constraint. (The latter terminology is standard, but the usage explicitforces seems newoften they are called applied forces, but this is confus-ing because they need not originate externally to the system, but also frominteractions between the particles.)By forces of constraint, Fcstk , we mean those imposed on the particlesby the rigid rods, joints, sliding planes etc. that make up the holonomicconstraints on the system. These forces simply adjust themselves to whatevervalues are required to maintain the geometric constraint equations and canbe regarded as private forces that, for most purposes, we do not need toknow1. Furthermore, we may not be able to tell what these forces need tobe until we have solved the equations of motion, so they cannot be assumedknown a priori.The explicit force on each particle, Fxplk , is the vector sum of any externallyimposed forces, such as those due to an external gravitational or electric eld,plus any interaction forces between particles such as those due to elastic1Of course, in practical engineering design contexts one should at some stage check thatthe constraint mechanism is capable of supplying the required force without deforming orbreaking!2.2. GENERALIZED NEWTONS 2ND LAW 19springs coupling point masses, or to electrostatic attractions between chargedparticles. If there is friction acting on the particle, including that due toconstraint mechanisms, then that must be included in Fxplk as well. Theseare the public forces that determine the dynamical evolution of the degreesof freedom of the system and are determined by the conguration q of thesystem at each instant of time, and perhaps by the generalized velocity q inthe case of velocity-dependent forces such as those due to friction and thoseacting on a charged particle moving in a magnetic eld.Figure 2.1 shows a simple system withNmgFFigure 2.1: A body onan inclined plane as de-scribed in the text.a holonomic constrainta particle slid-ing on a plane inclined at angle . It issubject to the force of gravity, mg, thenormal force N, and a friction force F inthe directions shown. The force of con-straint is N. It does no work because itis orthogonal to the direction of motion,and its magnitude is that required to nullout the normal component of the gravita-tional force, [N[ = mg cos , so as to giveno acceleration in the normal direction and thus maintain the constraint.We now make the crucial observation that, because the constraints areassumed to be provided by rigid, undeformable mechanisms, no work canbe done on the interior constraint subsystem by the virtual displacements.That is, no (net) virtual work is done against the nonfriction forces imposedby the particles on the constraint mechanisms in performing the variationsq. By Newtons Third Law, the nonfriction forces acting on the constraintmechanisms are equal and opposite to the forces of constraint, Fcstk . Thusthe sum over Fcstk xk vanishes and we can replace the total force Fk withFxplk in eqs. (2.2) and (2.3). Note: If there are friction forces associatedwith the constraints there is work done against these, but this fact does notnegate the above argument because we have included the friction forces inthe explicit forcesany work done against friction forces goes into heat whichis dissipated into the external world, not into mechanical energy within theconstraint subsystem.That is,Qi =Nk=1Fxplk xkqi. (2.4)For the purpose of calculating the generalized forces, this is a much morepractical expression than eq. (2.3) because the Fxplk are known in terms of20 CHAPTER 2. LAGRANGIAN MECHANICSthe instantaneous positions and velocities. Thus Qi = Qi(q, q, t).2.2.2 Generalized equation of motionWe now suppose that eq. (2.4) has been used to determine the Qas functionsof the q (and possible q and t if we have velocity-dependent forces and time-dependent constraints, respectively). Then we rewrite eq. (2.3) in the formNk=1Fkxkqi= Qi(q, q, t) . (2.5)To derive an equation of motion we use Newtons second law to replaceFk on the left-hand side of eq. (2.5) with mkrk,Nk=1mkrkxkqiNk=1mk ddt

rkxkqi

rk ddtxkqi

= Qi . (2.6)We now dierentiate rk = xk(q(t), t) with respect to t to nd the functionvk such that rk = vk(q, q, t): rk = xkt +nj=1 qjxkqj vk(q, q, t) . (2.7)Dierentiating vk wrt qi (treating q and q as independent variables in partialderivatives) we immediately have the lemmaxkqi= vk qi. (2.8)The second lemma about partial derivatives of vk that will be needed isddtxkqi= vkqi, (2.9)which follows because d/ dt can be replaced by /t + q/q, which com-mutes with /qi (cf. the interchange of delta and dot lemma in Sec. 1.4).Applying these two lemmas in eq. (2.6) we ndNk=1 ddt

mkvkvk qi

mkvkvkqi

=Nk=1 ddt qi

12mkv2k

qi

12mkv2k

= Qi , (2.10)2.3. LAGRANGES EQUATIONS (SCALAR POTENTIAL CASE) 21In terms of the total kinetic energy of the system, TT Nk=112mkv2k , (2.11)we write eq. (2.10) compactly as the generalized Newtons second lawddtT qi Tqi= Qi . (2.12)These n equations are sometimes called Lagranges equations of motion, butwe shall reserve this term for a later form [eq. (2.24)] arising when we assumea special (though very general) form for the Qi. They are also sometimescalled (e.g. Scheck p. 83) dAlemberts equations, but this may be historicallyinaccurate so is best avoided.2.2.3 Example: Motion in Cartesian coordinatesLet us check that we can recover Newtons equations of motion as a specialcase when q = x, y, z. In this caseT = 12m( x2+ y2+ z2) (2.13)soTx = Ty = Tz = 0 (2.14)andT x = m x , T y = m y , T z = m z . (2.15)Also, from eq. (2.4) we see that Qi Fi. Substituting in eq. (2.12) weimmediately recover Newtons 2nd Law in Cartesian formm x = Fx m y = Fy m z = Fy (2.16)as expected.2.3 Lagranges equations (scalar potential case)In many problems in physics the forces Fk are derivable from a potential ,V (r1, r2, , rN). For instance, in the classical N-body problem the parti-cles are assumed to interact pairwise via a two-body interaction potential22 CHAPTER 2. LAGRANGIAN MECHANICSVk,l(rk, rl) Uk,l([rkrl[) such that the force on particle k due to particle lis given byFk,l = kVk,l= (rkrl)[rkrl[U

k,l(rk,l) , (2.17)where the prime on U denotes the derivative with respect to its argument,the interparticle distance rk,l [rk rl[. Then the total force on particle kis found by summing the forces on it due to all the other particlesFk = l=kkVk,l= kV , (2.18)where the N-body potential V is the sum of all distinct two-body interactionsV Nk=1lkVk,l= 12Nk,l=1

Vk,l . (2.19)In the rst line we counted the interactions once and only once: notingthat Vk,l = Vl,k, so that the matrix of interactions is symmetric we havekept only those entries below the diagonal to avoid double counting. In thesecond, more symmetric, form we have summed all the o-diagonal entriesof the matrix but have compensated for the double counting by dividing by2. The exclusion of the diagonal self-interaction potentials is indicated byputting a prime on the.Physical examples of such an N-body system with binary interactionsare: An unmagnetized plasma, where Vk,l is the Coulomb interactionUk,l(r) = ekel

0r , (2.20)where the ek are the charges on the particles and 0 is the permittivityof free space. We could also allow for the eect of gravity by addingthe potentialk mkgzk to V , where zk is the height of the kth particlewith respect to a horizontal reference plane and mk is its mass.2.3. LAGRANGES EQUATIONS (SCALAR POTENTIAL CASE) 23 A globular cluster of stars, where Vk,l is the gravitational interactionUk,l(r) = Gmkmlr , (2.21)where G is the gravitational constant. A dilute monatomic gas, where Vk,l is the Van der Waals interaction.However, if the gas is too dense (or becomes a liquid) we would haveto include 3-body or higher interactions as the wave functions of morethan two atoms could overlap simultaneously.Even when the system is subjected to external forces, such as gravity,and/or holonomic constraints, we can often still assume that the explicitforces are derivable from a potentialFxplk = kV . (2.22)Taking into account the constraints, we see that the potential V (r1, r2, , rN)becomes a function, V (q, t), in the reduced conguration space. Then, fromeq. (2.4) we haveQi = Nk=1xkqikV= qiV (q, t) . (2.23)Substituting this form for Qi in eq. (2.12) we see that the generalizedNewtons equations of motion can be encapsulated in the very compact form(Lagranges equations of motion)ddt

L qi

Lqi= 0 , (2.24)where the function L(q, q, t), called the Lagrangian, is dened asL T V . (2.25)2.3.1 Hamiltons PrincipleComparing eq. (2.24) with eq. (1.17) we see that Lagranges equations ofmotion have exactly the same form as the EulerLagrange equations for thevariational principle S = 0, where the functional S[q], dened byS

t2t1dt L( q, q, t) , (2.26)24 CHAPTER 2. LAGRANGIAN MECHANICSis known as the action integral . Since the natural boundary conditionseq. (1.18) are not physical, the variational principle is one in which the end-points are to be kept xed.We can now state Hamiltons Principle: Physical paths in congurationspace are those for which the action integral is stationary against all innites-imal variations that keep the endpoints xed.By physical paths we mean those paths, out of all those that are consistentwith the constraints, that actually obey the equations of motion with thegiven Lagrangian.To go beyond the original Newtonian dynamics with a scalar potentialthat we used to motivate Lagranges equations, we can instead take Hamil-tons Principle, being such a simple and geometrically appealing result, as amore fundamental and natural starting point for Lagrangian dynamics.2.4 Lagrangians for some Physical Systems2.4.1 Example 1: 1-D motionthe pendulumOne of the simplest nonlinear systems is( 1 c o s ) l0mgz lFigure 2.2: Phys-ical pendulum.the one-dimensional physical pendulum (so calledto distinguish it from the linearized harmonicoscillator approximation). As depicted in Fig. 2.2,the pendulum consists of a light rigid rod oflength l, making an angle with the vertical,swinging from a xed pivot at one end andwith a bob of mass m attached at the other.The constraint l = const and the assump-tion of plane motion reduces the system to onedegree of freedom, described by the general-ized coordinate . (This system is also calledthe simple pendulum to distinguish it from thespherical pendulum and compound pendula,which have more than one degree of freedom.)The potential energy with respect to the equi-librium position = 0 is V () = mgl(1 cos ), where g is the accelerationdue to gravity, and the velocity of the bob is v = l, so that the kineticenergy T = 12mv2 = 12ml2 2. The Lagrangian, T V , is thusL(, ) = 12ml2 2mgl(1 cos ) . (2.27)2.4. LAGRANGIANS FOR SOME PHYSICAL SYSTEMS 25This is also essentially the Lagrangian for a particle moving in a sinusoidalspatial potential, so the physical pendulum provides a paradigm for problemssuch as the motion of an electron in a crystal lattice or of an ion or electronin a plasma wave.From eq. (2.27) L/ = ml2 and L/ = mgl sin . Thus, theLagrangian equation of motion isml2 = mgl sin . (2.28)Expanding the cosine up to quadratic order in gives the harmonic os-cillator oscillator approximation (see also Sec. 2.6.2)L Llin 12ml2 2 12mgl2, (2.29)for which the equation of motion is, dividing through by ml2, + 20 = 0,with 0

g/l.2.4.2 Example 2: 2-D motion in a central potentialLet us work in plane polar coordinates, q = r, , such thatx = r cos , y = r sin , (2.30)so that x = r cos r sin , y = r sin +r cos , (2.31)whence the kinetic energy T 12( x2+ y2) is found to beT = 12m

r2+r2 2

. (2.32)An alternative derivation of eq. (2.32) may be found by resolving v into thecomponents rer and re, where er is the unit vector in the radial directionand e is the unit vector in the azimuthal direction.We now consider the restricted two body problemone light particleorbiting about a massive particle which may be taken to be xed at r = 0(e.g. an electron orbiting about a proton in the Bohr model of the hydrogenatom, or a planet orbiting about the sun). Then the potential V = V (r)(given by eq. (2.20) or eq. (2.21)) is a function only of the radial distancefrom the central body and not of the angle.26 CHAPTER 2. LAGRANGIAN MECHANICSThen, from eq. (2.25) the Lagrangian isL = 12m

r2+r2 2

V (r) . (2.33)First we observe that L is independent of (in which case is said to beignorable). Then L/ 0 and the component of Lagranges equations,eq. (2.24) becomesddtL = 0 , (2.34)which we may immediately integrate once to get an integral of the motion,i.e. a dynamical quantity that is constant along the trajectoryL = const . (2.35)From eq. (2.33) we see that L/ = mr2 , which is the angular momentum.Thus eq. (2.33) expresses conservation of angular momentum.Turning now to the r-componenturFigure 2.3: Planar mo-tion with a time-varying cen-tripetal constraint as de-scribed in the text.of Lagranges equations, we see fromeq. (2.33)Lr = mr2V

(r) ,L r = m r , ddtL r = m r .(2.36)From eq. (2.24) we nd the radialequation of motion to bem r mr2= V

(r) . (2.37)2.4.3 Example 3: 2-D motion with time-varying con-straintInstead of free motion in a central potential, consider instead a weight ro-tating about the origin on a frictionless horizontal surface (see Fig. 2.3) andconstrained by a thread, initially of length a, that is being pulled steadilydownward at speed u through a hole at the origin so that the radius r = aut.Then the Lagrangian is, substituting for r in eq. (2.32),L = T = 12m

u2+ (a ut)2 2

. (2.38)2.4. LAGRANGIANS FOR SOME PHYSICAL SYSTEMS 27Now only is an unconstrained generalized coordinate. As before, it isignorable, and so we again have conservation of angular momentumm(a ut)2 l = const , (2.39)which equation can be integrated to give as a function of t, = 0 +(l/mu)[1/(a ut) 1/a] = 0 + lt/[ma(a ut)].Clearly angular momentum is conserved, because the purely radial stringcannot exert any torque on the weight. Thus Lagranges equation of motiongives the correct answer. However, the string is obviously doing work on thesystem because T = 12[mu2+ (l2/m)/(a ut)2] is not conserved. Have wenot therefore violated the postulate in Sec. 2.2.2 of no work being done bythe constraints? The answer is no because what we assumed in Sec. 2.2.2was that no virtual work was done by the constraints. The fact that theconstraint is time-dependent is irrelevant to this postulate, because virtualdisplacements are done instantaneously at any given time.2.4.4 Example 4: Atwoods machineConsider two weights of mass m1 and m2 xx0m1m2Figure 2.4: At-woods machine.suspended from a frictionless, inertialess pullyof radius a by a rope of xed length, as de-picted in Fig. 2.4. The height of weight 1 is xwith respect to the chosen origin and the holo-nomic constraint provided by the rope allowsus to express the height of weight 2 as x, sothat there is only one degree of freedom forthis system.The kinetic and potential energy are T =12(m1 + m2) x2and V = m1gx m2gx. ThusL = T V is given byL = 12(m1 +m2) x2(m1m2)gx (2.40)and its derivatives are L/x = (m1m2)gand L/ x = (m1+m2) x, so that the equation of motion d(L/ x) = L/xbecomes x = m1m2m1 +m2g . (2.41)28 CHAPTER 2. LAGRANGIAN MECHANICS2.4.5 Example 5: Particle in e.m. eldThe fact that Lagranges equations are the EulerLagrange equations for theextraordinarily simple and general Hamiltons Principle (see Sec. 2.3.1) sug-gests that Lagranges equations of motion may have a wider range of validitythan simply problems where the force is derivable from a scalar potential.Thus we do not dene L as T V , but rather postulate the universal validityof Lagranges equations of motion (or, equivalently, Hamiltons Principle), fordescribing non-dissipative classical dynamics and accept any Lagrangian asvalid that gives the physical equation of motion.In particular, it is obviously of great physical importance to nd a La-grangian for which Lagranges equations of motion eq. (2.24) reproduce theequation of motion of a charged particle in an electromagnetic eld, underthe inuence of the Lorentz force,mr = eE(r, t) +e rB(r, t) , (2.42)where e is the charge on the particle of mass m.We assume the electric and magnetic elds E and B, respectively, tobe given in terms of the scalar potential and vector potential A by thestandard relationsE = tA ,B = A . (2.43)The electrostatic potential energy is e, so we expect part of the La-grangian to be 12m r2 e, but how do we include the vector potential?Clearly we need to form a scalar since L is a scalar, so we need to dot Awith one of the naturally occurring vectors in the problem to create a scalar.The three vectors available are A itself, r and r. However we do not wish touse A, since AA in the Lagrangian would give an equation of motion thatis nonlinear in the electromagnetic eld, contrary to eq. (2.42). Thus we canonly use r and r. Comparing eqs. (2.42) and (2.43) we see that rA has thesame dimensions as , so let us try adding that to form the total LagrangianL = 12m r2e + e rA . (2.44)Taking q q1, q2, q3 = x, y, z we haveLqi= eqi+e3j=1 qjAjqi, (2.45)2.4. LAGRANGIANS FOR SOME PHYSICAL SYSTEMS 29andddtL qi= m qi +e dAidt = m qi + eAit +3j=1 qjAiqj . (2.46)Substituting eqs. (2.45) and (2.46) in eq. (2.24) we ndm qi = eqi Ait

+e3j=1 qjAjqi Aiqj

. (2.47)This is simply eq. (2.42) in Cartesian component form, so our guessed La-grangian is indeed correct.2.4.6 Example 6: Particle in ideal uidIn Sec. 1.3 we presented a uid as a system with an innite number of degreesof freedom. However, if we concentrate only on the motion of a single uidelement (a test particle), taking the pressure p and mass density as known,prescribed functions of r and t the problem becomes only three-dimensional.Dividing by we write the equation of motion of a uid element asdvdt = p V , (2.48)where V (r, t) is the potential energy (usually gravitational) per unit mass.To nd a Lagrangian for this motion we need to be able to combinethe pressure gradient and density into an eective potential. In an idealcompressible uid we have the equation of state p( dV )= p0( dV0), where is the ratio of specic heats, which we can write asp = const , (2.49)where the right-hand side is a constant of the motion for the given testparticle. If we further assume that it is the same constant for all uidelements in the neighbourhood of the test particle, then we can take thegradient of the log of eq. (2.49) to get (p)/p = ()/. Thusp =

p

p2=

p

p .30 CHAPTER 2. LAGRANGIAN MECHANICSSolving for p/ we getp = h (2.50)where the enthalpy (per unit mass) is dened byh 1p . (2.51)Using eq. (2.50) in eq. (2.48) we recognize it as the equation of motionfor a particle of unit mass with total potential energy h + V . Thus theLagrangian isL = 12 r2h V . (2.52)2.5 Averaged LagrangianFrom Sec. 2.3.1 we know that Lagrangian dynamics has a variational for-mulation, and so we expect that trial function methods (see Sec. 1.4.2) maybe useful as a way of generating approximate solutions of the Lagrangianequations of motion. In particular, suppose we know that the solutions areoscillatory functions of t with a frequency much higher than the inverse ofany characteristic time for slow changes in the parameters of the system (thechanges being then said to occur adiabatically). Then we may use a trialfunction of the formq(t) = q((t), A1(t), A2(t), . . .) (2.53)where q is a 2-periodic function of , the phase of the rapid oscillations, andthe Ak are a set of slowly varying amplitudes characterizing the waveform(e.g. see Problem 2.8.2). Thus, q = (t) q + A1 qA1+ A2 qA2+ , (2.54)where the instantaneous frequency is dened by(t) (t) . (2.55)Since d ln Ak/ dt, to a rst approximation we may keep only the rstterm in eq. (2.54). Thus our approximate L is a function of , but not ofA1, A2 etc.Now take the time integration in the action integral to be over a timelong compared with the period of oscillation, but short compared with the2.5. AVERAGED LAGRANGIAN 31timescale for changes in the system parameters. Thus only the phase-averageof L,L

20d2 L (2.56)contributes to the action,S

t2t1dt L(, A1(t), A2(t), . . .) (2.57)for the class of oscillatory physical solutions we seek. Note that the averagingin eq. (2.56) removes all direct dependence on , so L depends only on itstime derivative .Thus, we have a new, approximate form of Hamiltons Principle in whichthe averaged Lagrangian replaces the exact Lagrangian, and in which theset , A1, A2, . . . replaces the set q1, q2, . . . as the generalized variables.Requiring S to be stationary within the class of quasiperiodic trial functionsthen gives the new adiabatic EulerLagrange equationsddtL = 0 , (2.58)andLAk= 0 , k = 1, 2, . . . . (2.59)The rst equation expresses the conservation of the adiabatic invariant,J = const, where J is known as the oscillator actionJ L . (2.60)The second set of equations gives relations between and the Aks that givethe waveform and the instantaneous frequency.2.5.1 Example: Harmonic oscillatorConsider a weight of mass m at the end of a light spring with spring constantk = m20. Then the kinetic energy is T = 12m x2and the potential energy isV = 12kx2= 12m20x2. Thus the natural form of the Lagrangian, T V , isL = 12m( x220x2) . (2.61)Now consider that the length of the spring, or perhaps the spring constant,changes slowly with time, so that 0 = 0(t), with d ln 0/ dt 0).(c) Consider a charged particle constrained to move on a non-rotating smoothinsulating sphere, immersed in a uniform magnetic eld B = Bez, on whichthe electrostatic potential is a function of latitude and longitude. Write downthe Lagrangian in the same generalized coordinates as above and show it isthe same as that for the particle on the rotating planet with appropriateidentications of and V .2.8.2 Anharmonic oscillatorConsider the following potential V , corresponding to a particle of mass moscillating along the x-axis under the inuence of a nonideal spring (i.e. one2.8. PROBLEMS 39with a nonlinear restoring force),V (x) = m202

x2+ x4l20

,where the constant 0 is the angular frequency of oscillations having ampli-tude small compared with the characteristic length l0, and = 1 dependson whether the spring is soft ( = 1) or hard ( = +1).Consider the trial functionx = l0 [Acos t +Bcos 3t +C sin 3t] ,where A, B, C are the nondimensionalized amplitudes of the fundamentaland third harmonic, respectively, and is the nonlinearly shifted frequency.By using this trial function in the time-averaged Hamiltons Principle, ndimplicit relations giving approximate expressions for , B and C as functionsof A. Show that C 0. The trial function is strictly appropriate only to thecase A < 1, but plot /0 and B vs. A from 0 to 1 in the case of both ahard and a soft spring. (You are encouraged to use Maple or Mathematicaand/or MatLab in this problem.)40 CHAPTER 2. LAGRANGIAN MECHANICSChapter 3Hamiltonian Mechanics3.1 Introduction: Dynamical systemsMathematically, a continuous-time dynamical system is dened to be a sys-tem of rst order dierential equations z = f(z, t) , t R , (3.1)where f is known as the vector eld and R is the set of real numbers. Thespace in which the set of time-dependent variables z is dened is called phasespace.Sometimes we also talk about a discrete-time dynamical system. This isa recursion relation, dierence equation or iterated mapzt+1 = f(z, t) , t Z , (3.2)where f is known as the map (or mapping) and Z is the set of all integers. . . , 2, 1, 0, 1, 2, . . ..Typically, such systems exhibit long-time phenomena like attracting andrepelling xed points and limit cycles, or more complex structures such asstrange attractors. In this chapter we show how to reformulate nondissipativeLagrangian mechanics as a dynamical system, but shall nd that it is a veryspecial case where the above-mentioned phenomena cannot occur.3.2 Mechanics as a dynamical system3.2.1 Lagrangian methodLagranges equations do not form a dynamical system, because they implic-itly contain second-order derivatives, q. However, there is a standard way to4142 CHAPTER 3. HAMILTONIAN MECHANICSobtain a system of rst-order equations from a second-order system, whichis to double the size of the space of time-dependent variables by treatingthe generalized velocities u as independent of the generalized coordinates, sothat the dynamical system is q = u, u = q(q, u, t). Then the phase spaceis of dimension 2n. This trick is used very frequently in numerical problems,because the standard numerical integrators require the problem to be posedin terms of systems of rst-order dierential equations.In the particular case of Lagrangian mechanics, we have from eq. (2.24),expanding out the total derivative using the chain rule and moving all butthe highest-order time derivatives to the right-hand side,nj=12L qi qj qj = Lqi 2L qit nj=12L qiqj qj . (3.3)The matrix H acting on q, whose elements are given byHi,j 2L qi qj, (3.4)is called the Hessian matrix. It is a kind of generalized mass tensor (seeSec. 3.2.6), and for our method to work we require it to be nonsingular, sothat its inverse, H1, exists and we can nd q. Then our dynamical systembecomes q = u , u = H1Lq 2L qt 2L qq q

. (3.5)Remark 3.1 The Lagrangian method per se does not break down if the Hes-sian is singular, only our attempt to force it into the standard dynamicalsystem framework. We can still formally solve the dynamics in the follow-ing manner: Suppose H is singular, with rank n m. Then, within the n-dimensional linear vector space ^ on which H acts, there is an m-dimensionalsubspace (the nullspace) such that H q 0 for all q . We can solveeq. (3.3) for the component of q lying in the complementary subspace ^ `provided the right-hand side satises the solubility condition that it have nocomponent in . The component of q lying in cannot be found directly,but the solubility condition provides m constraints that complete the deter-mination of the dynamics.As a simple example, suppose L does not depend on one of the generalizedvelocities, qi. Then L/ qi 0 and the ith component of the Lagrangeequations of motion eq. (2.24) reduces to the constraint L/qi = 0. 23.2. MECHANICS AS A DYNAMICAL SYSTEM 433.2.2 Hamiltonian methodWe can achieve our aim of nding 2n rst-order dierential equations byusing many choices of auxiliary variables other than u. These will be morecomplicated functions of the generalized velocities, but the extra freedom ofchoice may also bring advantages.In particular, Hamilton realised that it is very natural to use as the newauxiliary variables the set p = pi[i = 1, . . . , n dened bypi qiL(q, q, t) , (3.6)where pi is called the generalized momentum canonically conjugate to qi.We shall always assume that eq. (3.6) can be solved to give q as a functionof q and p q = u(q, p, t) . (3.7)Remark 3.2 We have in eect changed variables from u to p, and such achange of variables can only be performed if the Jacobian matrix pi/uj =2L/ qi qj is nonsingular. From eq. (3.4) we recognize this matrix as beingthe Hessian we encountered in the Lagrangian approach to constructing adynamical system. Thus in either approach we require the Hessian to benonsingular (i.e. for its determinant to be nonzero). This condition is usuallytrivially satised, but there are physical problems (e.g. if the Lagrangian doesnot depend on one of the generalized velocities) when this is not the case.However, in the standard Hamiltonian theory covered in this course it isalways assumed to hold.The reason for dening p as in eq. (3.6) is that L/ q occurs explicitlyin Lagranges equations, eq. (2.24), so we immediately get an equation ofmotion for p p = L(q, q, t)q

q=u(q,p,t). (3.8)Equations (3.7) and (3.8) do indeed form a dynamical system, but so farit looks rather unsatisfactory: now u is dened only implicitly as a functionof the phase-space variables q and p, yet the right-hand side of eq. (3.8)involves a partial derivative in which the q-dependence of u is ignored!We can x the latter problem by holding p xed in partial derivativeswith respect to q (because it is an independent phase-space variable) butthen subtracting a correction term to cancel the contribution coming from44 CHAPTER 3. HAMILTONIAN MECHANICSthe q-dependence of u. Applying the chain rule and then using eqs. (3.6)and (3.7) we get p = L(q, u, t)q ni=1Luiuiq= L(q, u, t)q ni=1piuiq= q[L(q, u, t) pu] Hq , (3.9)where we have dened a new function to replace the Lagrangian, namely theHamiltonianH(q, p, t) pu L(q, u, t) . (3.10)Given the importance of H/q it is natural to investigate whetherH/p plays a signicant role as well. Dierentiating eq. (3.10) we getHp = u(q, p, t) +ni=1pi uiL(q, u, t)

uip= q , (3.11)where the vanishing of the expression in the square bracket and the identi-cation of u with q follows from eqs. (3.6) and (3.7).Summarizing eqs. (3.9) and (3.11), q = Hp p = Hq , (3.12)These equations are known as Hamiltons equations of motion. As withLagranges equations they express the dynamics of a system with an arbitrarynumber of degrees of freedom in terms of a single scalar function! Unlike theLagrangian dynamical system, the phase-space variables are treated on acompletely even footingin Hamiltonian mechanics both the conguration-space variables q and momentum-space variables p are generalized coordi-nates. We dene canonical coordinates as phase-space coordinates such thatthe equations of motion can be expressed in the form of eq. (3.12) and acanonical system as one for which canonical coordinates exist.3.2. MECHANICS AS A DYNAMICAL SYSTEM 45Remark 3.3 The transition from the Lagrangian to the Hamiltonian in or-der to handle the changed meaning of partial derivatives after a change ofvariable is a special case of a technique known as a Legendre transformation.It is encountered quite often in physics and physical chemistry, especially inthermodynamics.3.2.3 Example 1: Scalar potentialConsider the Lagrangian for a particle in Cartesian coordinates, so q =x, y, z may be replaced by r = xex +yey +zez. Also assume that it movesunder the inuence of a scalar potential V (r, t) so that the natural form ofthe Lagrangian isL = T V = 12m[ r[2V (r, t) . (3.13)Then, from eq. (3.6)p p L r = m r , (3.14)so that in this case the canonical momentum is the same as the ordinarykinematic momentum. Equation (3.14) is solved trivially to give q = u(p)where u(p) = p/m. Thus, from eq. (3.10) we haveH = [p[2m

[p[22m V (r, t)

= [p[22m +V (r, t)= T +V . (3.15)That is, the Hamiltonian is equal to the total energy of the system, ki-netic plus potential. The fact that the Hamiltonian is an important physicalquantity, whereas the physical meaning of the Lagrangian is more obscure,is one of the appealing features of the Hamiltonian approach. Both theLagrangian and Hamiltonian have the dimensions of energy, and both ap-proaches can be called energy methods. They are characterized by the useof scalar quantities rather than the vectors encountered in the direct use ofNewtons second law. This has both the theoretical advantage of leading tovery general formulations of mechanics and the practical benet of avoidingsome vector manipulations when changing between coordinate systems (infact, Lagrangian and Hamiltonian methods were developed before modernvector notation was invented).46 CHAPTER 3. HAMILTONIAN MECHANICSHarmonic OscillatorAn example is the harmonic oscillator Hamiltonian corresponding to theLagrangian, eq. (2.61)H = p22m + m0x22 . (3.16)From eq. (3.12) the Hamiltonian equations of motion are x = pm p = m0x . (3.17)Gauge-transformed Harmonic OscillatorNow consider the gauge-transformed harmonic oscillator Lagrangian eq. (2.75)L

= 12m( x2+ 20x x 20x2) .The canonical momentum is thusp = L

/ x = m( x +0x) (3.18)and we see that the gauge transformation has eected a transformation ofthe canonical momentum, even though the generalized coordinate x remainsthe same. This is an example of a canonical transformation, about which wewill have more to say later.Solving eq. (3.18) for x we nd u(p) = (p m0x)/m. HenceH = p(p m0x)m

(p m0x)22m +0x(p m0x) 12m20x2

= (p m0x)22m + 12m20x2= T +V . (3.19)Thus, even though L was not of the natural form T V in this case, theHamiltonian remains equal to the total energy, thus conrming that it is aquantity with a more direct physical signifance than the Lagrangian. (Thefunctional form of the Hamiltonian changes under the gauge transformationhowever, because the meaning of p changes.)3.2. MECHANICS AS A DYNAMICAL SYSTEM 473.2.4 Example 2: Physical pendulumA nonlinear one-dimensional case is provided by the physical pendulum, in-troduced in Sec. 2.4.1. Using eq. (2.27) in eq. (3.6) we get p = ml2 . Thus = p/ml2and so the Hamiltonian, H = p L, becomesH(, p) = p22ml2 + mgl(1 cos ) , (3.20)which again is of the form T +V .By conservation of energy (see also Sec. 3.3),0 0.2 0.4 0.6 0.8 1-3-2-10123/2p X O OFigure 3.1: Phasespace of the phys-ical pendulum.the Hamiltonian H = T + V is a constantof the motion so the nature of the orbits inphase space can be found simply by plottingthe contours of H as in Fig. 3.1 (which is inunits such that m = l = g = 1). We see thatthe structure of the phase space is rather morecomplicated than in the case of the harmonicoscillator since there are two topologically dis-tinct classes of orbit. One class is the rotatingorbits for which the pendulum has enough en-ergy, H > 2mgl, to swing entirely over the topso increases or decreases secularly (thoughthe physical position x = l sin does not).The other class is the librating orbits for H < 2mgl implying that thependulum is trapped in the gravitational potential well and oscillates like apair of scales (hence the name). For [[ < 1 we may expand the cosine soV mgl2/2 and the system is approximately a harmonic oscillator. Theequilibrium point = 0 or 2, labelled O in Fig. 3.1, is a xed point. Theorbits in its neighbourhood, like those of the harmonic oscillator, remain inthe neighbourhood for all time (i.e. the xed point is stable or elliptic).The dividing line H = 2mgl between the two topological classes of orbit iscalled the separatrix, and on the separatrix lies another xed point, labelledX in Fig. 3.1. This corresponds to the case where the pendulum is justbalanced upside down. Almost all orbits in the neighbourhood of an Xpoint eventually are repelled far away from it, and thus it is referred to asan unstable or hyperbolic xed point.48 CHAPTER 3. HAMILTONIAN MECHANICS3.2.5 Example 3: Motion in e.m. potentialsNow consider the case of a charged particle in an electromagnetic eld withmagnetic vector potential A and electrostatic potential . The Lagrangianis given by eq. (2.44) and thus, from eq. (3.6),p L r = m r +eA(r, t) . (3.21)Thus u(p) = p/meA/m and, from eq. (3.10) we haveH = (p eA)pm

[p eA[22m + e(p eA)Am e(r, t)

= [p eA(r, t)[22m +e(r, t)= T +V . (3.22)Thus we nd again that, although the Lagrangian cannot be put into thenatural form T V (r), the Hamiltonian is still the total energy, kinetic pluselectrostatic potential energy.3.2.6 Example 4: The generalized N-body systemLet us now revisit the general case which led to the original denition of theLagrangian in Sec. 2.3, the case of N particles interacting via a scalar N-bodypotential, possibly with constraints (which we here assume to be independentof time), so that the number of generalized coordinates is n 3N. Then, byeq. (2.25) and eq. (2.7) (assuming the function xk(q) to be independent oftime) the natural form of the Lagrangian isL = T V= 12ni,j=1Nk=1mk qixkqixkqj qjV (q)= 12ni,j=1 qii,j qjV (q) , (3.23)where the symmetric mass matrixi,j(q) Nk=1mkxkqixkqj(3.24)3.2. MECHANICS AS A DYNAMICAL SYSTEM 49is the metric tensor for a conguration-space mass-weighted length elementds dened by ( ds)2k mk dl2k. From eq. (3.4), we see that it is the Hessianmatrix for this system.Then, using eq. (3.23) in eq. (3.6)pi =nj=1i,j(q) qj . (3.25)Assuming none of the particles is massless, i,j is a positive-denite ma-trix, so its inverse 1i,j exists and we can formally solve eq. (3.25) for qi togiveui(q, p, t) =nj=11i,jpj . (3.26)Then, from eq. (3.10) we haveH =ni,j=1pi1i,jpj

12ni,j,i

,j

=1pi1i,i

i

,j 1j

,jpjV (q)

= 12ni,j=1pi1i,jpj +V (q)= T +V . (3.27)Thus, the Hamiltonian is again equal to the total energy of the system. Thisresult does not hold in the case of a time-dependent representation, xk(q, t).[See Problems 3.5.1(c) and 4.7.1.]Particle in a central potentialAs a simple, two-dimensional example of a problem in non-Cartesion coor-dinates we return to the problem of motion in a central potential, expressedin plane polar coordinates in Sec. 2.4.2. Recapitulating eq. (2.33),L = 12m

r2+r2 2

V (r) .Comparing with eq. (3.23) we see that the mass matrix is diagonal = m 00 mr2

, (3.28)and thus can be inverted simply taking the reciprocal of the diagonal ele-ments. Hence, from eq. (3.27)H = p2r2m + p22mr2 +V (r) . (3.29)50 CHAPTER 3. HAMILTONIAN MECHANICS3.3 Time-Dependent and Autonomous Hamil-tonian systemsAn autonomous dynamical system is one in which the vector eld f [seeeq. (3.1)] depends only on the phase-space coordinates and has no explicitdependence on t. (The reason for the name is that explicit time dependencewould come from external forcing of the system, whereas autonomousmeans independent or self-governing.) In the Hamiltonian case this meansthat H has no explicit time dependence. Conversely, if there is an externaltime-varying perturbation, then H = H(p, q, t).Consider the time rate of change of the Hamiltonian, H dH/ dt, follow-ing the phase-space trajectory. Using the Hamiltonian equations of motion,eq. (3.12),dHdt = Ht + qHq + pHp= Ht + HpHq HqHp Ht . (3.30)Thus, if H does not depend explicitly on time, so its partial time deriva-tive vanishes, then its total time derivative also vanishes. That is, in anautonomous Hamiltonian system the Hamiltonian is an integral of motion.We have seen above that H can often be identied as the total energy.When this is the case we may interpret the completely general result above asa statement of conservation of energy. More generally, comparing eq. (3.10)and eq. (2.84) we recognize H as the negative of the energy integral I forautonomous systems as predicted by Noethers theorem.3.4 Hamiltons Principle in phase spaceLet us express the action integral S in terms of q and p and show that wemay derive Hamiltons equations of motion by making S stationary undervariations of the trajectory in phase space, rather than in conguration space.To nd a suitable denition for the action integral, we rst rearrangeeq. (3.10) to get L on the left-hand side. Then we replace u(q, p, t) by q inthe term pu and thus dene the phase-space LagrangianLph(q, q, p, t) p q H(q, p, t) , (3.31)3.4. HAMILTONS PRINCIPLE IN PHASE SPACE 51If q = u(q, p, t) were identically satised, even on arbitrarily varied phase-space paths, then Lph would simply be L expressed in phase-space coordi-nates. However, one can easily construct a counter example to show thatthis is not the case: consider a variation of the path in which we can varythe direction of its tangent vector, at some point (q, p), while keeping thispoint xed (see Sec. 4.2 for an illustration of this). Then q changes, but uremains the same. Thus Lph and L are the same value only on the subset ofpaths (which includes the physical paths) for which p q = pu(q, p, t).Replacing L by Lph in eq. (2.26) we dene the phase-space action integralSph[q, p] =

t2t1dt Lph(q, p, q, t) . (3.32)We shall now show that requiring Sph = 0 for arbitrary variations about agiven path implies that that path is such that Hamiltons equations of motionare satised at all points along it. Since we concluded above that Lph and Lwere not the same on arbitrary nonphysical paths, this variational principleis subtly dierent from the Lagrangian version of Hamiltons principle, andis sometimes called the modifed Hamiltons Principle.We know from variational calculus that Sph is stationary under arbitraryvariations of the phase-space path (with endpoints xed) if and only if theEulerLagrange equation