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" Sun Franeisco Bostor New York Capetown Hong Kong London Madnd Mexico City Montreal Mumch Paris Singapore Sydney Tokyo Toronto Conterts 4 E The Kinematics of Rigid Body Motion 134 4.1 The Independent Coordinates of a Rigid Body 134 4.2 Orthogonal Transtormations 139 4.3 Formal Properties of the Transformation Matrix 1d4 44 The Euler Angles 150 4.5 The Cayley-Klein Parameters and Related Quantities 154 4.6 Euler's Theorem on the Motion ofa Rigid Body 155 4.7 Finite Rotations 161 4.8 Infinitesimal Rotations 163 49 Rate ofChange ofa Vector 171 4.10 The Coriolis Effect 174 5 H The Rigid Body Equations of Motion 184 5.1 Angular Momentum and Kmetic Energy ot Motion abouta Point 184 5.2 Tensors 188 5.3 The Inertia Tensor and the Moment of Inertia 191 5.4 The Eigenvalues of the Inertia Tensor and the Principal Axis Transformation 195 5.5 Solving Rigid Body Problems and the Euler Eguations of Motion 198 5.6 Torque-free Motion of a Rigid Body 200 5.7 The Heavy Symmetrical Top with One Point Fixed 208 5.8 Precession of the Equinoxes and of Satellite Orbits 223 5.9 Piccession of Systems of Charges in 2 Magnetic Field 230 6 E Oscillations 238 6.1 Formulation ofthe Problem 238 6.2 The Eigenvalue Equation and the Principal Axis Transformation 241 6.3 Frequencies of Free Vibration, and Normal Coordinates 250 6.4 Free Vibrations of a Linear Triatomic Molecule 253 6.5 Forced Vibrations and the Effect of Dissipative Forces 259 66 Beyond Small Oscillations: The Damped Driven Pendulum and the Josephson Junction 265 7 E The Classical Mechanics of the Special Theory of Relativity 276 7.1 Basic Postulates of the Special Theory 277 7.2 Lorentz Transformations 280 73 Velocity Addition and Thomas Precession 282 74 Vectors and the Metric Tensor 286 Contents vii 7.5 1-Forms and Tensors 289 7.6 Forces in the Special Theory; Electromagnetism 297 7.7 | Relativistic Kinematics of Collisions and Many-Particle Systems 300 7.8 Relativistic Angular Momentum 309 7.9 The Lagrangian Formulation of Relativistic Mechanics 312 7.10 Covariant Lagrangian Formulations 318 7.11 Introduction to the General Theory of Relativity 324 8 E The Hamilton Equations of Motion 334 9H 10H 8.1 Legendre Transformations and the Hamilton Equations of Motion 334 82 Cyelic Coordinates and Conservation Theorems 343 8.3 Routh's Procedure 347 84 The Hamiltonian Formularion of Relativistic Mechanics 340 85 —Derivation of Hamilton's Equations from a Variational Principle 353 8.6 The Principle of Least Action 356 Canonical Transformations 368 9.1 The Equations of Canonical Transformation 368 9.2 Examples of Canonical Transformations 375 9.3 The Harmonic Oscillator 377 9.4 The Symplectic Approach to Canonical Transformations 381 9.5 Poisson Brackets and Other Canonical Invariants 388 9.6 Equations of Motion, Infinitesimal Canonical Transformations, and Conservation Theorems in the Poisson Bracket Formulation 396 9.7 The Angular Momentum Poisson Bracket Relations 408 9.8 Symmetry Groups of Mechanical Systems 412 9.9 Liouville's Theorem 419 Hamilton-Jacobi Theory and Action-Angle Variables 430 10.1 The Hamilton-Jacobi Equation for Hamilton's Principal Function 430 10.2 The Harmonic Oscillator Problem as an Example of the Hamilton-Jacobi Method 434 10.3 The Hamilton-Jacobi Equation for Hamilton's Characteristic Function 440 10.4 Separation of Variables in the Hamilton-Jacobi Equation 444 10.5 Ignorable Coordinates and the Kepler Problem 445 10.6 Action-angle Variables in Systems of One Degree of Freedom 452 Preface to the Third Edition The first edition of this text appeared in 1950, and it was so well received that it went through a second printing the very next year. Throughout the next three decades it maintained its position as the acknowledged standard text for the intro- ductory Classical Mechanics course in graduate level physics curricula through- out the United States, and in many other countries around the world. Some major institutions also used it for senior level undergraduate Mechanics. Thirty years later, in 1980, a second edition appeared which was “a through-going revision of the first edition” The preface to the second edition contains the following state- ment: “1 have tried to retain, as much as possible, the advantages of the first edition while taking into account the developmenits of the subject itself, its position in the curriculum, and its applications to other fields.” This is the philosophy which has guided the preparation oí this third edition twenty more years later. The second edition introduced one additional chapter on Perturbation Theory, and changed the ordering of the chapter on Small Oscillations. In addition it added a significant amount of new material which increased the number of pages by about 68%. This third edition adds still one more new chapter on Nonlinear Dy- namics or Chaos, but counterbalances this by reducing the amount of material in several of the other chapters, by shortening the space allocated to appendices, by considerably reducing the bibliography, and by omitting the long lists of symbols. Thus the third edition is comparable in size to the second. In the chapter on relativity we have abandoned the complex Minkowski space in favor of the now standard real metric. Two of the authors prefer the complex metric because of its pedagogical advantages (HG) and because it fits in well with Clifford Algebra formulations of Physics (CPP), but the desire to prepare students who can easily move forward into other areas of theory such as field theory and general relativity dominated over personal preferences. Some modem notation such as 1-forms, mapping and the wedge produet is introduced in this chapter. The chapter on Chaos is a necessary addition because of the current interest in nonlinear dynamics which has begun to play a significant role in applications of classical dynamics. The majority of classical mechanics problems and apph- cations in the real world include nonlineantes, and rt 15 important for the student to have a grasp of the complexities involved, and of the new properties that can emerge. It is also important to realize the role of fractal dimensionality in chaos. New sections have been added and others combined cr eliminated here and there throughout the book, with the omissions to a great extent motivated by the desire not to extend the overall length beyond that of the second edition. A section ix Preface to the Third Edition was added on the Euler and Lagrange exact solutions to the three body problem. Tn several places phase space plots and Lissajous figures were appended to illus- trate solutions. The damped driven pendulum was discussed as an example that explains the workings of Josephson junctions. The symplectic approach was clar- ified by writing out some of the matrices. The harmonic oscillator was treated with anisotropy, and also in polar coordinates. The last chapter on continua and fields was formulated in the modem notation introduced in the relativity chap- ter. The significances of the special unitary group in two dimensions SU(2) and the special orthogonal group in three dimensions SO(3) were presented in more up-to-date notation, and an appendix was added on groups and algebras. Special tables were introduced to clarify properties of ellipses, vectors, vector fields and 1-forms, canonical transformations, and the relationships berween the spacetime and symplectic approaches. Several of the new features and approaches in this third edition had been men- tioned as possibilities im the preface to the second edition, such as properties of group theory, tensors in non-Euclidean spaces, and “new mathematics” of theoret- ical physics such as manifolds. The reference to “One area omitted that deserves special artention—nonlinear oscillation and associated stability questions” now constitutes the subject matter of our new Chapter 11 “Classical Chaos.” We de- bated whether to place this new chapter after Perturbation theory where it fits more logically, or before Perturbation theory where it is more likely to be covered in class, and we chose the latter. The referees who reviewed our manuscript were evenly divided on this question. The mathematical level of the present edition is about the same as that of the first two editions. Some of the mathematical physics, such as the discussions of hermitean and unitary matrices, was omitted because it pertains much more to quantum mechanics than it does to classical mechanics, and little used nota- tions like dyadics were curtailed. Space devoted to power law potentials, Cayley- Klein parameters, Routh”s procedure, time independent perturbation theory, and the stress-energy tensor was reduced. In some cases reference was made to the second edition for more details, The problems at the end of the chapters were divided into “derivations” and “exercises)” and some new ones were added, The authors are especially indebted to Michael A, Unseren and Forrest M. Hoffman of the Oak Ridge National laboratory for their 1993 compilation of errata in the second edition that they made available on the Internet. It is hoped that not too many new errors have slipped into this present revision. We wish to thank the students who used this text in courses with us, and made a number of useful suggestions that were incorporated into the manuscript. Professors Thomas Sayeita and the late Mike Schuette made helpful comments on the Chaos chapter, and Professors Joseph Johnson and James Knight helped to clarify our ídeas on Lie Algebras. The following professors reviewed the manuscript and made many helpful suggestions for improvements: Yoram Alhassid, Yale University; Dave Ellis, University of Toledo; John Gruber, San Jose State; Thomas Handler, University of Tennessee; Daniel Hong, Lehigh University: Kara Keeter, Idaho State University; Carolyn Lee; Yannick Meurice, University of Iowa: Daniel 11 E Survey of the Elementary Principles The motion of material bodies formed the subject of some of the earliest research pursued by the pioneers of physics. From their efforts there has evolved a vast field known as analytical mechanics or dynamics, or simply, mechanics. In the present century the term “classical mechanics” has come into wide use to denore this branch of physics in contradistinction to the newer physical theories, espe- cially quantum mechanics. We shall follow this usage, interpreting the name to include the type of mechanics arising out of the special theory of relativity. Itis the purpose of this book to develop the structure of classical mechanics and to outlins some of its applications of present-day interest in pure physics. Basic to any presentation of mechanics are a number of fundamental physical concepts, such as space, time, simultaneity, mass, and force. For the most part, however, these concepts will not be analyzed critically here: rather, they will be assumed às undefined terms whose meanings are familiar to the reader. MECHANICS OF A PARTICLE Tetr be the radius vector of a particle from some given ongin and v its vector velocity: =—. LI a: EL) The linear momentum p of the particle is defined as the product of the particle mass and its velocity: p=my. [024] In consequence of interactions with external objects and fields, the particle may experience forces of various types, e.g. gravitational or electrodynamic; the vec- tor sum of these forces exerted on the particle is the total force F. The mechanics of the particle is contained in Newton's second law of motion, which states that there exist frames of reference in which the motion of the particle is described by the differential equation F="=p, (13) 4 Chapter 1 Survey of the Elementary Principles or F= Emo. (1.4) In most instances, the mass of the particle 15 constant and Eg. (1,4) reduces to d F=m = = ma. (15) where a is the vector acceleration of the particle defined by dr a 1.6) “= qo 16) The equation of motion is thus a differential equation of second order, assuming F does not depend on higher-order derivatives. A reference frame in which Eq. (1.3) is valid is called an inertial or Galilean system. Even within classical mechanics the notion of an inertial system is some- thing of an idealization. In practice, however, it is usually feasible to set up à co- ordinate system that comes as close to the desired properties as may be required. For many purposes, a reference frame fixed in Earth (the “laboratory system”) is a sufficient approximation to an inertial system, while for some astronomical pur- poses it may be necessary to construct an inertial system by reference to distant galaxies. Many of the important conclusions of mechanics can be expressed in the form of conservation theorems, which indicate under what conditions various mechan- ical quantities are constant in time. Equation (1.3) directly fumishes the first of these, the Conservation Theorem for the Linear Momentum af a Particle: Ifthe total force, F, is zero, then p = O and the linear momertum, p, is conserved. The angular momentum of the particle about point O, denoted by L, is defined as L=rxp, (7 where r is the radius vector from O to the particle. Notice that the order of the factors is important, We now define the moment of force or torque about O as N=rxF. €1,8) The equation analogous te (1.3) for N is obtained by forming the cross product of r with Eg. (1.4): d =N= — . IR rxF=N rx cetmv) 9 Chapter 1 Survey of the Elementary Principles Physically it is clear that a system cannot be conservative if iriction or other dis- sipation forces are present, because F + ds due to friction is always positive and the integral cannot vanish. By a well-known theorem of vector analysis, a necessary and sufficient condi- tion that the work, Wj2, be independent of the physical path taken by the particle is that F be the gradient of some scalar function of position: F=-VV(, (1.15) where V is called the potenrial, or potential energy. The existence of V can be inferred imtutively by a simple argument, Tf Wiz is independent of the path of integration between the end points 1 and 2, it should be possible to express Wiz as the change in a quantity that depends only upon the positions of the end points, This quantity may be designated by —V, so that for a differential path length we have the relation F.ds=-dVv or ov és" F= which is equivalent to Eq. (1.16). Note that in Eg. (1.16) we can add to V any quantity constant in space, without affecting the results. Hence the zero level of V is arbitrary. For a conservative system, the wotk done by the forces is Wo =Vi—vo. (17) Combining Eq. (1.17) with Eq. (1.14), we have the result n+W=B+V, (1.18) which states in symbols the Energy Conservation Theorem for a Particle: If the forces acting on a particle are conservative, then the totul energy of the particle, T + V, is conserved. The force applied to a particle may in some circumstances be given by the gradient of a scalar function that depends explicitly on both the position of the particle and the time. However, the work done on the particle when it travels a distance ds, Feds= q ds, is then no longer the total change in —Y during the displacement, since V also changes explicitly with time as the particle moves. Hence, the work done as the 1.28 1.2 Mechanics of a System of Particles 5 particle goes from point 1 to point 2 is no longer the difference in the function V between those points. While a total enerpy T + V may still be defined, it is not conserved during the course of the particle's motion. MECHANICS OF A SYSTEM OF PARTICLES In generalizing the ideas of the previous section to systems of many particles, we must distinguish between the external forces acting on the particles due to sources outside the system, and internal forces on, say, some particle i due to all other particles in the system. Thus, the equation of motion (Newton's second law) for the Fth particle is written as DEu+FO =p, (119) 4 where Fº stands for an external force, and E, is the internal force on the ith particle due to the jth particle (F,,, naturally, is zero). We shall assume that the F,, (like the 1) obey Newton's third law of motion in its original form: that the forces two particles exert on each other are equal and opposite. This assumption (which does not hold for all types of forces) is sometimes referred to as the weak law af action and reaction Summed over all particles, Eg. (1.19) takes the form a Smr = pr? + DE (1.20) és The first sum on the right is simply the total external force F(9, while the second term vanishes, since the law of action and reaction states that each pair F,; +F,, is zero. To reduce the left-hand side, we define a vector R as the average of the radii vectors of the particles, weighted in proportion to their mass: R- mir, - Lmr; Sm; Moo The vector R defines a point known as the center of mass, or more loosely as the center of gravity, of the system (cf. Fig. 1.1). With this definition, (1.20) reduces to (1.21) na = pEº= = Fº, (122) which states that the center of mass moves as if the total external force were acting on the entire mass of the system concentrated at the center of mass. Purely internal forces, if the obey Newton's third law, therefore have no effect on the 1.2 Mechanics of a System of Particles 7 FIGURE 1.2 The vector r;; between the ith and jth particles. using the equality of action and reaction But r; — r, is identical with the vector E; from j to é (cf, Fig. 1.2), so that the right-hand side of Eq. (1.25) can be wntten as Fry XF,. Jf the internal forces between two particles, in addition to being equal and oppo- site, also lie along the line joining the particles—a condition known as the strong law of action and reaction—then all of these cross products vanish. The sum over paírs is zero under this assumption and Eg. (1.24) may be written im the form dL — =Nº, 1.26 FT (1.26) The time derivative of the total angular momentum is thus equal to the moment of the external force about the given point. Corresponding to Eg. (1.26) is the Conservation Theorem for Total Angular Momentum: L is constant in time if the applied (external) torque is zero. (K is perhaps vonkwiile to emphasize that this is à vector theorem; Le, L; will be conserved if Nº is zero, even ENO and nt are not zero.) Note that the conservation of linear rmomentum in the absence of applied forces assumes that the weak law of action and reaction is valid for the internal forces. The conservation of the total angular momentum of the system in the absence of applied torques requires the validity of the strong law of action and reaction—that the internal forces in addition be central. Many of the familiar physical forces, such as that of gravity, satisfy the strong form of the law. But it is possible to find forces for which action and reaction are equal even though the forces are not central (see below). In a system involving moving charges, the forces between the charges predicted by the Biot-Savart law may indeed violate both forms of Chapter 1 Survey of the Elementary Principles the action and reaction law.* Equations (1.23) and (1.26), and their corresponding conservation theorems, are not applicable in such cases, at least in the form given here. Usually it is then possible to find some generalization of P or L that is conserved. Thus, in an isolated system of moving charges it is the sum of the mechanical angular momentum and the electromagnetic “angular momentum” of the field that is conserved. Equation (1.23) states that the total linear momentum of the system is the same as if the entire mass were concentrated at the center of mass and moving with it. The analogous theorem for angular momentum is more complicated. With the ongin O as reference point, the total angular momentum of the system is L=5r xp. t Let R be the radius vector from O to the center of mass, and let 1 be the radius vector from the center of mass to the ith particle, Then we have (cf. Fig. 1.3) nr=E+R (127) and v=v+v where - R dt Center of mass FIGURE 1.3 The vectors involved in the sluft of reference point for the angular momen- tum. *IÉ two charges are moving unformly with parallel velocity vectois that are not perpendicular to the line joining the charges, then tac net mutual forces are equal and opposite but do not le along the vector between the charges. Consider, further, two charges moving (instantaneously) so as to “cross lhe Tº 1.., onc charge moving directly at the other, wluch in tum 1s moving at right angles to the first Then the second charge exerts a nonvanishing magnetic force on lhe first, without experencing any magnetic reaction torce at that mstant. Chapter | Survey of the Elementary Principles Making use of the transformations to center-of-mass coordinates, given in Eg. (1,27), we may also write T as =5 mtv) err) 1 1 d = qbmvt mtv, (Dm) and by the reasoning already employed in calculating the angular momentum, the last term Yanishes, leaving 1 ! T= Mv? +; ma? (1.31) T The kinetic energy, like the angular momentum, thus also consists of two parts: the kmetic energy obtained if all the mass were concentrated at the center of mass, plus the kinetic energy of motion about the center of mass. Consider now the right-hand side of Eq. (1.29). In the special case that the external forces are derivable in terms of the gradient of a potential, the first term can be written as 2 W4 Lf Eds =—5 | Wu -dy=-30 where the subscript i on the del operator indicates that the derivatives are with respect to the components of r,. If the internal forces are also conservative, then the mutual forces between the ith and jth particles, F;, and F,,, can be obtained from a potential function V,,. To satisfy the strong law of action and reaction, V,, can be a function only of the distance between the particles; 2 , 1 Voy = Yolr—r,D. (137) The two forces are then automatically equal and opposite, F,=-VNVMy=+V,Vy=—Fy. (1.33) and lie along the line joining the two particles, VVilm—rD=(m—r)/f (1.34) where j is some scalar function. If V,, were also a function of the difference of some other pair of vectors associated with the particles, such as their velocities or (to step into the domain of modem physics) their intrinsic “spin” angular mo- menta, then the forces would still be equal and opposite, but would not necessarily lie along the direction between the particles. 1.2 Mechanics of a System of Particles “ When the forces are all conservative, the second term m Eg. (1.29) can be rewritten as à sum over pairs of particles, the terms for each pair being of the form 2 -f (VVj eds + V)Vy ds). I Tf the difference vector r; — r, is denoted by r;;, and if V;; stands for the gradient with respect to r,,, then VV = VyPy= VV. and ds —ds,=dr, —dr, = dr, so that the term for the i; pair has the form - frsv + dr. The total work arising from internal forces then reduces to 2 1 2 1 52h Vi Vi dry ==30%y . ] I vês ráj (135) The factor i appears in Eg. (1.35) because in summing over both i and j each member of a given pair 's included twice, first in the i summation and then in the 4 summation. From these considerations, it is clear that 1í the external and internal forces are both derivable from potentials it is possible to define a totai potential energy, V, of the system, l V= D+. (1.36) 1) such that the total energy T + V is conserved, the analog of the conservation theorem (1.18) for a single particle, The second term on the right in Eq. (1.36) will be called the internal potential energy of the system. In general, it need not be zero and, more important, it may vary as the system changes with time. Only for the particular class of systems known as rigid bodies will the internal potential always be constant. Formally, a rigid body can be defined as a system of particles in which the distances r, are fixed and cannot vary with time. In such case, the vectors dr;; can only be perpendicular to the corresponding r,,, and therefore to the F;;. Therefore, m a rigid body the internal forces do no work, and the internal potential must remain 
