This book is written with the belief that classical mecha nics, as a theoretical discipline, possesses an inherent beauty, depth, and richness that far t ranscend its immediate a pplications in mecha nical systems, a lt hough these a re no doubt important in t heir own right. These properties a re ma nifested , by and large,through the cohe rence a nd elegance of the mathematical structure underlying the discipline, and, at least in the opinion of t his author, a reeminently worthy of being communicated to physics students at the earliest stage possible. The present text is therefore add ressed mainly to advanced undergraduate and beginning graduate physics students who are interested in an appreciation of the relevance of modern mathematical methods in classical mechanics, in particular, those derived from t he much intertwined fields of topology and differential geometry, and also to the occasional mathematics student who is interested in important physics applications of these a reas of mathematics. Its chief purpose is to offer an introductory and broad glimpse of the ma jestic edifice of the mathematical theory of classical dynamics, not only in the time-honored analytical tradition of Newton, Laplace, Lagrange, Hamilton , Jacobi , and Whittaker,but also the more topological/geometrical one established by Poincare, and enriched by Birkhoff, Lyapunov, Smale, Siegel, Kolmogorov, Arnold, and Moser(as well as ma ny others). The latter t radition has been somewhat inexplicably and politely ignored for many decades in the 20th century within the realm of physics instruction, and has only rela tively recently regained favor in some physics textbooks under the guise of t he more fashionable topics of chaos and complexity in dynamical systems theory. T his unfortunate circumstance may perhaps in hindsight be attributa ble to rela ted historical events: the rise of quantum mechanics just as Poincare 's contributions in celestial mechanic were coming to the fore at the dawn of the 20th century, and the subsequent competition for limited "curricular space" in physics instruction between class ical and quantum mechanics. The irony from a historical perspective, of comse, is that it was precisely the Hamilton-Jacobi theory within the Hamiltonian formulation of classical mecha nics and its relationship to wave optics that precipitated the development of non-relativistic quantum theory, and that the formalism of action-angle va riables in Hamiltonian mechanics, through the Bol1r-Sommerfeld quantization rules, proved to be the royal road to the "old quantum theory". In addition, it was the Lagrangian formulation of classical mecha nics that provided the groundwork for the Feynman path approach in quantum field theory.
We hope that the present text will make a modest contribution, a long with many other excellent ones that have already appeared , towards the rehabilitation of Poincare's tradition in the mainstream physics curricula. Because of the pervasive topological and geometrical cha racter of this tradition, it is inevitable that a coherent, if omewhat spotty, introductory exposit ion ( without straying too far into the broader field of dynamical systems) of a key group of concepts and tool of topology and differential geometry will have to be a central feature of the presenta tion, as well a an earnest attempt to convince the reader of the fundamental relevance of these mathema tical tools in classical mechanics. These objectives give rise to the adoption of a "dual-track" character of the book, alternating between physics and mathematics. Our hope is to strike a delicate balance between the int uitive/ physically specific and the abstract/mathematica lly general. T he presentation will be mainly at a heuristic level, la rgely uubmdened by a rigorous, systematic, and lengthy presenta tion of the mathematical background. The inevitable loss of generality, rigor, and even accuracy entailed by such a style of "physical" presentation will hopefully be compensated for by a sound degree of continuity from the physics to the mathematics (and vice versa) , and by ample explicit calculations based on carefully chosen a pplications of the abstract mathematical machinery.
conventional analytical treatment of the subject (based on vector calculus , multivariable calculus, and the solutions of differential equa tions) as presented in almost all undergradua te texts and most graduate ones, and substantially enrich the exposition with a healt hy dose of the modern language and tools of topology and differential geometry, especially those related to the basic notions of differentiable manifolds, tangent and cotangent spaces, the exterior differential calculus, homotopy, homology and cohomology, connections on fiber bundles,Riemannian geometry, symplectic geometry, and Lie groups and algebras. The main pedagogical route to the introduction and elucidation of most of these mathematical topics will be the use of Cartan's Method of Moving Frames in the classical mechanical context of rigid-body dynamics, as promulgated by the geometer S. S. Chern. Historically, the mathematical development of the differential geometric theory of moving frames was in fact originally motivated by this classical mechanics problem. We seek to erect a pedagogical bridge, as it were,between the time-honored texts of, for example, Goldstein's " Classical Jvlechanics'on the one hand and Abraham and Marsden's "Foundations of Mechanics"on the other - allbeit at a more elementary and much less rigorous level. In this vein , the classic work " Mathematical Methods of Classical Mechanics" by Arnold immediately comes to mind as a sort of "gold standa rd"; but a.gain,our aim is more modest: not only do we seek to build a bridge between the physical/analytical and the mathematical/geometrical ,but a lso one between the undergraduate and graduate curricula. Due to the immense richness and depth of the field , however, we cannot lay claim to any degree of completeness or origina lity of treatment.
At the risk of alienat ing the more mathematically oriented reader - this is,after all, primarily a physics text - I have laid down the following guidelines for the writing of this book in regard to the sometimes tortured relationship between the physics and the mathematics contained therein.
Except on occasion, when clarity, conciseness, and precision trump intuitive appeal, the usual a brupt definit ion- theorem- proof sequence encountered in the mathematical literature will be replaced by a more casual build-up with physical motivation and justification leading to the relevant mathematical concepts and facts. For example, the discussion of the mathema tical equivalence of gauge fields and connect ions on fi ber bundles is essentia lly bui lt up from the elementa ry mecha nical context of rigid bodies moving in Euclidean space. As another example, the discussion of the releva nce of symplectic struct ures on ma nifolds and the crucial importance of Da rboux 's Theorem regarding symplectic ma nifolds will be preceded by a good amount of justification in more famil iar analytical language. Once the informa l build-up has achieved its purpose, however,we will not shy away from exp loit ing the logical sharpness, elega nce and beauty of abstract concepts to a rrive a t results quickly, for example, in demonstra ting the existence of certain integral invariants under canonicalt ransformations (symplectomorphisms) in phase space.
The analytical mode of development (because of its familiarity and concreteness)will almost always be presented first , as a precursor to and motivation for t he more a bstract and unfamiliar geome trical development.For example, Poisson brackets will be int roduced first in the traditional"physics" manner (in terms of par tial derivatives with respect to canonical coordina tes and momenta) before being defined in terms of the value of the symplectic form on two vector fields on phase space. Once the geometrical not ions have been firm ly grounded in ana lytical representa tions, however,the strong interplay between the analy tical (local) and the geome trical(globa l) viewpoints will be stressed .
Within the geometrical exposition , (local) coordina tes will almost always be used first in favor of t he coordinate-free a pproach usua lly preferred bymathema ticia ns. Darboux's Theorem guarantees that this can be done with impunity in the case of Hamiltonian mecha nics, but in many other situation , such as the use of moving frames for rigid-body dynamics a precursor to the int roduction of frame bundles and the introduct ion of connections on principal bundles (gauge field ), the d escription in terms of local coordina tes is usually much more int uitive. In addition, it yields easily interpre ta ble analytical formulas for calcula tions , even though they may only be locally valid.
When precise mathematical definitions a re unavoidable, we strive to avoid layering unfamiliar , abstract, and overly technical ones in quick succession.If possible, specific examples will be used to motivate definitions of a bstract concepts. For example, the general notion of connections on vector bundles is introduced t hrough the much more familiar kinematical.notion of angular velocities. As another example, the intuitive picture of two-dimensional KAM tori interlaced in three-dimensional energy by persurfaces of fom-dimensional phase space is used to motivate the notion of foliated spaces in general.
.Whole chapters serving as mathema tical interludes dealing with specia lized topics a re placed at strategic positions within the text, rather than being lumped together entirely aL the beginning of the book, or relegated to a set of appendices at the end. (We plead guilty - partially - to violating this rule at the eginning of the text, where we feel that it is necessary to engage in a higher concentration of mathematical presentation; buteven t here, important physics is linked to the mathematics whenever possible.We hope that this practice will avoid having the patience of the non- mathematical reader taxed excessively, a nd preserve our stylistic goal of interweaving the physics with the mathematics. Examples are chapters dealing with the following topics: Exterior Calculus, Vector Calculus by Different ial Forms, Lie Groups and foving Frames, Connections on Fiber Bundles, Riemannian Cmvatw-e, and Symplectic Geometry.
Full proofs of mathematical theorems are usually not presented, especially those involving a large amount of technical details. P roofs a re given only when they are rela tively direct, simple and illustrative of useful calculational techniques.
The physics prerequis ites for this book can be simply stated: a solid lowerdivision course in classical mechanics. The mathematical prerequisites, however,are more nebulous, and, as can be expected , a bit more demanding. '0le hope the reader wiJJ have a good working knowledge of multivariable calculus, vectorcalculus (div, grad, cw·! a nd all tha t), some linear algebra (matrices and eigenvalue problems) , some complex va ria ble theory, elements of ordinary and partial differential equations, and finally, some expo ure to intductory tensorana lysis. Beyond t hese, everyt hing is essent ially developed from the ground up in the text , a lthough, as we stipulated before, not comprehensively and rigorously.A certain degree of inna te cur iosity and what is often loosely referred to as ma thema tical ma tw-ity will also be useful.
The material contained in this book, its organization, and even the ped agogicals trategies adopted in the delicate dance between t he physics and the mathema tics presenta tions, might be gleaned from the Chapter titles and their ordering. There is proba bly more than enough material for a year-long syllabus,but students and instructors should be able to adopt select port ions according to their own inte rests and preferences without too much difficulty. The individual chapters are intended to be as t hemat ically, and even on many occasions as technically, self-conta ined as po ible. But no less important, we have also endeavored to render t he entire sequencing into a coherent and logical flow. Numerous xercise problems, many of them suppUed with generous and systematichints, are strat egically located within the running narrative of the text. It goes without saying that these form an integral part of the text, and the reader is urged to attempt as many of them as possible.
Special care has been devoted to cross-referencing of material for easy look up by the reader, especially with rega rd to possibly unfamiliar physical or mathematical concepts.Within the text t his will be clone by thorough and careful references (forward and backward) through specifi c equation numbers, theorem and definition numbers, or direction of t he reader to specific locations in the text. Outs ide of the text, every effort has been made to make the index as serviceable as possible.PA/About 40% of the material in this book, in le s developed and organized form, were used at one time or another as class notes made available to students over the past decade or so when I taught t he upper-division Classical Mechanics sequence at Cal Poly Pomona. Even though these tended to comprise the more physical parts of the text, t he a pproach taken was perhaps more mathematical than customa ry and at t imes may have been a bit idiosyncratic.But throughout t his period and for the most part, my students have been kind enough to put up with it a nd indeed provided honest and constructive feedback,and even occasional encouragement. It is to them that I would like to first and foremost express my gratitude.No less important to the task of writing was the collegial and congenial atmo phere afforded by my home department that I am fortu na te enough to count myself a member of for the better part of the last three decades. Dr. Stephen McCauley, my current Department Chair in particular, was helpful and encouraging in every way possible. Amidst the inevitable heavy teaching duties at times, t he Faculty Sabbatical Program of the California State University provided much needed focused periods of time to be devoted to this book project . In the matter of intellectual debt, I would like to single out and expre s my sincere gratitude to the late eminent differential geometer Professor S. S. Chern, who graciously and generously included me in a d iffe rential geometry book t rans lation and expansion project some 15 years ago, and thus paved t he way for my own modest book projects aimed at familiarizing physics students with that subject, of which the present text is the latest one. Much of t he mathematical material on differential geometry in this book will bear the imprint of the valuable teachings and insights of Professor Chern, while the errors and inaccuracie remain of course entirely my own.Ms.E . H. Chionh of World Scientific Publishing Company, my able editor for my last publis hed text, has again shepherded me with nothing but the highest standards of professionalism through t he present one. To her I again extend my a ppreciation. Finally, my dear wife, Dr. Bonnie Buratti , and otu- three sons.Nathan, Reuben , and Arron, have all been, in their diferent ways, the bed.rock as well as the inspirational muse of my life th rough the ma ny ups and downs while this book was in progress over the past several years. To them I would like to say, as always and simply, thank you.

1 Vectors, Tensors, and Linear Transformations 1

2 Exterior Algebra: Determinants, Oriented Frames and Oriented Volumes 19

3 The Hodge-Star Operator and the Vector Cross Product 33

4 Kine matics and Moving Frames:From the Angular Velocity to Gauge Fields 41

5 Differentiable Manifolds:The Tangent and Cotangent Bundles 53

6 Exterior Calculus: Differential Forms 63

7 Vector Calculus by Differential Forms 73

8 The Stokes Theorem 77

9 Cartan's Method of Moving Frames:Curvilinear Coordinates in IR3 91

10 Mechanical Constraints: The Frobenius Theorem 103

11 Flows and Lie Derivatives 109

12 Newton's Laws: Inertial and Non-inertial Frames 117

13 Simple Applications of Newton's Laws 127

14 Potential Theory: Newtonian Gravitation 147

15 Centrifugal and Corio lis Forces 163

16 Harmonic Os cillato1's: Fourier Transforms and Green's Functions 171

17 Classical Model of the Atom: Power Spectra 183

18 Dynamical Systems and their Stabilities 191

19 Many-Particle Systems and the Conservation Principles 209

20 Rigid-Body Dynamics:The Euler-Poisson Equations of Motion 217

21 Topology and Systems with Holonomic Constraints:Homology and de Rham Cohomology 231

22 Connections on Vector Bundles :Affine Connections on Tangent Bundles 241

23 The Parallel Translation of Vectors: The Foucault Pendulum 253

24 Geometric Phases, Gauge Fie lds, and the M echanics of Deformable Bodies: The "Falling Cat" Problem 261

25 Force and Curvature 273

26 The Gauss-Bonnet-Chern Theorem and Holonomy 291

27 The Curvature Te n sor in Riemannian G eometry 299

28 Frame Bundles and Principal Bundles,Connections on Principal Bundles 317

29 Calculus of Variations, the Euler-Lagrange Equations,the First Variation of Arclength and Geodesics 329

30 The Second Vai-iation of Arclength, Inde x Forms ,and Jacobi Fields 345

31 The Lagrangian Formulation of Classical Mechanics:Hamilton's Principle of Least Action, Lagrange Multipliers in Constrained Motion 357

32 Small Oscillations and Normal Modes 367

33 The Hamiltonian Formulation of Classical Mechanics :Hamilton's Equations of Motion 381

34 Symmetry and Conservation 399

35 Symme tric Tops 403

36 Canonical Transformations and the Symplec tic Group 411

37 Generating Functions and the Hamilton-Jacobi Equation 423

38 Integrability, Invariant Tori, Action-Angle Variables 445

39 Symplectic Geome try in Hamiltonian Dynamics,Hamiltonian Flows, and Poincare-Cartan Integral Invariants 467

40 Darboux's Theorem in Symplectic Geometry 479

41 The Kolmogorov-Arnold-Moser (KAM) Theorem 487

42 The Homoclinic Ta ngle and Instability, Shifts as Subsystems 521

43 The Restricted Three-Body Problem 547

References 561

Index 565

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