differential geometry with applications to

DIFFERENTIAL GEOMETRY WITHAPPLICATIONS TOMECHANICS AND PHYSICS

PURE AND APPLIED MATHEMATICS

A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS

Earl J. TaftRutgers University

New Brunswick, New Jersey

EDITORIAL BOARD

Zuhair NashedUniversity ofDelaware

Newark. Delaware

M S. BaouendiUniversity ofCalifornia,

San Diego

Jane CroninRutgers University

JackK. HaleGeorgia Institute ofTechnology

S. KobayashiUniversity ofCalifornia,

Berkeley

Marvin MarcusUniversity ofCalifornia.

Santa Barbara

w. S. MasseyYale University

AnilNerodeCornell University

Donald PassmanUniversity ofWisconsin,Madison

Fred S. RobertsRutgers University

David L. RussellVirginia Polytechnic Instituteand State University

Walter SchemppUniversitiit Siegen

Mark TeplyUniversity ofWisconsin.Milwaukee

MONOGRAPHS AND TEXTBOOKS INPURE AND APPLIED MATHEMATICS

1. K. Yano, Integral Formulas In Riemannian Geometry (1970)2. S. Kobayashi, Hyperbolic Manifolds and Holomorphlc Mappings (1970)3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; A. Littlewood,

trans.) (1970)4. B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation

ed.; K. Makowski, trans.) (1971)5. L. Narici et al., Functional Analysis and Valuation Theory (1971)6. S. S. Passman, Infinite Group Rings (1971)7. L. Domhoff, Group Representation Theory. Part A: Ordinary Representation Theory.

Part B: Modular Representation Theory (1971,1972)8. W Boothby and G. L. Weiss, eds., Symmetric Spaces (1972)9. Y. Matsushima, Differentiable Manifolds (E. T. Kobayashi, trans.) (1972)

10. L. E. Ward, Jr., Topology (1972)11. A. Babakhanian, Cohomological Methods in Group Theory (1972)12. R. Gilmer, Multiplicative Ideal Theory (1972)13. J. Yeh, Stochastic Processes and the Wiener Integral (1973)14. J. Barros-Neto, Introduction to the Theory of Distributions (1973)15. R. Larsen, Functional Analysis (1973)16. K. Yano and S. Ishihara, Tangent and Cotangent Bundles (1973)17. C. Procesi, Rings with Polynomial Identities (1973)18. R. Hermann, Geometry, Physics, and Systems (1973)19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973)20. J. Dieudonne, Introduction to the Theory of Formal Groups (1973)21. I. Vaisman, Cohomology and Differential Forms (1973)22. B.-Y. Chen, Geometry of Submanlfolds (1973)23. M. Marcus, Finite Dimensional Multilinear Algebra (In two parts) (1973,1975)24. R. Larsen, Banach Algebras (1973)25. R. O. Kujala and A. L. Vitter, eds., Value Distribution Theory: Part A; Part B: Deficit

and Bezout Estimates by Wilhelm Stoll (1973)26. K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974)27. A. R. Magid, The Separable Galois Theory of Commutative Rings (1974)28. B. R. McDonald, Finite Rings with Identity (1974)29. J. Satake, Linear Algebra (5. Koh et aI., trans.) (1975)30. J. S. Golan, Localization of Noncommutative Rings (1975)31. G. Klambauer, Mathematical Analysis (1975)32. M. K. Agoston, Algebraic Topology (1976)33. K. R. Goodearl, Ring Theory (1976)34. L. E. Mansfield, Linear Algebra with Geometric Applications (1976)35. N. J. Pullman, Matrix Theory and tts Applications (1976)36. B. R. McDonald, Geometric Algebra Over Local Rings (1976)37. C. W. Groetsch, Generalized Inverses of Linear Operators (1977)38. J. E. Kuczkowski and J. L. Gersting, Abstract Algebra (1977)39. C. O. Christenson and W. L. Voxman, Aspects of Topology (1977)40. M. Nagata, Field Theory (1977)41. R. L. Long, Algebraic Number Theory (1977)42. W. F. Pfeffer, Integrals and Measures (1977)43. R. L. Wheeden and A. Zygmund, Measure and Integral (1977)44. J. H. Curtiss, Introduction to Functions of a Complex Variable (1978)45. K. Hrbacek and T. Jech, Introduction to Set Theory (1978)46. W. S. Massey, Homology and Cohomology Theory (1978)47. M. Marcus, Introduction to Modem Algebra (1978)48. E. C. Young, Vector and Tensor Analysis (1978)49. S. B. Nadler, Jr., Hyperspaces of Sets (1978)SO. S. K Segal, Topics in Group Kings (1978)51. A. C. M. van Rooij, Non-Archimedean Functional Analysis (1978)52. L. Corwin and R. Szczarba, Calculus In Vector Spaces (1979)53. C. Sadosky, Interpolation of Operators and Singular Integrals (1979)54. J. Cronin, Differential Equations (1980)55. C. W Groetsch, Elements of Applicable Functional Analysis (1980)

56. I. Vaisman, Foundations ofThree-Dimensional Euclidean Geometry (1980)57. H. I. Freedan, Deterministic Mathematical Models in Population Ecology (1980)58. S. B. Chae, Lebesgue Integration (1980)59. C. S. Rees et al., Theory and Applications of Fourier Analysis (1981)60. L. Nachbin, Introduction to Functional Analysis (R. M. Aron, trans.) (1981)61. G. Orzech and M. Orzech, Plane Algebraic Curves (1981)62. R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysis

(1981)63. W. L. Voxman and R. H. Goetschel, Advanced Calculus (1981)64. L. J. Corwin and R. H. Szczarba, Multivariable Calculus (1982)65. V. I. Istra,tescu, Introduction to Linear Operator Theory (1981)66. R. D. Jiirvinen, Finite and Infinite Dimensional Linear Spaces (1981)67. J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry (1981)68. D. L. Annacost, The Structure of Locally Compact Abelian Groups (1981)69. J. W. Brewer and M. K. Smith, eds., Emmy Noether: A Tribute (1981)70. K. H. Kim, Boolean Matrix Theory and Applications (1982)71. T. W. Wieting, The Mathematical Theory of Chromatic Plane Omaments (1982)72. D. B.Gauld, Differential Topology (1982)73. R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983)74. M. Canneli, Statistical Theory and Random Matrices (1983)75. J. H. Carruth et al., The Theory of Topological Semigroups (1983)76. R. L. Faber, Differential Geometry and Relativity Theory (1983)77. S. Bamett, Polynomials and Linear Control Systems (1983)78. G. Karpilovsky, Commutative Group Algebras (1983)79. F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings (1983)80. I. Vaisman, A First Course in Differential Geometry (1984)81. G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1984)82. T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1984)83. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive

Mappings (1984)84. T. Albu and C. Nasfasescu, Relative Finiteness in Module Theory (1984)85. K. Hrbacek and T. Jech, Introduction to Set Theory: Second Edition (1984)86. F. Van Oystaeyen andA. Verschoren, Relative Invariants of Rings (1984)87. B. R. McDonald, Linear Algebra Over Commutative Rings (1984)88. M. Namba, Geometry of Projective Algebraic Curves (1984)89. G. F. Webb, Theory of Nonlinear Age-Dependent PopUlation Dynamics (1985)90. M. R. Bremner et al., Tables of Dominant Weight Multiplicities for Representations of

Simple Lie Algebras (1985)91. A. E. Fekete, Real Linear Algebra (1985)92. S. B. Chae, Holomorphy and Calculus in Normed Spaces (1985)93. A. J. Jerri, Introduction to Integral Equations with Applications (1985)94. G. Karpilovsky, Projective Representations of Finite Groups (1985)95. L. Narici and E. Beckenstein, Topological Vector Spaces (1985)96. J. Weeks, The Shape of Space (1985)97. P. R. Gribik and K. O. Kortanek, Extremal Methods of Operations Research (1985)98. J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic Analysis

(1986)99. G. D. Crown et al., Abstract Algebra (1986)

100. J. H. Carruth et al., The Theory ofTopological Semigroups, Volume 2 (1986)101. R. S. Doran and V. A. Belfl, Characterizations of C·-Algebras (1986)102. M. W. Jeter, Mathematical Programming (1986)103. M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations with

Applications (1986)104. A. Verschoren, Relative Invariants of Sheaves (1987)105. R. A. Usmani, Applied Linear Algebra (1987)106. P. Blass and J. Lang, Zariski Surfaces and Differential Equations In Characteristic p >

0(1987)107. J. A. Reneke et al., Structured Hereditary Systems (1987)108. H. Busemann and B. B. Phadke, Spaces with Distinguished Geodesics (1987)109. R. Harte, Invertibility and Singularity for Bounded Linear Operators (1988)110. G. S. Ladde et al., Oscillation Theory of Differential Equations with Deviating Argu­

ments (1987)111. L. Dudkin et al., Iterative Aggregation Theory (1987)112. T. Okubo, Differential Geometry (1987)

113. D. L. Stancl and M. L. Stanc/, Real Analysis with Point-Set Topology (1987)114. T. C. Gard, Introduction to Stochastic Differential Equations (1988)115. S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux (1988)116. H. Strade and R. Famsteiner, Modular Lie Algebras and Their Representations (1988)117. J. A. Huckaba, Commutative Rings with Zero Divisors (1988)118. W. D. Wallis, Combinatorial Designs (1988)119. W. Wi~slaw, Topological Fields (1988)120. G. Karpilovsky, Field Theory (1988)121. S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded

Rings (1989)122. IN. Kozlowski, Modular Function Spaces (1988)123. E. Lowen-Co/ebunders, Function Classes of Cauchy Continuous Maps (1989)124. M. Pavel, Fundamentals of Pattem Recognition (1989)125. V. Lakshmikantham et al., Stability Analysis of Nonlinear Systems (1989)126. R. Sivaramakrishnan, The Classical Theory of Arithmetic Functions (1989)127. N. A. Watson, Parabolic Equations on an Infinite Strip (1989)128. K. J. Hastings, Introduction to the Mathematics of Operations Research (1989)129. B. Fine, Algebraic Theory of the Bianchi Groups (1989)130. D. N. Dikranjan et a/., Topological Groups (1989)131. J. C. Morgan II, Point Set Theory (1990)132. P. Biler and A. Witkowski, Problems in Mathematical Analysis (1990)133. H. J. Sussmann, Nonlinear Controllability and Optimal Control (1990)134. J.-P. Florens et al., Elements of Bayesian Statistics (1990)135. N. Shell, Topological Fields and Near Valuations (1990)136. B. F. Doolin and C. F. Martin, Introduction to Differential Geometry for Engineers

(1990)137. S. S. Holland, Jr., Applied Analysis by the Hilbert Space Method (1990)138. J. Oknfnski, Semigroup Algebras (1990)139. K. Zhu, Operator Theory in Function Spaces (1990)140. G. B. Price, An Introduction to Multicomplex Spaces and Functions (1991)141. R. B. Darst, Introduction to Linear Programming (1991)142. P. L Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991)143. T. Husain, Orthogonal Schauder Bases (1991)144. J. Foran, Fundamentals of Real Analysis (1991)145. W. C. Brown, Matrices and Vector Spaces (1991)146. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces (1991)147. J. S. Golan and T. Head, Modules and the Structures of Rings (1991)148. C. Small, Arithmetic of Finite Fields (1991)149. K. Yang, Complex Algebraic Geometry (1991)150. D. G. Hoffman et al., Coding Theory (1991)151. M. O. GonzfJlez, Classical Complex Analysis (1992)152. M. O. GonzfJlez, Complex Analysis (1992)153. L. W. Baggett, Functional Analysis (1992)154. M. Sniedovich, Dynamic Programming (1992)155. R. P. Agarwal, Difference Equations and Inequalities (1992)156. C. Brezinski, Biorthogonality and Its Applications to Numerical Analysis (1992)157. C. Swartz, An Introduction to Functional Analysis (1992)158. S. B. Nadler, Jr., Continuum Theory (1992)159. M. A. AI-Gwaiz, Theory of Distributions (1992)160. E. Perry, Geometry: Axiomatic Developments with Problem Solving (1992)161. E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and

Engineering (1992)162. A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analysis

(1992)163. A. Charlieret al., Tensors and the Clifford Algebra (1992)164. P. Biler and T. Nadzieja, Problems and Examples in Differential Equations (1992)165. E. Hansen, Global Optimization Using Interval Analysis (1992)166. S. Guerre-Delabriere, Classical Sequences in Banach Spaces (1992)167. Y. C. Wong, Introductory Theory ofTopological Vector Spaces (1992)168. S. H. Kulkami and B. V. Limaye, Real Function Algebras (1992)169. W. C. Brown, Matrices Over Commutative Rings (1993)170. J. Loustau and M. Dilfon, Linear Geometry with Computer Graphics (1993)171. W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential

Equations (1993)

172. E. C. Young, Vector and Tensor Analysis: Second Edition (1993)173. T. A. Bick, Elementary Boundary Value Problems (1993)174. M. Pavel, Fundamentals of Pattem Recognition: Second Edition (1993)175. S. A. Albeverio et al., Noncommutative Distributions (1993)176. W. Fulks, Complex Variables (1993)177. M. M. Rao, Conditional Measures and Applications (1993)178. A. Janicki and A. Weron, Simulation and Chaotic Behavior of ti-Stable Stochastic

Processes (1994)179. P. NeittaanmlJki and D. 7iba, Optimal Control of Nonlinear Parabolic Systems (1994)180. J. Cronin, Differential Equations: Introduction and Qualitative Theory, Second Edition

(1994)181. S. Heikki/lJ and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous

Nonlinear Differential Equations (1994)182. X. Mao, Exponential Stability of Stochastic Differential Equations (1994)183. B. S. Thomson, Symmetric Properties of Real Functions (1994)184. J. E. Rubio, Optimization and Nonstandard Analysis (1994)185. J. L. Bueso et al., Compatibility, Stability, and Sheaves (1995)186. A. N. Michel and K. Wang, Qualitative Theory of Dynamical Systems (1995)187. M. R. Damel, Theory of Lattice-Ordered Groups (1995)188. Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational

Inequalities and Applications (1995)189. L. J. Corwin and R. H. Szczarba, calculus in Vector Spaces: Second Edition (1995)190. L. H. Erbe et al., Oscillation Theory for Functional Differential Equations (1995)191. S. Agaian et al., Binary Polynomial Transforms and Nonlinear Digital Filters (1995)192. M. I. Gil', Norm Estimations for Operation-Valued Functions and Applications (1995)193. P. A. Grillet, Semigroups: An Introduction to the Structure Theory (1995)194. S. Kichenassamy, Nonlinear Wave Equations (1996)195. V. F. Krotov, Global Methods in Optimal Control Theory (1996)196. K. I. Beidaret al., Rings with Generalized Identities (1996)197. V. I. Amautov et al., Introduction to the Theory of Topological Rings and Modules

(1996)198. G. Sierksma, Linear and Integer Programming (1996)199. R. Lasser, Introduction to Fourier Series (1996)200. V. Sima, Algorithms for Linear-Quadratic Optimization (1996)201. D. Redmond, Number Theory (1996)202. J. K. Beem et al., Global Lorentzian Geometry: Second Edition (1996)203. M. Fontana et al., prafer Domains (1997)204. H. Tanabe, Functional Analytic Methods for Partial Differential Equations (1997)205. C. Q. Zhang, Integer Flows and Cycle Covers of Graphs (1997)206. E. Spiegel and C. J. O'Donnell, Incidence Algebras (1997)207. B. Jakubczyk and W. Respondek, Geometry of Feedback and Optimal Control (1998)208. T. W. Haynes et al., Fundamentals of Domination in Graphs (1998)209. T. W. Haynes et al., Domination In Graphs: Advanced Topics (1998)210. L. A. D'Alotto et al., A Unified Signal Algebra Approach to Two-Dimensional Parallel

Digital Signal Processing (1998)211. F. Halter-Koch, Ideal Systems (1998)212. N. K. Govil et al., Approximation Theory (1998)213. R. Cross, Multivalued Linear Operators (1998)214. A. A. Martynyuk, Stability by Liapunov's Matrix Function Method with Applications

(1998)215. A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces (1999)216. A. //lanes and S. Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances

(1999)217. G. Kato and D. Struppa, Fundamentals of Algebraic Mlcrolocal Analysis (1999)218. G. x.-Z. Yuan, KKM Theory and Applications in Nonlinear Analysis (1999)219. D. Motreanu and N. H. Pavel, Tangency, Flow Invariance for Differential Equations,

and Optimization Problems (1999)220. K. Hrbacek and T. Jech, Introduction to Set Theory, Third Edition (1999)221. G. E. Ko/osov, Optimal Design of Control Systems (1999)222. N. L. Johnson, Subplane Covered Nets (2000)223. B. Fine and G. Rosenberger, Algebraic Generalizations of Discrete Groups (1999)224. M. VlJth, Volterra and Integral Equations of Vector Functions (2000)225. S. S. Miller and P. T. Mocanu, Differential Subordinations (2000)

DIFFERENTIAL GEOMETRY WITHAPPLICATIONS TOMECHANICS AND PHYSICS

Yves TalpaertOuagadougou UniversityOuagadougou, Burkina Faso

0 CRC Pressell! Taylor & Francis Group

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PREFACE

Differential geometry is a mathematical discipline which in a decisive manner contributes tomodern developments of theoretical physics and mechanics; many books relating to these areeither too abstract since aimed at mathematicians, too quickly applied to particular physicsbranches when aimed at physicists.

Most of the text comes from Master's-level courses I taught at several' Africanuniversities and aims to make differential geometry accessible to physics and engineeringmajors.

The first seven lectures rather faithfully translate lessons of my French book "GeometrieDifferentielle et Mecanique Analytique," but contain additional examples. The last threelectures have been completely revised and several new subjects exceed the Master's degree.

The text sets out, for an eclectic audience, a methodology paving the road to analyticalmechanics, fluid-dynamics, special relativity, general relativity, thermodynamics, cosmology,electromagnetism, stellar dynamics, and quantum physics.

The theory and the 133 solved exercises will be of interest to other disciplines and willalso allow mathematicians to find many examples and concepts. The introduced notionsshould be known by students when beginning a Ph.D. in mathematics applied to theoreticalphysics and mechanics.

The chapters illustrate the imaginative and unifying characters of differential geometry. Ameasured and logical progression towards (sometimes tricky) ideas, gives this book itsoriginality. All the proofs and exercises are detailed. The important propositions and theformulae to be framed are shown by <T and 6V'.

Two introduced methods (in fluid-mechanics and calculus of variations) deserve furtherstudy.

There is no doubt that engineers could overcome difficulties by using differentialgeometry methods to meet technological challenges.

Acknowledgements. I am grateful to Professor Michel N. Boyom (Montpellier University)who allowed me to improve the French version and to Professor Emeritus Raymond Coutrez(Brussels University) who taught me advanced mathematical methods of mechanics andastronomy.Many thanks to my former African students who let me expound on the material that resultedin this book.All my affection to Moira who drew the figures.

v

vi Preface

I wish to express my gratitude to Marcel Dekker, Inc. for helpful remarks and suggestions.

Yves Talpaert

CONTENTS

Preface v

Lecture O. TOPOLOGY AND DIFFERENTIAL CALCULUS REQUIREMENTS 1

Banach space .Differential calculus in Banach spaces ..Differentiation ofR n into Banach ..Differentiation ofR n into R n •••••••••••••••••••••••••••••••••••••.•••••••

Differentiation ofR n into R n ..

2.12.22.32.42.5

1. Topology 1

1.1 Topological space 11.2 Topological space basis 21.3 Haussdorff space 41.4 Homeomorphism 51.5 Connected spaces 61.6 Compact spaces 61.7 Partition of unity 72. Differential calculus in Banach spaces 8

810171922

3. Exercises 30

Lecture 1. MANIFOLDS 37

Introduction 37

1. Differentiable manifolds 40

1.1 Chart and local coordinates " 401.2 Differentiable manifold structure 411.3 Differentiable manifolds 43

2. Differentiable mappings 50

2.1 Generalities on differentiable mappings 502.2 Particular differentiable mappings 552.3 Pull-back of function 57

3. Submanifolds 59

3.1 Submanifolds ofR n 593.2 Submanifold of manifold 64

4. Exercises. . ..... . .. ... . .. .. .... .... ... ... ...... ... .. ... . .. . .. .... .. . . . . . . ... .... ... 65

vii

viii Contents

Lecture 2. TANGENT VECTOR SPACE 71

1. Tangent vector 71

1.1 Tangent curves 711.2 Tangent vector 74

2. Tangent space 80

2.1 Definition of a tangent space 802.2 Basis of tangent space 812.3 Change ofbasis 82

3. Differential at a point 83

3.1 Definitions .. 843.2 The image in local coordinates 853.3 Diferential of a function 86

4. Exercises 87

Lecture 3. TANGENT BUNDLE - VECTOR FIELD - ONE-PARAMETERGROUP LIE ALGEBRA 91Introduction 91

1. Tangent bundle 93

1.1 Natural manifold TM 931.2 Extension and commutative diagram 94

2. Vector field on manifold 96

2.1 Definitions 962.2 Properties of vector fields 96

3. Lie algebra structure 97

3.1 Bracket 973.2 Lie algebra 1003.3 Lie derivative . .. .. .. . .. .. .. .. .. .. .. . .. . .. . .. . .. .. . . . . 101

4. One-parameter group of diffeomorphisms 102

4.1 Differential equations in Banach 1024.2 One-parameter group ofdiffeomorphisms 104

5. Exercises 111

Lecture 4. COTANGENT BUNDLE - VECTOR BUNDLE OF TENSORS 125

1. Cotangent bundle and covector field 125

1.1 I-form 1251.2 Cotangent bundle . . .. . . . . . .. .. . . . . . .. .. . .. . .. 1291.3 Field of covectors 130

Contents ix

2. Tensor algebra 130

2.1 Tensor at a point and tensor algebra 1302.2 Tensor fields and tensor algebra 137

3. Exercises 144

Lecture S. EXTERIOR DIFFERENTIAL FORMS 153

1. Exterior form at a point 153

1.1 Definition ofap-fonn 1531.2 Exterior product of I-fonns 1551.3 Expression of a p-fonn 1561.4 Exterior product offonns 1581.5 Exterior algebra 159

2. Differential forms on a manifold 162

2.1 Exterior algebra (Grassmann algebra) 1622.2 Change of basis 165

3. Pull-back of a differential form 167

3.1 Definition and representation 1673.2 Pull-back properties 168

4. Exterior differentiation 170

4.1 Definition 1704.2 Exterior differential and pull-back 173

5. Orientable manifolds 174

6. Exercises 178

Lecture 6. LIE DERIVATIVE - LIE GROUP 185

1. Lie derivative 186

1.1 First presentation ofLie derivative 1861.2 Alternative interpretation ofLie derivative 195

2. Inner product and Lie derivative 199

2.1 Definition and properties 1992.2 Fundamental theorem 201

3. Frobenius theorem 204

4. Exterior differential systems 207

4.1 Generalities 2074.2 Pfaff systems and Frobenius theorem 208

x Contents

5. Invariance of tensor fields 211

5.1 Definitions 2115.2 Invariance ofdifferential fonns 2125.3 Lie algebra 214

6. Lie group and algebra 214

6.1 Lie group definition 2156.2 Lie algebra of Lie group 2156.3 Invariant differential fonns on G 2176.4 One-parameter subgroup ofa Lie group 218

7. Exercises 224

Lecture 7. INTEGRATION OF FORMS: Stokes' Theorem, Cohomology andIntegral Invariants 235

1. n-form integration on n-manifold 235

1.1 Integration definition 2351.2 Pull-back of a fonn and integral evaluation 237

2. Integral over a chain 239

2.1 Integral over a chain element 2392.2 Integral over a chain 239

3. Stokes' theorem 240

3.1 Stokes' fonnula for a closedp-interval 2403.2 Stokes' fonnula for a chain 242

4. An introduction to cohomology theory 243

4.1 Closed and exact fonns - Cohomology 2434.2 Poincare lemma 2444.3 Cycle - Boundary - Homology 247

5. Integral invariants 248

5.1 Absolute integral invariant : 2485.2 Relative integral invariant 252

6. Exercises 253

Lecture 8. RIEMANNIAN GEOMETRY 257

1. Riemannian manifolds 257

1.1 Metric tensor and manifolds 2571.2 Canonical isomorphism and conjugate tensor 2621.3 Orthononnal bases 2661.4 Hyperbolic manifold and special relativity 267

Contents xi

1.5 Killing vector field 2741.6 Volume 2751.7 The Hodge operator and adjoint 2771.8 Special relativity and Maxwell equations 2801.9 Induced metric and isometry 283

2. Affine connection 285

2.1 Affine connection definition. . .. . .. . .. .. . . . .. .. . 2852.2 Christoffel symbols 2862.3 Interpretation of the covariant derivative 2882.4 Torsion 2912.5 Levi-Civita (or Riemannian) connection 2912.6 Gradient - Divergence - Laplace operators .........•................. 293

3. Geodesic and Euler equation 300

4. Curvatures - Ricci tensor - Bianchi identity - Einstein equations 302

4.1 Curvature tensor 3024.2 Ricci tensor 3054.3 Bianchi identity 3084.4 Einstein equations 309

5. Exercises . . .. . .. . .. . . . . .. . .. . . . . .. . .. . .. . . . . .. . .. . . . . .. . .. . 310

Lecture 9 LAGRANGE AND HAMILTON MECHANICS 325

1. Classical mechanics spaces and metric 325

1.1 Generalized coordinates and spaces 3251.2 Kinetic energy and Riemannian manifold 327

2. Hamilton principle, Motion equations, Phase space 329

2.1 Lagrangian 3292.2 Principle ofleast action 3292.3 Lagrange equations 3312.4 Canonical equations ofHamilton 3322.5 Phase space 337

3. D'Alembert-Lagrange principle - Lagrange equations 338

3.1 D'Alembert-Lagrangeprinciple 3383.2 Lagrange equations 3403.3 Euler-Noether theorem 3413.4 Motion equations on Riemannian manifolds 343

4. Canonical transformations and integral invariants 344

4.1 Diffeomorphisms on phase spacetime 3444.2 Integral invariants 3464.3 Integral invariants and canonical transformations 348

xii

4.4

Contents

Liouville theorem 352

5. The N-body problem and a problem of statistical mechanics 352

5.1 N-bodyproblem and fundamental equations 3535.2 A problem ofstatistical mechanics 358

6. Isolating integrals 369

6.1 Definition and examples 3696.2 Jeans theorem 3726.3 Stellar trajectories in the galaxy 3736.4 The third integral 3756.5 Invariant curve and third integral existence 379

7. Exercises 381

Lecture 10. SYMPLECTIC GEOMETRY-Hamilton-Jacobi Mechanics 385

Preliminaries 385

1. Symplectic geometry 388

1.1 Darboux theorem and symplectic matrix 3881.2 Canonical isomorphism 3911.3 Poisson bracket ofone-forms 3931.4 Poisson bracket of functions 3961.5 Symplectic mapping and canonical transformation 399

2. Canonical transformations in mechanics 404

2.1 Hamilton vector field 4042.2 Canonical transformations - Lagrange brackets 4082.3 Generating functions 412

3. Hamilton-Jacobi equation 415

3.1 Hamilton-Jacobi equation and Jacobi theorem 4153.2 Separability 419

4. A variational principle of analytical mechanics 422

4.1 Variational principle (with one degree of freedom) 4234.2 Variational principle (with n degrees of freedom) 427

5. Exercises 429

Bibliography 443

Glossary. ... . .. . .. . .. 445

Contents xiii

Index 449

LECTURE 0

TOPOLOGY AND DIFFERENTIAL CALCULUS

REQUIREMENTS

1. TOPOLOGY

This section presents the required basic notions of topology.

1.1 TOPOLOGICAL SPACE

D A topological space S is a set with a topology.A topology on S is a collection 0 of subsets, called open sets 1, such that:

the union of any collection ofopen sets is open,- any finite intersection ofopen sets is open,

space S and empty space 0 are open sets.

Let UeO be an open set.

D The complement of U with respect to S is said to be a closed set, namely:

CU=S-U={seS I sliOU}.

D An open neighborhood of a point x in a topological space S is an open set Ucontaining x.

Afterwards, any open neighborhood will be simply called neighborhood.

Let P be a subset of space S.

D The relative topology on P is defined by

Op = { Un P I U EO} .

D A point xeS is a contact point of P if every neighborhood of x contains at least apoint of P.

I Afterwards, "open set" will often be simply called "open. "

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D A point xeS is an accumulation point or limit point ofP if every neighborhood ofxcontains (at least) one point of P different from x.

D A point xeP is an isolatedpoint of P ifx has a neighborhood which does not containany point of P different from x.

In other words, an isolated point ofP is a point ofP, which is no accumulation point.

PRJ Every accumulation point is a contact point, the opposite is evidently false.

PR2 Any contact point is either an isolated point or an accumulation point.

D The closure of P, denoted cl(P), is the set ofcontact points of P.

PR3 The closure of P is a closed set, it's the smallest closed set containing P.

D A subset P of S is (everywhere) dense in S if

cl(P) = S.

The definition means every point ofS is a contact point ofP.

1.2 TOPOLOGICAL SPACE BASIS

Let S be a topological space.

1.2.1 Definition

D A basis B for the topology on S is a collection of open sets such that every open setof S is a union ofelements of B.In other words:

(Ui)ieI is a basis of open sets of Sifevery open set of Sis UUj UeJcl).j

1.2.2 Example of the metric space

D A metric space M is a set provided with a distance.A distance on M is a function

d: Mx M ~R+: (XJl)H d(xJl)

satisfying the following conditions:

'if x,y,z e M: d(x,y)= 0 ~ x=y'if x,y e M: d(x,y)=d(y,x)'if x,y,z eM: d(x,z):;; d(x,y) + d(y,z) .

Example. The standard distance on J(' is defined by

n

d(x,y) = ~)X' _ /)2I-I

Topology and Differential Calculus Requirements

where x = (x1,...,xn) and y=(yl,...,yn).

We introduce a topology on M related to the distance notion and called metric topology.

D An open sphere about aeM in metric space M is

B(a,r)= {xeM I d(a,x)<r, reR+}.

D A subset P ofa metric space M is an open set of M ifeither P=0or \Ix e P: 3B(x,r) c P (r:F 0) .

PR4 Every open sphere ofa metric space M is an open set of M

Proof The open sphere B(a,0)=0 is an open set.If the case is not trivial, we prove the existence ofan open sphere included in B(a,r), namely:

B(x,r-d(a,x» c B(a,r).

Indeed, every point yeB(x,r-d(a,x» is so that:

d(x,y) < r-d(a,x)

which implies:

d(a,y) S; d(a,x) + d(x,y) < r;

then any point y e B(x,r-d(a,x» necessarily belongs to B(a,r).

3

To conclude, there exists an open sphere B(x,r-d(a,x»cB(a,r) and, by definition, B(a,r) is anopen set of M

The previous proposition implies every union ofopen spheres ofa metric space M is open.

Reciprocally, every open set of M is the union of open spheres. That follows from the openset definition. Then, we can express:

PR5 The open spheres ofa metric space make up a basis for a topology called metric spacetopology.

Let Sand T be topological spaces.

D The product topology on SxT is the collection of subsets that are unions of opens asU xV, such that U and V are opens respectively in S and T.

Thus open rectangles form a basis for the topology.

1.2.3 Separable space

D A topological space S is said to be a space with countable basis if there is (at least)one basis in S consisting of a countable number of elements, countable meaning finiteor denumerable.

4 Lecture 0

D A topological space S containing a (everywhere) dense countable set is calledseparable.

PR6 Every topological space S with countable basis is separable.

Proof. Let B = {V; lieN} be a countable basis of S. We prove the space S contains a set

D = {Xi e V; lieN} (everywhere) dense in S.

Every neighborhood Vy of yeS includes an open set containing y. This open set is the union

Uv; of a certain number of V;. ThusI

VynD#0because, in particular, this set contains Xi e V;. This conclusion is true for every point yeS,thus every point y is a contact point of D and hence D is dense in S.

Example. The space It' is separable with the topology defined by

B = {B(x,r) I xeQn, reQ }

where Q is the set of rational numbers.

1.3 HAUSSDORFF SPACE

D A topological space S is called a type Tl space if for any two distinct points X and y ofS exist a neighborhood Ux; of X with y not belonging to Ux; and a neighborhood Uy

with x not belonging to Uy.

D A topological space S is called a type Tz space or a Haussdorffspace if for any twodistinct points x and y of S exist a neighborhood Ux; of x and a neighborhood Uy of ysuch that:

Example. The real straight line with two origins o} and 02 is a type T} space but not aHaussdorff space.

91----- .-----°2

Figure 1.

IR

Indeed, a topology is defined using:

- the usual open intervals in R (for the semi-lines),{{o.}U(-e,O)U(O,e)} (foro}),

- {{02}U(-e',O)U(O,e')} (for 02).

In the last two cases, the intersection of two open sets (of o) and 02) is necessarily not empty.Thus it's not a Haussdorff space example.

Later, Haussdorff spaces (any two distinct points x and y of the space have disjointneighborhoods) will playa fundamental part.

PR7 Every metric space is a Haussdorff space.

Topology and Differential Calculus Requirements 5

Proof Let x andy be two points ofa metric space M and let r be the distance d(x,y). Clearly,the open spheres B(x,r/2) and B(y,r/2) are disjoint.

PRS Every subspace ofa Haussdorff space is a Haussdorff space.

This proposition is immediate.

1.4 HOMEOMORPIDSM

Let Sand T be topological spaces.

D A mappingf:S~ T:XHy

is said to be continuous at point xeS if, for every neighborhood Vy of f(x), r l (Vy)

is a neighborhood of x in S.

D A mapping f of S into T is continuous on S if it is continuous at each point of S.In an equivalent manner:A mapping f of S into T is continuous on S if, for every open set Win T, r 1(W)is an open set in S.

PR9 A mapping f of S into T is continuous on S if, for every closed set A in T, r 1(A) is aclosed set in S.

Proof The explanation is immediate sincer1(CA) = C.rI(A).

Notation. The set ofcontinuous mappings of S into T is denoted

CO( S;T).

We know that two topological spaces Sand T are homeomorphic if there is a bijectionf: S~ T "exchanging" the open sets, i.e. to each open V in S corresponds an open f(V) in

T and to each open Win T corresponds an open r l (W) in S.

D (I'" A homeomorphism f of S onto T is a bicontinuous bijection, namely a bijectionsuch that fand /"1 are continuous.

This definition is logical because iff and r 1are continuous, then the inverse image ofeveryopen set of T is open and the image ofevery open set of S is open.

Examples.n

1. The open sphere {xeR" I })X')2 < r , reR+} is homeomorphic to R"'.j$l

2. The space {xeR" I a< i (X' )2 < b ; a,beR+} is not homeomorphic to K.'~1

6 Lecture 0

1.5 CONNECTED SPACES

D A topological space S is connected if every partition of S into two open sets A and Bimplies A=0 or B=0.

In other words:

if ll(A,B)eOxO: [A~0,B~0,AnB=0 and AUB::::SJ.

This definition means 0 and S are the only subsets of S that are both open and closed.

D A topological space S is locally connected at point x €S ifx has a basis of connectedneighborhoods.

D A topological space S is locally connected if it is locally connected at each point.

D A topological space S is arcwise connected if for every two points a and b in S thereis a continuous mapping lof a closed interval [a,p]c R into 8 such that I(a):::: aand f(P)=b.

That is:If every two points in 8 can be joined by an arc in S.

1.6 COMPACT SPACES

Let Sand T be topological spaces.

D A topological space 8 is compact 1 if for every covering of S by open sets (that isUU1 :::: 8) there is a finite subcovering.

I

The reader will easily prove the following proposition.

PR10 Any closed subset ofa compact space is compact.

PR11 Any compact subset ofa Haussdorff space S is closed in S.

Proof. Let A be a compact subset of 8. Since 8 is a HaussdorfT space, there are disjointneighborhoods of xeA and of yeCA. In addition A is compact and thus there are disjointneighborhoods of y and A. Therefore CA is open.

PRl2 Any continuous mapping I of a compact space S into a Haussdorff space T impliesthe subset 1(8) of T is compact.

Proof First observe that the subspace 1(8) of T is Haussdorff because T is a Haussdorffspace. Secondly, let {U;}iEI be an arbitrary open covering of I(S). Since I is continuous,{f-JU;}iEJ is an open covering of S. But the space S is compact, then from {j"IU;}Jel there isafinitesubcovering {f-lU;}ieJ. Since l(rIU/)cU/ then {U;}ieJ is also a covering of I(S).

This implies I(S) is a compact space.

1 In this definition, the Haussdorff separation condition could be added in order to avoid the spaces with trivialtopology {0,S}.

Topology and Differential Calculus Requirements 7

PR13 Every continuous bijection f of a compact space S onto a Haussdorff space T is ahomeomorphism.

Proof It is sufficient to prove thatf-1 is continuous on T or in a similar manner (from PR9)that:

[ '<;fA closed in S => f(A) closed in T] .We have the following sequence:

[ PRIO] => [A closed in S => A compact][PRI2] => [f(A) in Haussdorff T :::> f(A) compact]

[PRll] :::> [f(A) closed in T].

D An open covering {U;}iEJ of a topological space S is called a refinement ofa covering{Yj}j61 iffor every U; there is (at lest) an open Yj such that U,c Yj.

D An open covering {U;} of S is called locally finite if each point xeS has aneighborhood Vx which intersects only a finite number of U; :

#{ U; I u;nv. *0} < 00.

D A topological space S is called paracompact if:(i) S is Haussdorff,(ii) every open covering {U;} of S has a locally finite refinement {~}.

D A topological space is called locally compact if each point has a neighborhood whoseclosure is compact.

1.7 PARTITION OF UNITY

The partition of unity notion is important in differential geometry because it allows toreduce the study of global problems to local problems as seen later in the integration context.

Let S be a topological space.

D The support ofa real-valued functiong:S~R

is the closure of the set ofpoints xeS such as g(x)~O.

We denotesuppg = cl {XES I g(x)~O}.

D Apartitlon ofunity on S is a family {gil ofcontinuous functions (even ofclass ("1)g, : S ~ R+ : x H g,(x)

such that:(i) {supp gil is a locally finite covering of S •(ii) '<;fxeS: Lg,(x) = 1.

I

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