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Summary
Table of Contents
Preface | p. v |
Topology and Differential Calculus Requirements | p. 1 |
Topology | p. 1 |
Topological space | p. 1 |
Topological space basis | p. 2 |
Haussdorff space | p. 4 |
Homeomorphism | p. 5 |
Connected spaces | p. 6 |
Compact spaces | p. 6 |
Partition of unity | p. 7 |
Differential calculus in Banach spaces | p. 8 |
Banach space | p. 8 |
Differential calculus in Banach spaces | p. 10 |
Differentiation of R[superscript n] into Banach | p. 17 |
Differentiation of R[superscript n] into R[superscript n] | p. 19 |
Differentiation of R[superscript n] into R[superscript n] | p. 22 |
Exercises | p. 30 |
Manifolds | p. 37 |
Introduction | p. 37 |
Differentiable manifolds | p. 40 |
Chart and local coordinates | p. 40 |
Differentiable manifold structure | p. 41 |
Differentiable manifolds | p. 43 |
Differentiable mappings | p. 50 |
Generalities on differentiable mappings | p. 50 |
Particular differentiable mappings | p. 55 |
Pull-back of function | p. 57 |
Submanifolds | p. 59 |
Submanifolds of R[superscript n] | p. 59 |
Submanifold of manifold | p. 64 |
Exercises | p. 65 |
Tangent Vector Space | p. 71 |
Tangent vector | p. 71 |
Tangent curves | p. 71 |
Tangent vector | p. 74 |
Tangent space | p. 80 |
Definition of a tangent space | p. 80 |
Basis of tangent space | p. 81 |
Change of basis | p. 82 |
Differential at a point | p. 83 |
Definitions | p. 84 |
The image in local coordinates | p. 85 |
Differential of a function | p. 86 |
Exercises | p. 87 |
Tangent Bundle--Vector Field--One-Parameter Group Lie Algebra | p. 91 |
Introduction | p. 91 |
Tangent bundle | p. 93 |
Natural manifold TM | p. 93 |
Extension and commutative diagram | p. 94 |
Vector field on manifold | p. 96 |
Definitions | p. 96 |
Properties of vector fields | p. 96 |
Lie algebra structure | p. 97 |
Bracket | p. 97 |
Lie algebra | p. 100 |
Lie derivative | p. 101 |
One-parameter group of diffeomorphisms | p. 102 |
Differential equations in Banach | p. 102 |
One-parameter group of diffeomorphisms | p. 104 |
Exercises | p. 111 |
Cotangent Bundle--Vector Bundle of Tensors | p. 125 |
Cotangent bundle and covector field | p. 125 |
1-form | p. 125 |
Cotangent bundle | p. 129 |
Field of covectors | p. 130 |
Tensor algebra | p. 130 |
Tensor at a point and tensor algebra | p. 130 |
Tensor fields and tensor algebra | p. 137 |
Exercises | p. 144 |
Exterior Differential Forms | p. 153 |
Exterior form at a point | p. 153 |
Definition of a p-form | p. 153 |
Exterior product of 1-forms | p. 155 |
Expression of a p-form | p. 156 |
Exterior product of forms | p. 158 |
Exterior algebra | p. 159 |
Differential forms on a manifold | p. 162 |
Exterior algebra (Grassmann algebra) | p. 162 |
Change of basis | p. 165 |
Pull-back of a differential form | p. 167 |
Definition and representation | p. 167 |
Pull-back properties | p. 168 |
Exterior differentiation | p. 170 |
Definition | p. 170 |
Exterior differential and pull-back | p. 173 |
Orientable manifolds | p. 174 |
Exercises | p. 178 |
Lie Derivative--Lie Group | p. 185 |
Lie derivative | p. 186 |
First presentation of Lie derivative | p. 186 |
Alternative interpretation of Lie derivative | p. 195 |
Inner product and Lie derivative | p. 199 |
Definition and properties | p. 199 |
Fundamental theorem | p. 201 |
Frobenius theorem | p. 204 |
Exterior differential systems | p. 207 |
Generalities | p. 207 |
Pfaff systems and Frobenius theorem | p. 208 |
Invariance of tensor fields | p. 211 |
Definitions | p. 211 |
Invariance of differential forms | p. 212 |
Lie algebra | p. 214 |
Lie group and algebra | p. 214 |
Lie group definition | p. 215 |
Lie algebra of Lie group | p. 215 |
Invariant differential forms on G | p. 217 |
One-parameter subgroup of a Lie group | p. 218 |
Exercises | p. 224 |
Integration of Forms: Stokes' Theorem, Cohomology and Integral Invariants | p. 235 |
n-form integration on n-manifold | p. 235 |
Integration definition | p. 235 |
Pull-back of a form and integral evaluation | p. 237 |
Integral over a chain | p. 239 |
Integral over a chain element | p. 239 |
Integral over a chain | p. 239 |
Stokes' theorem | p. 240 |
Stokes' formula for a closed p-interval | p. 240 |
Stokes' formula for a chain | p. 242 |
An introduction to cohomology theory | p. 243 |
Closed and exact forms--Cohomology | p. 243 |
Poincare lemma | p. 244 |
Cycle--Boundary--Homology | p. 247 |
Integral invariants | p. 248 |
Absolute integral invariant | p. 248 |
Relative integral invariant | p. 252 |
Exercises | p. 253 |
Riemannian Geometry | p. 257 |
Riemannian manifolds | p. 257 |
Metric tensor and manifolds | p. 257 |
Canonical isomorphism and conjugate tensor | p. 262 |
Orthonormal bases | p. 266 |
Hyperbolic manifold and special relativity | p. 267 |
Killing vector field | p. 274 |
Volume | p. 275 |
The Hodge operator and adjoint | p. 277 |
Special relativity and Maxwell equations | p. 280 |
Induced metric and isometry | p. 283 |
Affine connection | p. 285 |
Affine connection definition | p. 285 |
Christoffel symbols | p. 286 |
Interpretation of the covariant derivative | p. 288 |
Torsion | p. 291 |
Levi-Civita (or Riemannian) connection | p. 291 |
Gradient--Divergence--Laplace operators | p. 293 |
Geodesic and Euler equation | p. 300 |
Curvatures--Ricci tensor--Bianchi identity--Einstein equations | p. 302 |
Curvature tensor | p. 302 |
Ricci tensor | p. 305 |
Bianchi identity | p. 308 |
Einstein equations | p. 309 |
Exercises | p. 310 |
Lagrange and Hamilton Mechanics | p. 325 |
Classical mechanics spaces and metric | p. 325 |
Generalized coordinates and spaces | p. 325 |
Kinetic energy and Riemannian manifold | p. 327 |
Hamilton principle, Motion equations, Phase space | p. 329 |
Lagrangian | p. 329 |
Principle of least action | p. 329 |
Lagrange equations | p. 331 |
Canonical equations of Hamilton | p. 332 |
Phase space | p. 337 |
D'Alembert-Lagrange principle--Lagrange equations | p. 338 |
D'Alembert-Lagrange principle | p. 338 |
Lagrange equations | p. 340 |
Euler-Noether theorem | p. 341 |
Motion equations on Riemannian manifolds | p. 343 |
Canonical transformations and integral invariants | p. 344 |
Diffeomorphisms on phase spacetime | p. 344 |
Integral invariants | p. 346 |
Integral invariants and canonical transformations | p. 348 |
Liouville theorem | p. 352 |
The N-body problem and a problem of statistical mechanics | p. 352 |
N-body problem and fundamental equations | p. 353 |
A problem of statistical mechanics | p. 358 |
Isolating integrals | p. 369 |
Definition and examples | p. 369 |
Jeans theorem | p. 372 |
Stellar trajectories in the galaxy | p. 373 |
The third integral | p. 375 |
Invariant curve and third integral existence | p. 379 |
Exercises | p. 381 |
Symplectic Geometry--Hamilton--Jacobi Mechanics | p. 385 |
Preliminaries | p. 385 |
Symplectic geometry | p. 388 |
Darboux theorem and symplectic matrix | p. 388 |
Canonical isomorphism | p. 391 |
Poisson bracket of one-forms | p. 393 |
Poisson bracket of functions | p. 396 |
Symplectic mapping and canonical transformation | p. 399 |
Canonical transformations in mechanics | p. 404 |
Hamilton vector field | p. 404 |
Canonical transformations--Lagrange brackets | p. 408 |
Generating functions | p. 412 |
Hamilton-Jacobi equation | p. 415 |
Hamilton-Jacobi equation and Jacobi theorem | p. 415 |
Separability | p. 419 |
A variational principle of analytical mechanics | p. 422 |
Variational principle (with one degree of freedom) | p. 423 |
Variational principle (with n degrees of freedom) | p. 427 |
Exercises | p. 429 |
Bibliography | p. 443 |
Glossary | p. 445 |
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