The differential geometric formulation of analytical mechanics not only offers a new insight into Mechanics, but also provides a more rigorous formulation of its physical content from a mathematical viewpoint. Topics covered in this volume include differential forms, the differential geometry of tangent and cotangent bundles, almost tangent geometry, symplectic and pre-symplectic Lagrangian and Hamiltonian formalisms, tensors and connections on manifolds, and geometrical aspects of variational and constraint theories. The book may be considered as a self-contained text and only presupposes that readers are acquainted with linear and multilinear algebra as well as advanced calculus.
Author(s): M. de León, P.R. Rodrigues
Series: Mathematics Studies
Publisher: Elsevier Science Ltd
Year: 1989
Language: English
Pages: 495
Methods of Differential Geometry in Analytical Mechanics......Page 4
Copyright Page......Page 5
Contents......Page 8
Preface......Page 12
1.1 Some main results in Calculus on Rn......Page 14
1.2 Differentiable manifolds......Page 16
1.3 Differentiable mappings. Rank Theorem......Page 19
1.4 Partitions of unity......Page 20
1.5 Immersions and submanifold......Page 22
1.6 Submersions and quotient manifolds......Page 24
1.7 Tangent spaces. Vector fields......Page 26
1.8 Fibred manifolds. Vector bundles......Page 33
1.9 Tangent and cotangent bundles......Page 37
1.10 Tensor fields. The tensorial algebra. Riemannian metrics......Page 41
1.11 Differential forms. The exterior algebra......Page 49
1.12 Exterior differentiation......Page 58
1.13 Interior product......Page 62
1.14 The Lie derivative......Page 63
1.15 Distributions. Frobenius theorem......Page 66
1.16 Orientable manifolds. Integration. Stokes theorem......Page 72
1.17 de Rham cohomology. Poincare lemma......Page 82
1.18 Linear connections. Riemannian connections......Page 86
1.19 Lie groups......Page 92
1.20 Principal bundles. Frame bundles......Page 102
1.21 G-structures......Page 111
1.22 Exercises......Page 116
2.1 Almost tangent structures on manifolds......Page 122
2.2 Examples. The canonical almost tangent structure of the tangent bundle......Page 125
2.3 Integrability......Page 127
2.4 Almost tangent connections......Page 130
2.5 Vertical and complete lifts of tensor fields to the tangent bundle......Page 131
2.6 Complete lifts of linear connections to the tangent bundle......Page 137
2.7 Horizontal lifts of tensor fields and connections......Page 140
2.8 Sasaki metric on the tangent bundle......Page 146
2.9 Affine bundles......Page 149
2.10 Integrable almost tangent structures which define fibrations......Page 150
2.11 Exercises......Page 155
3.1 Almost product structures......Page 158
3.2 Almost complex manifolds......Page 162
3.3 Almost complex connections......Page 167
3.4 Kahler manifolds......Page 172
3.5 Almost complex structures on tangent bundles (I)......Page 176
3.6 Almost contact structures......Page 180
3.7 f–structures......Page 187
3.8 Exercises......Page 191
4.1 Differential calculus on TM......Page 192
4.2 Homogeneous and semibasic forms......Page 197
4.3 Semisprays. Sprays. Potentials......Page 204
4.4 Connections in fibred manifolds......Page 208
4.5 Connections in tangent bundles......Page 210
4.6 Semisprays and connections......Page 217
4.7 Weak and strong torsion......Page 222
4.8 Decomposition theorem......Page 224
4.9 Curvature......Page 227
4.10 Almost complex structures on tangent bundles (II)......Page 229
4.11 Connection in principal bundles......Page 232
4.12 Exercises......Page 235
5.1 Symplectic vector spaces......Page 238
5.2 Symplectic manifolds......Page 245
5.3 The canonical symplectic structure......Page 248
5.4 Lifts of tensor fields to the cotangent bundle......Page 251
5.5 Almost product and almost complex structures......Page 256
5.6 Darboux Theorem......Page 260
5.7 Almost cotangent structures......Page 264
5.8 Integrable almost cotangent structures which define fibrations......Page 269
5.9 Exercises......Page 272
6.1 Hamiltonian vector fields......Page 274
6.2 Poisson brackets......Page 278
6.3 First integrals......Page 283
6.4 Lagrangian submanifolds......Page 286
6.5 Poisson manifolds......Page 293
6.6 Generalized Liouville dynamics and Poisson brackets......Page 298
6.7 Contact manifolds and non–autonomous Hamiltonian systems......Page 300
6.8 Hamiltonian systems with constraints......Page 306
6.9 Exercises......Page 308
7.1 Lagrangian systems and almost tangent geometry......Page 312
7.2 Homogeneous Lagrangians......Page 317
7.3 Connection and Lagrangian systems......Page 319
7.4 Semisprays and Lagrangian systems......Page 328
7.5 A geometrical version of the inverse problem......Page 334
7.6 The Legendre transformation......Page 337
7.7 Non–autonomous Lagrangians......Page 341
7.8 Dynamical connections......Page 347
7.9 Dynamical connections and non–autonomous Lagrangians......Page 355
7.10 The variational approach......Page 358
7.11 Special symplectic manifolds......Page 368
7.12 Noether's theorem. Symmetries......Page 373
7.13 Lagrangian and Hamiltonian mechanical systems with constraints......Page 378
7.14 Euler–Lagrange equations on T*M + TM......Page 381
7.15 More about semisprays......Page 387
7.16 Generalized Caplygin systems......Page 402
7.17 Exercises......Page 406
8.1 The first-order problem and the Hamiltonian formalism......Page 410
8.2 The second-order problem and the Lagrangian formalism......Page 420
8.3 Exercises......Page 447
A.1. Newtonian Mechanics......Page 450
A.2. Classical Mechanics: Lagrangian and Hamiltonian formalisms......Page 454
B.1. Jets of mappings (in one independent variable)......Page 462
B.2. Higher order tangent bundles......Page 463
B.4. The higher-order Poincare-Cartan form......Page 465
Bibliography......Page 468
Index......Page 482
