Methods of Differential Geometry in Analytical Mechanics

495 Pages·1989·13.84 MB·English

by  M. de León

METHODS OF
DIFFERENTIAL GEOMETRY IN
ANALYTICAL MECHANICS
NORTH-HOLLAND MATHEMATICS STUDIES 158
(Continuation of the Notas de Matematica)
Editor: Leopoldo NACHBIN
Centro Brasileiro de Pesquisas Fisicas
Rio de Janeiro, Brazil
and
University of Rochester
New York, U.S.A.
NORTH-HOLLAND -AMSTERDAM NEW YORK OXFORD TOKYO
METHODS OF
DIFFERENTIAL GEOMETRY IN
ANALYTICAL MECHANICS
Manuel de LEON
CECIME
Consejo Superior de lnvestigaciones Cientificas
Madrid, Spain
Paulo R. RODRIGUES
Departamento de Geometria
lnstituto de Ma tema tica
Universidade Federal Fluminense
Niteroi, Brazil
1989
NORTH-HOLLAND -AMSTERDAM NEW VORK OXFORD TOKYO
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vii
Contents
Preface 1
1 Differential Geometry 3
1.1 Some main results in Calculus on Rn . . . . . . . . . . . . . . 3
1.2 Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . 5
1.3 Differentiable mappings . Rank Theorem . . . . . . . . . . . . 8
1.4 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Immersions and submanifold . . . . . . . . . . . . . . . . . . 11
1.6 Submersions and quotient manifolds . . . . . . . . . . . . . . 13
1.7 Tangent spaces . Vector fields . . . . . . . . . . . . . . . . . . 15
1.8 Fibred manifolds . Vector bundles . . . . . . . . . . . . . . . . 22
1.9 Tangent and cotangent bundles . . . . . . . . . . . . . . . . . 26
1.10 Tensor fields. The tensorial algebra . Riemannian metrics . . 30
1.11 Differential forms . The exterior algebra . . . . . . . . . . . . 38
1.12 Exterior differentiation . . . . . . . . . . . . . . . . . . . . . . 47
1.13 Interior product . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.14 The Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . 52
1.15 Distributions . Frobenius theorem . . . . . . . . . . . . . . . . 55
1.16 Orientable manifolds . Integration . Stokes theorem . . . . . . 61
1.17 de Rham cohomology . PoincarC lemma . . . . . . . . . . . . . 71
1.18 Linear connections . Riemannian connections . . . . . . . . . 75
1.19 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
1.20 Principal bundles . Frame bundles . . . . . . . . . . . . . . . . 91
1.21 G-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
1.22 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2 Almost tangent structures and tangent bundles 111
2.1 Almost tangent structures on manifolds . . . . . . . . . . . . 1 11
viii Con tents
2.2 Examples . The canonical almost tangent structure of the tan-
gent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2.3 Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.4 Almost tangent connections . . . . . . . . . . . . . . . . . . . 119
2.5 Vertical and complete lifts of tensor fields to the tangent bun-
dle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
2.6 Complete lifts of linear connections to the tangent bundle . . 126
2.7 Horizontal lifts of tensor fields and connections . . . . . . . . 129
2.8 Sasaki metric on the tangent bundle . . . . . . . . . . . . . . 1 35
2.9 Affine bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2.10 Integrable almost tangent structures which define fibrations . 139
2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
3 Structures on manifolds 147
3.1 Almost product structures . . . . . . . . . . . . . . . . . . . . 147
3.2 Almost complex manifolds . . . . . . . . . . . . . . . . . . . . 151
3.3 Almost complex connections . . . . . . . . . . . . . . . . . . . 1 56
3.4 Kahler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 161
3.5 Almost complex structures on tangent bundles (I) . . . . . . 165
3.5.1 Complete lifts . . . . . . . . . . . . . . . . . . . . . . . 165
3.5.2 Horizontal lifts . . . . . . . . . . . . . . . . . . . . . . 166
3.5.3 Almost complex structure on the tangent bundle of a
Riemannian manifold . . . . . . . . . . . . . . . . . . 1 67
3.6 Almost contact structures . . . . . . . . . . . . . . . . . . . . 169
3.7 f-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4 Connections in tangent bundles 181
4.1 Differential calculus on TM . . . . . . . . . . . . . . . . . . .1 81
4.1.1 Vertical derivation . . . . . . . . . . . . . . . . . . . . 183
4.1.2 Vertical differentiation . . . . . . . . . . . . . . . . . . 184
4.2 Homogeneous and semibasic forms . . . . . . . . . . . . . . . 186
4.2.1 Homogeneous forms . . . . . . . . . . . . . . . . . . . 1 86
4.2.2 Semibasic forms . . . . . . . . . . . . . . . . . . . . . 190
4.3 Semisprays. Sprays. Potentials . . . . . . . . . . . . . . . . . 193
4.4 Connections in fibred manifolds . . . . . . . . . . . . . . . . . 197
4.5 Connections in tangent bundles . . . . . . . . . . . . . . . . .1 99
4.6 Semisprays and connections . . . . . . . . . . . . . . . . . . . 2 06
4.7 Weak and strong torsion . . . . . . . . . . . . . . . . . . . . . 211
Con tents ix
4.8 Decomposition theorem . . . . . . . . . . . . . . . . . . . . . 213
4.9 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
4.10 Almost complex structures on tangent bundles (11) . . . . . . 2 18
4.11 Connection in principal bundles . . . . . . . . . . . . . . . . . 2 21
4.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
5 Symplectic manifolds and cotangent bundles 227
5.1 Symplectic vector spaces . . . . . . . . . . . . . . . . . . . . . 227
5.2 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . 234
5.3 The canonical symplectic structure . . . . . . . . . . . . . . . 237
5.4 Lifts of tensor fields to the cotangent bundle . . . . . . . . . . 240
5.5 Almost product and almost complex structures . . . . . . . . 2 45
5.6 Darboux Theorem . . . . . . . . . . . . . . . . . . . . . . . . 249
5.7 Almost cotangent structures . . . . . . . . . . . . . . . . . . .2 53
5.8 Integrable almost cotangent structures which define fibrations 258
5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
6 Hamiltonian systems 263
6.1 Hamiltonian vector fields . . . . . . . . . . . . . . . . . . . . 263
6.2 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . 267
6.3 First integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
6.4 Lagrangian submanifolds . . . . . . . . . . . . . . . . . . . . . 275
6.5 Poisson manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 282
6.6 Generalized Liouville dynamics and Poisson brackets . . . . . 287
6.7 Contact manifolds and non-autonomous Hamiltonian systems 289
6.8 Hamiltonian systems with constraints . . . . . . . . . . . . .2 95
6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
7 Lagrangian systems 301
7.1 Lagrangian systems and almost tangent geometry . . . . . . . 301
7.2 Homogeneous Lagrangians . . . . . . . . . . . . . . . . . . . . 306
7.3 Connection and Lagrangian systems . . . . . . . . . . . . . . 308
7.4 Semisprays and Lagrangian systems . . . . . . . . . . . . . . 317
7.5 A geometrical version of the inverse problem . . . . . . . . . 323
7.6 The Legendre transformation . . . . . . . . . . . . . . . . . . 326
7.7 Non-autonomous Lagrangians . . . . . . . . . . . . . . . . . . 3 30
7.8 Dynamical connections . . . . . . . . . . . . . . . . . . . . . . 336
7.9 Dynamical connections and non-autonomous Lagrangians . . 344
7.10 The variational approach . . . . . . . . . . . . . . . . . . . . 347