Methods of Differential Geometry in Analytical Mechanics (Mathematics Studies)
By Manuel De Leon, Paulo R. Rodrigues
* Publisher: Elsevier
* Number Of Pages: 494
* Publication Date: 1989-08
* ISBN-10 / ASIN: 0444880178
* ISBN-13 / EAN: 9780444880178
Preface
The purpose of this book is to make a contribution to the modern development
of Lagrangian and Hamiltonian formalisms of Classical Mechanics
in terms of differential-geometric methods on differentiable manifolds. The
text is addressed to mathematicians, mathematical physicists concerned with
differential geometry and its applications, and graduate students.
It. is
included in the text to state its main properties and to help the reader in
subsequent chapters.
Chapters 2 and 3 are devoted to the study of several geometric structures
which are closely related to Lagrangian mechanics. Almost tangent
structures and tangent bundles are examined in Chapter 2. The theory of
vertical, complete and horizontal lifts of tensor fields and connections to
tangent bundles are also included.
In Chapter 4 we study the differential calculus on the tangent bundle of
a manifold given by its canonical almost tangent structure. Connections in
tangent bundles, in the sense of Grifone, are examined and other approaches
to connections are briefly considered.
In Chapter 5 we study symplectic structures and cotangent bundles. In
fact, the canonical symplectic structure of the cotangent bundle of a manifold
is the (local) model for symplectic structures (Darboux theorem). Lifts of
tensor fields and connections to cotangent bundles are also included.
In Chapter 6 we examine Hamiltonian systems. As there are many specialized
books where this topic is extensively dealt with we decided to reduce
the material to some essential results. This chapter may be considered as
an introduction to the subject.
Chapter 7 is devoted to Lagrangian systems on manifolds. We apply the
main results of our previous chapters to Lagrangian systems. It is usual to
find in the literature regular Lagrangian systems obtained by pulling back
to the tangent bundle the canonical symplectic form of the cotangent bun-
dle of a given manifold, using for this the fiber derivative of the Lagrangian
function. In this vein we do not need to use the tangent bundle geometry.
Nevertheless there is an alternative approach for Lagrangian systems which
consists of using the structures directly underlying the tangent bundle manifold.
This gives an independent approach, i.e., an independent formulation
of the Hamiltonian theory. This point of view is that of J. Klein which
was adopted in the French book of C. Godbillon (1969). More recently some
points which use this kind of geometric formulation have also been presented
in the book of G. Marmo et al. (1985). We think that this viewpoint gives
a more powerful and elegant exposition of the subject. In fact we may say
that almost tangent geometry has a similar role in Lagrangian theories to
the role of symplectic geometry in Hamiltonian theories.
Chapter 8 is concerned with presymplectic structures. As the reader
will see in Chapter 7 the almost tangent formulation of classical lagrangian
systems does not require regularity conditions on the Lagrangian functions.
Thus, in general, if we wish the Euler-Lagrange equations to define a vector
field describing the dynamics (as it occurs in the regular case) we are
lead into constrained Lagrangians. Presymplectic forms also appear in the
Hamiltonian formalism, originated, for example, by degenerate Lagrangians,
and lead to the so-called Dirac-Bergmann constraint theory. In this chapter
we describe the geometric tools for such situations which have been inspired
by many authors.
One is concerned with
Particle Mechanics in local coordinates and is addressed to students who
are not very familiar with the classical approach. The other is devoted to a
brief summary on the theory of Jet-bundles, an important topic in modern
differential geometry.
We would like to express our gratitude to the Conselho Nacional de
Desenvolvimento Cientifico e Tecnologico, CNPq (Brazil) Proc. 31.1 115/79,
the FundaCao de Amparo a Pesquisa do Rio de Janeiro (FAPERJ), Proc. E-
29/170.662/88 and the Consejo Superior de Investigaciones Cientificas, CSIC
(Spain) for their financial support during the preparation of the manuscript.
We thank Pilar Criado for her very careful typing of the text on a microcomputer
using m. Our thanks are also due to Luis A. Corder0 and
Alfred Gray who helped us to use this typesetting system and to John Butterfield
for his valuable suggestions. To the Editor of Notas de Matematica,
Leopoldo Nachbin and to the Mathematics Acquisitions Editor of Elsevier
Science Publishers B.V./Physical Sciences and Engineering Division, Drs.
Arjen Sevenster, our thanks for including this volume in their series.
We conclude the book with two Appendices.
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
1.8 Fibred manifolds . Vector bundles . . . . . . . . . . . . . . . . 22
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
1.7 Tangent spaces . Vector fields . . . . . . . . . . . . . . . . . . 15
1.9 Tangent and cotangent bundles . . . . . . . . . . . . . . . . . 26
. .
2 Almost tangent structures and tangent bundles 111
2.1 Almost tangent structures on manifolds . . . . . . . . . . . . 111
2.2 Examples . The canonical almost tangent structure of the tangent
bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2.3 Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.4 Almost tangent connections . . . . . . . . . . . . . . . . . . . 119
2.5 Vertical and complete lifts of tensor fields to the tangent bundle
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . 135
2.9 Affine bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2.10 Integrable almost tangent structures which define fibrations . 139
2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Structures on manifolds 147
3.1 Almost product structures . . . . . . . . . . . . . . . . . . . . 147
3.2 Almost complex manifolds . . . . . . . . . . . . . . . . . . . . 151
3.3 Almost complex connections . . . . . . . . . . . . . . . . . . . 156
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 . . . . . . . . . . . . . . . . . . 167
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 . . . . . . . . . . . . . . . . . . . 181
4.1.1 Vertical derivation . . . . . . . . . . . . . . . . . . . . 183
4.1.2 Vertical differentiation . . . . . . . . . . . . . . . . . . 184
4.2 Homogeneous and semibasic forms . . . . . . . . . . . . . . . 186
4.2.1 Homogeneous forms . . . . . . . . . . . . . . . . . . . 186
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 . . . . . . . . . . . . . . . . . 199
4.6 Semisprays and connections . . . . . . . . . . . . . . . . . . . 206
4.7 Weak and strong torsion . . . . . . . . . . . . . . . . . . . . . 211
4.8 Decomposition theorem . . . . . . . . . . . . . . . . . . . . . 213
4.9 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
4.10 Almost complex structures on tangent bundles (11) . . . . . . 218
4.11 Connection in principal bundles . . . . . . . . . . . . . . . . . 221
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 . . . . . . . . 245
5.6 Darboux Theorem . . . . . . . . . . . . . . . . . . . . . . . . 249
5.7 Almost cotangent structures . . . . . . . . . . . . . . . . . . . 253
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 . . . . . . . . . . . . . 295
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 . . . . . . . . . . . . . . . . . . 330
7.8 Dynamical connections . . . . . . . . . . . . . . . . . . . . . . 336
7.10 The variational approach . . . . . . . . . . . . . . . . . . . . 347
7.11 Special symplectic manifolds . . . . . . . . . . . . . . . . . . 357
7.12 Noether’s theorem . Symmetries . . . . . . . . . . . . . . . . . 362
7.13 Lagrangian and Hamiltonian mechanical systems with constraints
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
7.14 Euler-Lagrange equations on T*M @ TM . . . . . . . . . . . 370
7.15 More about semisprays . . . . . . . . . . . . . . . . . . . . . . 376
7.16 Generalized Caplygin systems . . . . . . . . . . . . . . . . . . 391
7.17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
8 Presymplectic mechanical systems 399
8.1 The first-order problem and the Hamiltonian formalism . . . 399
8.1.1 The presymplectic constraint algorithm . . . . . . . . 400
8.1.2 Relation to the Dirac-Bergmann theory of constraints 404
8.2 The second-order problem and the Lagrangian formalism . . 409
8.2.1 The constraint algorithm and the Legendre transformation
. . . . . . . . . . . . . . . . . . . . . . . . . . . 409
8.2.2 Almost tangent geometry and degenerate Lagrangians 413
8.2.3 Other approaches . . . . . . . . . . . . . . . . . . . . . 428
8.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
A A brief summary of particle mechanics in local coordinates439
A.l Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . 439
A.l.l Elementary principles . . . . . . . . . . . . . . . . . . 439
A.1.2 Energies . . . . . . . . . . . . . . . . . . . . . . . . . . 441
A.2 Classical Mechanics: Lagrangian and Hamiltonian formalisms 443
A.2.1 Generalized coordinates . . . . . . . . . . . . . . . . . 443
A.2.2 Euler-Lagrange and Hamilton equations . . . . . . . . 445
B Higher order tangent bundles . Generalities 45 1
B.l Jets of mappings (in one independent variable) . . . . . . . . 451
B.2 Higher order tangent bundles . . . . . . . . . . . . . . . . . . 452
B.3 The canonical almost tangent structure of order k . . . . . . 454
B.4 The higher-order PoincarB-Cartan form . . . . . . . . . . . . 454
Bibliography 457
Index 471
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