Classical mechanics, one of the oldest branches of science, has undergone a long evolution, developing hand in hand with many areas of mathematics, including calculus, differential geometry, and the theory of Lie groups and Lie algebras. The modern formulations of Lagrangian and Hamiltonian mechanics, in the coordinate-free language of differential geometry, are elegant and general. They provide a unifying framework for many seemingly disparate physical systems, such as n-particle systems, rigid bodies, fluids and other continua, and electromagnetic and quantum systems.Geometric Mechanics and Symmetry is a friendly and fast-paced introduction to the geometric approach to classical mechanics, suitable for a one- or two- semester course for beginning graduate students or advanced undergraduates. It fills a gap between traditional classical mechanics texts and advanced modern mathematical treatments of the subject. After a summary of the necessary elements of calculus on smooth manifolds and basic Lie group theory, the main body of the text considers how symmetry reduction of Hamilton's principle allows one to derive and analyze the Euler-Poincar? equations for dynamics on Lie groups.Additional topics deal with rigid and pseudo-rigid bodies, the heavy top, shallow water waves, geophysical fluid dynamics and computational anatomy. The text ends with a discussion of the semidirect-product Euler-Poincar? reduction theorem for ideal fluid dynamics.A variety of examples and figures illustrate the material, while the many exercises, both solved and unsolved, make the book a valuable class text.
Author(s): Darryl D. Holm, Tanya Schmah, Cristina Stoica
Series: Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts
Publisher: Wiley-Interscience
Year: 2009
Language: English
Pages: 537
0199212902......Page 1
Contents......Page 14
Part I......Page 18
1.1 Newtonian mechanics......Page 20
1.2 Lagrangian mechanics......Page 30
1.3 Constraints......Page 35
1.4 The Legendre transform and Hamiltonian mechanics......Page 41
1.5 Rigid bodies......Page 47
2.1 Submanifolds of R[sup(n)]......Page 60
2.2 Tangent vectors and derivatives......Page 74
2.3 Differentials and cotangent vectors......Page 86
2.4 Matrix groups as submanifolds......Page 95
2.5 Abstract manifolds......Page 100
3.1 Vector fields......Page 116
3.2 Differential 1-forms......Page 129
3.3 Tensors......Page 134
3.4 Riemannian geometry......Page 145
3.5 Symplectic geometry......Page 156
4.1 Lagrangian mechanics on manifolds......Page 172
4.2 The Legendre transform and Hamilton’s equations......Page 177
4.3 Hamiltonian mechanics on Poisson manifolds......Page 183
4.4 A brief look at symmetry, reduction and conserved quantities......Page 192
5.1 Matrix Lie groups and Lie algebras......Page 204
5.2 Abstract Lie groups and Lie algebras......Page 210
5.3 Isomorphisms of Lie groups and Lie algebras......Page 216
5.4 The exponential map......Page 220
6.1 Lie group actions......Page 226
6.2 Actions of a Lie group on itself......Page 237
6.3 Quotient spaces......Page 247
6.4 Poisson reduction......Page 250
7.1 Rigid body dynamics......Page 258
7.2 Euler–Poincaré reduction: the rigid body......Page 265
7.3 Euler–Poincaré reduction theorem......Page 272
7.4 Modelling heavy-top dynamics......Page 278
7.5 Euler–Poincaré systems with advected parameters......Page 287
8.1 Definition and examples......Page 298
8.2 Properties of momentum maps......Page 308
9 Lie–Poisson reduction......Page 312
9.1 The reduced Legendre transform......Page 313
9.2 Lie–Poisson reduction: geometry......Page 318
9.3 Lie–Poisson reduction: dynamics......Page 324
9.4 Momentum maps revisited......Page 327
9.5 Co-Adjoint orbits......Page 332
9.6 Lie–Poisson brackets on semidirect products......Page 335
10.1 Modelling......Page 342
10.2 Euler–Poincaré reduction......Page 347
10.3 Lie–Poisson reduction......Page 352
10.4 Momentum maps: angular momentum and circulation......Page 354
Part II......Page 368
11.1 Brief history of geometric ideal continuum motion......Page 370
11.2 Geometric setting of ideal continuum motion......Page 372
11.3 Euler–Poincaré reduction for continua......Page 376
11.4 EPDiff: Euler–Poincaré equation on the diffeomorphisms......Page 377
12.1 Introduction......Page 384
12.2 Shallow-water background for peakons......Page 388
12.3 Peakons and pulsons......Page 395
13 Integrability of EPDiff in 1D......Page 402
13.1 The CH equation is bi-Hamiltonian......Page 403
13.2 The CH equation is isospectral......Page 406
14.1 Singular momentum solutions of the EPDiff equation for geodesic motion in higher dimensions......Page 412
14.2 Singular solution momentum map J[sub(Sing)]......Page 416
14.3 The geometry of the momentum map......Page 423
14.4 Numerical simulations of EPDiff in two dimensions......Page 427
15.1 Introduction to computational anatomy (CA)......Page 436
15.2 Mathematical formulation of template matching for CA......Page 440
15.3 Outline matching and momentum measures......Page 442
15.4 Numerical examples of outline matching......Page 444
16.1 Overview......Page 450
16.2 Notation and Lagrangian formulation......Page 451
16.3 Symmetry-reduced Euler equations......Page 453
16.4 Euler–Poincaré reduction......Page 455
16.5 Semidirect-product examples......Page 459
17.1 Kelvin–Stokes theorem for ideal fluids......Page 470
17.2 Introduction to advected quantities......Page 473
17.3 Euler–Poincaré theorem......Page 478
18.1 Kelvin circulation theorem for GFD......Page 486
18.3 Equations of 2D geophysical fluid motion......Page 493
18.4 Equations of 3D geophysical fluid motion......Page 498
18.5 Variational principle for fluids in three dimensions......Page 503
18.6 Euler’s equations for a rotating stratified ideal incompressible fluid......Page 506
18.7 Well-posedness, ill-posedness, discretization and regularization......Page 517
Bibliography......Page 520
C......Page 526
E......Page 527
I......Page 528
M......Page 529
P......Page 530
S......Page 531
W......Page 532
