Spis treści: Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions


Part I

1     Lagrangian and Hamiltonian mechanics, s. 3

1.1      Newtonian mechanics, s. 3
1.2      Lagrangian mechanics, s. 13
1.3      Constraints, s. 18
1.4      The Legendre transform and Hamiltonian mechanics, s. 24
1.5      Rigid bodies, s. 30

2     Manifolds, s. 43

2.1      Submanifolds of R", s. 43
2.2      Tangent vectors and derivatives, s. 57
2.3      Differentials and cotangent vectors, s. 69
2.4      Matrix groups as submanifolds, s. 78
2.5      Abstract manifolds, s. 83

3     Geometry on manifolds, s. 99

3.1      Vector fields, s. 99
3.2      Differential 1-forms, s. 112
3.3      Tensors, s. 117
3.4      Riemannian geometry, s. 128
3.5      Symplectic geometry, s. 139

4   Mechanics on manifolds, s. 155

4.1      Lagrangian mechanics on manifolds, s. 155
4.2      The Legendre transform and Hamilton's equations, s. 160
4.3      Hamiltonian mechanics on Poisson manifolds, s. 166
4.4      A brief look at symmetry, reduction and conserved quantities, s. 175

5    Lie groups and Lie algebras, s. 187

5.1      Matrix Lie groups and Lie algebras, s. 187
5.2      Abstract Lie groups and Lie algebras, s. 193
5.3      Isomorphisms of Lie groups and Lie algebras, s. 199
5.4      The exponential map, s. 203

6    Group actions, symmetries and reduction, s. 209

6.1      Lie group action, s. 209
6.2      Actions of a Lie group on itself, s. 220
6.3      Quotient spaces, s. 230
6.4      Poisson reduction, s. 233

7    Euler-Poincare reduction: Rigid body and heavy top, s. 241

7.1      Rigid body dynamics, s. 241
7.2      Euler-Poincare reduction: the rigid body, s. 248
7.3      Euler-Poincare reduction theorem, s. 255
7.4      Modelling heavy-top dynamics, s. 261
7.5      Euler-Poincare systems with advected parameters, s. 270

8    Momentum maps, s. 281

8.1      Definition and examples, s. 281
8.2      Properties of momentum maps, s. 291

9    Lie-Poisson reduction, s. 295

9.1      The reduced Legendre transform, s. 296
9.2      Lie-Poisson reduction: geometry, s. 301
9.3      Lie-Poisson reduction: dynamics, s. 307
9.4      Momentum maps revisited, s. 310
9.5      Co-Adjoint orbits, s. 315
9.6      Lie-Poisson brackets on semidirect products, s. 318

10 Pseudo-rigid bodies, s. 325

10.1    Modelling, s. 325
10.2    Euler-Poincare reduction, s. 330
10.3    Lie-Poisson reduction, s. 335
10.4    Momentum maps: angular momentum and circulation, s. 337


Part II, s. 351


11   EPDiff, s. 353

11.1    Brief history of geometric ideal continuum motion, s. 353
11.2    Geometric setting of ideal continuum motion, s. 355
11.3    Euler-Poincare reduction for continua, s. 359
11.4    EPDiff: Euler-Poincare equation on the diffeomorphisms, s. 360

12  EPDiff solution behaviour, s. 367

12.1    Introduction, s. 367
12.2    Shallow-water background for peakons, s. 371
12.3    Peakons and pulsons, s. 378

13  Integrability of EPDiff in 1D, s. 385

13.1    The CH equation is bi-Hamiltonian, s. 386
13.2    The CH equation is isospectral, s. 389

14  EPDiff in n dimensions, s. 395

14.1    Singular momentum solutions of the EPDiff equation for geodesic motion in higher dimensions, s. 395
14.2    Singular solution momentum map Jsing, s. 399
14.3    The geometry of the momentum map, s. 406
14.4    Numerical simulations of EPDiff in two dimensions, s. 410

15 Computational anatomy: contour matching using EPDiff, s. 419

15.1    Introduction to computational anatomy (CA), s. 419
15.2    Mathematical formulation of template matching for CA, s. 423
15.3    Outline matching and momentum measures, s. 425
15.4    Numerical examples of outline matching, s. 427

Góra
varia 20.jpg

Zapytaj bibliotekarza

Telewizja UŚ

Więc Jestem. Studencki serwis rozwoju