The synthesis of symplectic geometry, the calculus of variations and control theory offered in this book provides a crucial foundation for the understanding of many problems in applied mathematics. Focusing on the theory of integrable systems, this book introduces a class of optimal control problems on Lie groups, whose Hamiltonians, obtained through the Maximum Principle of optimality, shed new light on the theory of integrable systems. These Hamiltonians provide an original and unified account of the existing theory of integrable systems. The book particularly explains much of the mystery surrounding the Kepler problem, the Jacobi problem and the Kovalevskaya Top. It also reveals the ubiquitous presence of elastic curves in integrable systems up to the soliton solutions of the non-linear Schroedinger's equation. Containing a useful blend of theory and applications, this is an indispensable guide for graduates and researchers in many fields, from mathematical physics to space control.
Author(s): Velimir Jurdjevic
Series: Cambridge Studies in Advanced Mathematics 154
Publisher: Cambridge University Press
Year: 2016
Language: English
Pages: 436
Contents......Page 5
Acknowledgments......Page 10
Introduction......Page 11
Chapter 1 The Orbit Theorem and Lie determined systems......Page 22
1.1 Vector fields and differential forms......Page 23
1.2 Flows and diffeomorphisms......Page 27
1.3 Families of vector fields: the Orbit theorem......Page 30
1.4 Distributions and Lie determined systems......Page 35
2.1 Control systems and families of vector fields......Page 40
2.2 The Lie saturate......Page 46
Chapter 3 Lie groups and homogeneous spaces......Page 50
3.1 The Lie algebra and the exponential map......Page 52
3.2 Lie subgroups......Page 55
3.3 Families of left-invariant vector fields and accessibility......Page 60
3.4 Homogeneous spaces......Page 62
4.1 Symplectic vector spaces......Page 65
4.2 The cotangent bundle of a vector space......Page 68
4.3 Symplectic manifolds......Page 70
5.1 Poisson manifolds and Poisson vector fields......Page 76
5.2 The cotangent bundle of a Lie group: coadjoint orbits......Page 78
Chapter 6 Hamiltonians and optimality: the Maximum Principle......Page 85
6.1 Extremal trajectories......Page 86
6.2 Optimal control and the calculus of variations......Page 89
6.3 The Maximum Principle......Page 92
6.4 The Maximum Principle in the presence of symmetries......Page 102
6.5 Abnormal extremals......Page 106
6.6 The Maximum Principle and Weierstrass’ excess function......Page 112
7.1 Hyperbolic geometry......Page 117
7.2 Elliptic geometry......Page 122
7.3 Sub-Riemannian view......Page 127
7.4 Elastic curves......Page 134
7.5 Complex overview and integration......Page 136
8.1 Lie groups with an involutive automorphism......Page 139
8.2 Symmetric Riemannian pairs......Page 141
8.3 The sub-Riemannian problem......Page 152
8.4 Sub-Riemannian and Riemannian geodesics......Page 156
8.5 Jacobi curves and the curvature......Page 159
8.6 Spaces of constant curvature......Page 163
Chapter 9 Affine-quadratic problem......Page 168
9.1 Affine-quadratic Hamiltonians......Page 173
9.2 Isospectral representations......Page 176
9.3 Integrability......Page 180
Chapter 10 Cotangent bundles of homogeneous spaces as coadjoint orbits......Page 188
10.1 Spheres, hyperboloids, Stiefel and Grassmannian manifolds......Page 189
10.2 Canonical affine Hamiltonians on rank one orbits: Kepler and Newmann......Page 198
10.3 Degenerate case and Kepler’s problem......Page 200
10.4 Mechanical problem of C. Newmann......Page 208
10.5 The group of upper triangular matrices and Toda lattices......Page 213
Chapter 11 Elliptic geodesic problem on the sphere......Page 219
11.1 Elliptic Hamiltonian on semi-direct rank one orbits......Page 220
11.2 The Maximum Principle in ambient coordinates......Page 224
11.3 Elliptic problem on the sphere and Jacobi’s problem on the ellipsoid......Page 232
11.4 Elliptic coordinates on the sphere......Page 234
Chapter 12 Rigid body and its generalizations......Page 241
12.1 The Euler top and geodesic problems on SOn(R)......Page 242
12.2 Tops in the presence of Newtonian potentials......Page 252
Chapter 13 Isometry groups of space forms and affine systems: Kirchhoff’s elastic problem......Page 259
13.1 Elastic curves and the pendulum......Page 262
13.2 Parallel and Serret–Frenet frames and elastic curves......Page 269
13.3 Serret–Frenet frames and the elastic problem......Page 272
13.4 Kichhoff’s elastic problem......Page 277
Chapter 14 Kowalewski–Lyapunov criteria......Page 283
14.1 Complex quaternions and SO4(C)......Page 285
14.2 Complex Poisson structure and left-invariant Hamiltonians......Page 291
14.3 Affine Hamiltonians on SO4(C) and meromorphic solutions......Page 294
14.4 Kirchhoff–Lagrange equation and its solution......Page 308
Chapter 15 Kirchhoff–Kowalewski equation......Page 317
15.1 Eulers’ solutions and addition formulas of A. Weil......Page 325
15.2 The hyperelliptic curve......Page 331
15.3 Kowalewski gyrostat in two constant fields......Page 334
16.1 The curvature problem......Page 347
16.2 Elastic problem revisited – Dubins–Delauney on space forms......Page 352
16.3 Curvature problem on symmetric spaces......Page 371
16.4 Elastic curves and the rolling sphere problem......Page 379
Chapter 17 The non-linear Schroedinger’s equation and Heisenberg’s magnetic equation–solitons......Page 389
17.1 Horizontal Darboux curves......Page 390
17.2 Darboux curves and symplectic Fr ´ echet manifolds......Page 397
17.3 Geometric invariants of curves and their Hamiltonian vector fields......Page 407
17.4 Affine Hamiltonians and solitons......Page 420
Concluding remarks......Page 425
References......Page 427
Index......Page 434
