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.

- Coverpage
- Halftitle page
- Series page
- Title page
- Copyright
- Acknowledgments
- Introduction
- Chapter 1 The Orbit Theorem and Lie determined systems
- Chapter 2 Control systems: accessibility and controllability
- Chapter 3 Lie groups and homogeneous spaces
- Chapter 4 Symplectic manifolds: Hamiltonian vector fields
- Chapter 5 Poisson manifolds, Lie algebras, and coadjoint orbits
- Chapter 6 Hamiltonians and optimality: the Maximum Principle
- Chapter 7 Hamiltonian view of classic geometry
- Chapter 8 Symmetric spaces and sub-Riemannian problems
- Chapter 9 Affine-quadratic problem
- Chapter 10 Cotangent bundles of homogeneous spaces as coadjoint orbits
- Chapter 11 Elliptic geodesic problem on the sphere
- Chapter 12 Rigid body and its generalizations
- Chapter 13 Isometry groups of space forms and affine systems: Kirchhoff’s elastic problem
- Chapter 14 Kowalewski–Lyapunov criteria
- Chapter 15 Kirchhoff–Kowalewski equation
- Chapter 16 Elastic problems on symmetric spaces: the Delauney–Dubins problem
- Chapter 17 The non-linear Schroedinger’s equation and Heisenberg’s magnetic equation–solitons
- References
- Index