MAT 1312S
LIE GROUPS, SYMMETRIC SPACES, SUB-RIEMANNIAN GEOMETRY AND THE EQUATIONS OF PHYSICS
V. Jurdjevic

This course will be driven by two fundamental observations; the first observation is that the dual of the Lie algebra g of a Lie group G is a Poisson manifold and that much of the structure and the theory of Lie groups and their symmetric spaces can be directly deduced from this fact alone, and the second observation is that the orbit theorem with related accessibility results together with the Maximum Principle and the associated Hamiltonian formalism of optimal control theory are effective tools in carryng out the program suggested by the first observation. The symplectic outlook on Lie groups and their homogeneous spaces offers an additional advantage in that it ellucidates the importance of Lie groups for the equations of applied mathematics. The course will be a synthesis of the following topics:

  1. Symmetric Spaces, Sub-Riemannian Problems and Principal Bundles.
  2. Kepler's Problem and Non-Euclidean Geometry.
  3. Kirchhoff's Elastic Problem, Elasticae and Mechanical Tops.
  4. The Rolling Sphere Problems.
  5. Integrable Hamiltonian Systems. Kowalewski- Lyapunov Integrability Criteria.
  6. Complex Symplectic Geometry. Real Forms. Applications to Geometry and Mechanics.
  7. Problems of Quantum Control.
  8. Schroedinger's Equation and Infinite Dimensional Hamiltonian Systems. Solitons.
Much of this material will be taken from [J2] and the graduate monograph of mine in progress that will be ( I hope) finished by the time the course is offered.

Prerequisite:
The orbit theorem and the theory of Lie determined systems (Chapters 2 and 3 of [J1]).
The Maximum Principle of optimal control. ([J1] and [AS]).
Basics from differential and symplectic geometry. Some knowledge of principal bundles and their connections is also desirable. For instance, the first few pages of [S] on the Geometry of G-structures are enough.

Reference:
[AS] A. Agrachev and Y. Sachkov, Control Theory from the Geometric Point of View, Encyclopedia of Mathematical Sciences, vol. 87, Springer-Verlag, New York, 2004.
[H] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.
[J1] V. Jurdjevic, Geometric Control Theory, Studies in Advanced Math, vol. 52, Cambridge Univ. Press, New York, 1997.
[J2] V. Jurdjevic, Hamiltonian Systems on Complex Lie groups and their Homogeneous Spaces, Memoirs AMS Number 838 178 (2005).
[S] S. Sternberg, Lectures on Differential Geometry, Prentice- Hall, Inc, Englewood Cliffs, N.J., 1964.