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:
- Symmetric Spaces, Sub-Riemannian Problems and Principal Bundles.
- Kepler's Problem and Non-Euclidean Geometry.
- Kirchhoff's Elastic Problem, Elasticae and Mechanical Tops.
- The Rolling Sphere Problems.
- Integrable Hamiltonian Systems. Kowalewski- Lyapunov Integrability Criteria.
- Complex Symplectic Geometry. Real Forms. Applications to Geometry and Mechanics.
- Problems of Quantum Control.
- 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.