MATS199 Advanced Differential Geometry (4 cr)
Learning outcomes
Geometric control theory and Geometric Mechanics
Study methods
2 take-home written exams.
One in the middle of the course. One at the end.
Content
Basics:
vector fields, existence and uniqueness of ODE (following Coddington & Levinson),
flow of linear vector fields, Lie brackets, Cayley-Hamilton theorem, Constant Rank Theorem.
Orbits of families of vector fields:
Integrable distributions, Frobenius Theorem, Bracket generation, Reachability, Orbit theorem, Hermann-Nagano Theorem, Chart Theorem for path space (with no proof).
Elements of Symplectic Geometry and Geometric mechanics:
Tautological form, Symplectic form, Lagrangian function, Hamiltonian function, Hamiltonian vector field, Legendre transform, Poisson bracket, Euler-Lagrange equations, N枚ther Theorem.
Extras:
Cartan鈥檚 approach, G-structures, metric on bundles
Materials
Main references:
Arnold. Mathematical Methods of Classical Mechanics (2nd ed.), 1989
Jurdjevic. Geometric control theory. Cambridge University Press, 1997.
Extra references:
A. Agrachev and Y. Sachkov. Control Theory from the Geometric Viewpoint
R. Montgomery. A tour of subriemannian geometries, their geodesics and applications, 2001.
H. Nijmeijer and A. van der Schaft. Nonlinear dynamical control systems. Springer-Verlag, 1990.
S. Shankar Sastry. Nonlinear systems: analysis, stability, and control. Springer-Verlag, 1999.
Prerequisites
Differential Geometry (MATS197)