MATS199 Advanced Differential Geometry (4 cr)

Study level:
Advanced studies
Grading scale:
0-5
Language:
English
Responsible organisation:
Department of Mathematics and Statistics
Curriculum periods:
2017-2018, 2018-2019, 2019-2020

Description

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, Nöther Theorem.

Extras:
Cartan’s approach, G-structures, metric on bundles

Completion methods

2 take-home written exams.
One in the middle of the course. One at the end.

Learning outcomes

Geometric control theory and Geometric Mechanics

Description of prerequisites

Differential Geometry (MATS197)

Study 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.

Completion methods

Method 1

Select all marked parts
Parts of the completion methods
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Unpublished assessment item