MATS199 Valittuja kohtia differentiaaligeometriaa (4 op)
Kuvaus
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.
Osaamistavoitteet
Esitietojen kuvaus
Oppimateriaalit
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.