MATS199 Valittuja kohtia differentiaaligeometriaa (4 op)

Opinnon taso:
Syventävät opinnot
Arviointiasteikko:
0-5
Suorituskieli:
englanti
Vastuuorganisaatio:
Matematiikan ja tilastotieteen laitos
Opetussuunnitelmakaudet:
2017-2018, 2018-2019, 2019-2020

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

Geometric control theory and Geometric Mechanics

Esitietojen kuvaus

Differential Geometry (MATS197)

Oppimateriaalit

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.

Suoritustavat

Tapa 1

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