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Symplectic geometry and classical mechanics Semester
SoSe 2023
Lecturer
Bernd Ammann
Type of course (Veranstaltungsart)
Vorlesung
German title
Symplektische Geometrie und klassische Mechanik
Contents
The lecture will be given in English as soon as there is at least one Non-German speaking participant.
- Differentiable manifolds, tangential and cotangential vectors, differential forms, Lie derivative, flows of vector fields
- symplectic vector spaces and relations to complex structures
- symplectic manifolds
- the cotangent space a symplectic manifold
- Darboux' theorem and generalization (e.g. Weinstein's tubular neighborhood theorem)
- symplectic gradients and Hamiltonian systems
- Liouville's theorem
- action functional and Lagrangian mechanics, Legendre transformation
- prinziple of Maupertuis: Solutions of the Hamiltonian equations are reparametrizations of geodesics for a suitable Riemannian metric
- Poisson brackets
- preserved quantities (e.g. preservation of energy)
- integrable systems
- Some applications, e.g. to celestical mechanics, rotating rigid bodies, etc.
- Advanced topics such as stability, Kolmogorow-Arnold-Moser theorem, symplectic reduction
The precise content of the lecture will depend on the previous knowledge and the interests of the audience, see also the web page.
Literature
- Kai Cieliebak, Symplectic Geometry, Lecture Notes
- Rolf Berndt: An Introduction to Symplectic Geometry, AMS
More literature will be given shortly on the lecture's web page.
Recommended previous knowledge
Analysis I bis IV,
Lineare Algebra I und II
oder alternativ Kenntnisse über klassische Mechanik aus der Physik
Time/Date
Monday and Thursday: 10-12; Exercises: probably Wednesday 16-18
Location
Lectures (Mo+Th) in M101; Exercises probably in M103
Additional question session
Time/Date:
Location:
Course homepage
http://www.mathematik.uni-regensburg.de/ammann/symplectic
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)
Registration- Registration for the exercise classes: In the first week of lectures, within the lecture
- Please register for this lecture in GRIPS as we are sending emails via this system
- Registration for course work/examination/ECTS: FlexNow
Course work (Studienleistungen)- Successful participation in the exercise classes: 50% of the maximal points in the exercise
sheets, presentation of one solution
Examination (Prüfungsleistungen)- Oral exam: Duration: 30 minutes, Date: by individual appointment, re-exam: Date: by individual
appointment, duration: 30 minutes
Modules
BV, MV, MGAGeo, LA-GyGeo
ECTS
9
Notice Lehramt Gymnasium
Eines der drei Module LGyGeo, LGyNum, LGyStoch kann unbenotet abgeschlossen werden. Für das unbenotete Modul ist die o.a. Studienleistung (Erfolgreiche Teilnahme am Übungsbetrieb) und als weitere Studienleistung die Klausur oder ein Fachgespräch notwendig (siehe auch Modulkatalog)
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