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Institut für Mathematik

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## Mathematical Physics I: Dynamical Systems and Classical Mechanics   (WS 13/14)

 Name Office hours Room Prof. Dr. Yuri B. Suris MA 827 Dr. Matteo Petrera Monday 10:00 - 11:00 and on app. MA 819

This is a course of the Berlin Mathematical School held in English.

### Schedule

 Lectures Tutorial Prof. Dr. Yuri B. Suris Tue 10:15 - 11:45 MA 749 Prof. Dr. Yuri B. Suris Thu 10:15 - 11:45 MA 749 Dr. Matteo Petrera Tue 12:15 - 13:45 MA 749

### Contents

Initial value problems, Dynamical systems, Stability theory, Bifurcation theory, Lagrangian mechanics, Hamiltonian mechanics, Mechanics on smooth manifolds.

### News

• Last Tutorial on 04.02.14.

• Last Homework will be Exercise Sheet 14 (Due: Tutorial on 04.02.14).

• Remember that to get the certificate for the tutorial (necessary condition for the oral exam) you need to collect 60% of the total sum of points, i.e., 252 points out of 14x30=420 points.

### Tutorial classes

• 22.10.13. IVPs, Existence and uniqueness of local solutions, Maximal solutions, Examples, Exercises.
• 29.10.13. Definition of dynamical system (continuous and discrete), Flows, Examples, Exercises.
• 05.11.13. Flows and vector fields, Lie series, Lie derivative of a function, Integrals of motion, Examples, Exercises.
• 12.11.13. Integrals of motion, Commuting flows, Stability of fixed points, Linear systems, Examples, Exercises.
• 19.11.13. Poincarè-Lyapunov theorem, Lyapunov functions, Examples, Exercises.
• 26.11.13. No Tutorial.
• 03.12.13. Non-autonoumous linear IVPs, Periodic linear IVPs, Examples, Exercises.
• 10.12.13. Stable, unstable and center manifolds, Examples, Exercises.
• 19.12.13. Mechanical systems, Conservative mechanical systems, Lagrangian mechanics, Examples, Exercises.
• 07.01.14. Euler-Lagrange equations, Symmetries of Lagrangians, Noether Theorem, Examples, Exercises.
• 14.01.14. Canonical Hamiltonian mechanics, canonical Hamilton equations, canonical Poisson brackets, Examples, Exercises.
• 21.01.14. Canonical and symplectic transformations, Examples, Exercises.
• 28.01.14. Smooth manifolds, motivations and some definitions, Examples, Exercises.
• 04.02.14. Vector fields and differential forms on manifolds, Examples, Exercises.

### Homework policy

• To get a certificate for the tutorial you need to satisfactorily complete 60% of the homework assignments.
• Homework assignments are due weekly. They may be turned in directly to Dr. Petrera at the beginning of the Tuesday Tutorial (12:15) or left in the letter box of Dr. Petrera (MA 701, Frau J. Downes)

 Matteo Petrera . 28.01.2014