Arbeitsgruppe Geometrie

Mathematical Physics I: Classical Mechanics   (WS 10/11)

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

Contents

Ordinary differential equations, Dynamical Systems, Linear dynamical systems, Stability theory, Calculus of variations, Legendre transformations, Lagrangian and Hamiltonian systems, Geodesic motions, Differentiable manifolds, Symplectic manifolds, Poisson brackets, Canonical transformations, Generating functions.

Literature

• Yu.B. Suris, Skript (in German).
• V.I. Arnold, Mathematical Methods of Classical Mechanics, (Graduate Texts in Mathematics) Springer-Verlag (1989).
• V.I. Arnold, Ordinary Differential Equations, The MIT Press (1978).

Schedule

News

Homework sheets

• Worksheet 1 (Due: Tutorial on 27.10.10)
• Worksheet 2 (Due: Tutorial on 03.11.10)
• Worksheet 3 (Due: Tutorial on 10.11.10)
• Worksheet 4 (Due: Tutorial on 17.11.10)
• Worksheet 5 (Due: Tutorial on 24.11.10)
• Worksheet 6 (Due: Tutorial on 01.12.10)
• Worksheet 7 (Due: Tutorial on 08.12.10)
• Worksheet 8 (Due: 05.01.11)
• Worksheet 9 (Due: Tutorial on 19.01.11)
• Worksheet 10 (Due: Tutorial on 26.01.11)
• Worksheet 11 (Due: Tutorial on 02.02.11)
• Worksheet 12 (Due: Tutorial on 09.02.11)

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Class Exercises (discussed and solved in class)

• Tutorial 1 (20.10.10) (Examples of first-order ODEs, method of separation of variables, maximal solutions)
• Tutorial 2 (27.10.10) (Examples of ODEs, existence and uniqueness of solutions, Picard iteration)
• Tutorial 3 (03.11.10) (Examples of dynamical systems, fixed points, phase portraits, periodic solutions)
• Tutorial 4 (10.11.10) (Lyapunov method, stability of linear systems)
• Tutorial 5 (17.11.10) (Applications of Poincaré-Lyapunov Theorem, stability by Lyapunov functions)
• Tutorial 6 (24.11.10) (Topological conjugation, applications of Poincaré-Lyapunov Theorem, bifurcations)
• Tutorial 7 (01.12.10) (Bifurcations, invariant manifolds, center manifolds)
• Tutorial 8 (15.12.10) (Legendre transformations, Calculus of variations)
• Extra Tutorial (17.12.10) (Strange attractors, Lorenz system, Hénon map). Maple worksheets
• Tutorial 9 (12.01.11) (The Kepler problem)
• Tutorial 10 (19.01.11) (Canonical transformations, generating functions, example of an integrable Poisson map)
• Tutorial 11 (26.01.11) (Symplectic integrators, examples of Lagrangian and Hamiltonian systems, discrete Lagrangian mechanics)
• Tutorial 12 (02.02.11) (Embeddings, manifolds, applications of Regular value Theorem)
• Tutorial 13 (09.02.11) (Lagrangian mechanics on manifolds, geodesics on the sphere and on the torus)
• Tutorial 14 (16.02.11) (Symmetries and Noether's Theorem)

Homework policy

• To get a certificate for the tutorial, you need to satisfactorily complete 60% of the homework assignments.
• The homeworks are due weekly at the beginning of the Wednesday lecture (8:30). No homework will be accepted after the deadline has passed.
• Homeworks may be turned in directly to the assistant at the lecture, or left with the Sekretariat in MA 7-2 (Room 701).