Arbeitsgruppe Geometrie

Mathematical Physics I: Dynamical Systems and Classical Mechanics   (WS 11/12)

Schedule

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

Office hours

Contents

• Ordinary differential equations, Existence and uniqueness theorems, Dependence on initial conditions and parameters, Prolongation of solutions.

• Dynamical systems (with continuous and discrete time), Flows and vector fields, Fixed points, Stability theorems, Linear dynamical systems, Linearization, Local bifurcation theory.

• Lagrangian mechanics in R^n, Legendre Transformation, Hamiltonian mechanics in R^n, Symplectic structure of the phase space, Poisson brackets, Canonical transformations, Hamilton-Jacobi theory, Symplectic integrators, Differentiable manifolds, Mechanics on manifolds, Symmetries and Noether theorem, Symplectic geometry, Poisson geometry, Rigid body equations of motion.

Literature

• M. Petrera, Lecture Notes (download the latest version!).
• Yu.B. Suris, Lecture Notes WS10/11 (hand-written, in German).

News

Homework sheets

• Worksheet 1 (Due: Tutorial on 25.10.11)
• Worksheet 2 (Due: Tutorial on 01.11.11)
• Worksheet 3 (Due: Tutorial on 08.11.11)
• Worksheet 4 (Due: Tutorial on 15.11.11)
• Worksheet 5 (Due: Tutorial on 22.11.11)
• Worksheet 6 (Due: Tutorial on 29.11.11)
• Worksheet 7 (Due: Tutorial on 06.12.11)
• Worksheet 8 (Due: Tutorial on 13.12.11)
• Worksheet 9+10 (Due: Tutorial on 10.01.12)
• Worksheet 11 (Due: Tutorial on 17.01.12)
• Worksheet 12 (Due: Tutorial on 24.01.12)
• Worksheet 13 (Due: Tutorial on 31.01.12)
• Worksheet 14 (Due: on 14.02.12, optional)

Class Exercises (discussed and solved in class)

• Tutorial 1 (18.10.11) (Examples of ODEs, construction of explicit solutions)
• Tutorial 2 (25.10.11) (Initial value problems, existence and uniqueness Theorem, maximal solutions)
• Tutorial 3 (01.10.11) (Qualitative analysis, planar systems od ODEs, integrals of motion, integrale curves)
• Tutorial 4 (08.11.11) (Periodic systems, discrete dynamical systems, integrals of motion, commuting flows)
• Tutorial 5 (15.11.11) (Linear autonomous and homogeneous systems, Lyapunov functions)
• Tutorial 6 (22.11.11) (Lyapunov functions, Poincaré-Lyapunov Theorem, Topological conjugacy)
• Tutorial 7 (29.11.11) (Center manifolds, Bifurcations)
• Tutorial 8 (06.12.11) (Conservative systems with one degree of freedom)
• Tutorial 9+10 (13.12.11) (Variational problems, Hamiltonian and Lagrangian mechanics, motion in central potential)
• Tutorial 11 (10.01.12) (Canonical transformations, Hamilton-Jacobi theory).
• Tutorial 12 (17.01.12) (Differentiable manifolds, differentiable maps, Regular value Theorem)
• Tutorial 13 (24.01.12) (Geodesics, Noether Theorem, symmetries)
• Tutorial 14 (31.01.12) (Lie groups, Lie algebras, exponential map, examples)
• Tutorial 15 (07.02.12) (Action-angle variables)

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 Tuesday lecture (16:00). No homework will be accepted after the deadline has passed.
• Homeworks may be turned in directly to the assistant at the lecture.