 Fakultät II
Institut für Mathematik

# Arbeitsgruppe Geometrie Geometry Group

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

### Schedule

 Lectures Tutorial Dr. Matteo Petrera Mon 12:15 - 14:00 MA 750 Dr. Matteo Petrera Fri 12:15 - 14:00 MA 750 Dr. Pasquale Zito Tue 16:15 - 18:00 MA 749

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

### Office hours

 Name Office hours Room Dr. Matteo Petrera Wed 10:00 - 12:00 MA 819 Dr. Pasquale Zito By appointment, Fri 15:30 - 17:30

### 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.

### 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.

 Matteo Petrera . 09.03.2012