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Mathematical Physics I: Dynamical Systems and Classical Mechanics (WS 11/12)
Schedule
Lectures |
Dr. Matteo Petrera |
Mon |
12:15 - 14:00 |
MA 750 |
Dr. Matteo Petrera |
Fri |
12:15 - 14:00 |
MA 750 |
Tutorial |
Dr. Pasquale Zito |
Tue |
16:15 - 18:00 |
MA 749 |
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
News
Homework sheets
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.
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