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Mathematical Physics 1: Dynamical Systems and
Classical Mechanics (WS 2019/2020)



Name 
Office hours 
Room 
Lectures 
Dr. Matteo Petrera 
Monday 12:00  13:00 
MA 819 
Tutorial 
Rene Zander 
Monday 12:00  13:00 
MA 806 
Lectures 
Dr. Matteo Petrera 
Wednesday 
12:15  13:45 
MA 721 
Dr. Matteo Petrera 
Thursday 
10:15  11:45 
MA 642 
Tutorial 
Rene Zander 
Tuesday 
14:15  15:45 
MA 749 
This is a course of the
Berlin Mathematical School
held in English.
Contents
Initial value problems, smooth dynamical systems, stability theory, bifurcation theory,
Lagrangian mechanics, Hamiltonian mechanics, mechanics on manifolds
News

17.02.20: You can pick up your Übungsschein in MA 701, Frau J. Downes
Literature

M. Petrera, Mathematical Physics 1. Dynamical systems and
classical mechanics. Lecture Notes, Logos, 2013.

V. Arnold, Mathematical methods of classical mechanics, Springer, 1989.

C. Chicone, Ordinary differential equations with applications, Springer, 2006.

M.W. Hirsch, S. Smale, Differential equations, dynamical systems and
linear algebra, Academic Press, 1974.

J.E. Marsden, T. Ratiu,
Introduction to mechanics and symmetry, Springer, 1999.
Exercise sheets
Lectures
 Week 1. Initial value problems, existence and uniqueness of solutions, dependence on initial data
 Week 2. Dependence on parameters, prolongation of solutions, definition of dynamical systems, orbits,
fixed point, invariant sets and functions, flows of IVPs
 Week 3. Flows and vector fields, invariant functions, Lie bracket, Lie algebra of vector fields, commutativity of flows
 Week 4. Evolution of phasespace volumes, Poincaré theorem, Lyapunov functions, linear systems
 Week 5. Stability of linear systems, Lyapunov matrix equation, linearization, PoincaréLyapunov theorem
 Week 6. Topological equivalence, HartmanGrobman theorem, invariant manifolds, nonautonomous linear systems
 Week 7. Periodic nonautonomous linear systems, Floquet theory, local bifurcation theory
 Week 8. Saddlenode bifurcations, Hopf bifurcations, Newton equations, conservative mechanical system, action functional
 Week 9. EulerLagrange equations, conservative Lagrangian systems, stability of Lagrangian systems, symmetries, Noether theorem
 Week 10. Legendre transformation, Hamiltonian function, Hamilton equations, Hamiltonian flows, symplectic structure of the phase space
 Week 11. Canonical Poisson brackets, canonical and symplectic tranformations
 Week 12. Differential forms, canonical symplectic 2form, generating functions for canonical transformations
 Week 13. Crash course on differential geometry
 Week 14. Poisson geometry, Hamiltonian mechanics on Poisson manifolds
 Week 15. Hamiltonian mechanics on symplectic manifolds
Homework policy

To get a certificate for the tutorial you need to obtain an average grade of
60% on the homework assignments in both halves of the semester.

Homework assignments are due weekly. They may be turned in at the beginning of the tutorial or left in the letter box
of Rene Zander
(MA 701, Frau J. Downes) before that time.
