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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 phase-space volumes, Poincaré theorem, Lyapunov functions, linear systems
  • Week 5. Stability of linear systems, Lyapunov matrix equation, linearization, Poincaré-Lyapunov theorem
  • Week 6. Topological equivalence, Hartman-Grobman theorem, invariant manifolds, nonautonomous linear systems
  • Week 7. Periodic nonautonomous linear systems, Floquet theory, local bifurcation theory
  • Week 8. Saddle-node bifurcations, Hopf bifurcations, Newton equations, conservative mechanical system, action functional
  • Week 9. Euler-Lagrange 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 2-form, 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.

Rene Zander . 17.02.2020.