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Mathematical Physics 1: Dynamical Systems and Classical Mechanics (WS 2015/2016)

Lectures Dr. Matteo Petrera Tuesday 12:15 - 13:45 MA 749
Thursday 12:15 - 13:45 MA 749
Tutorials Dr. Raphael Boll Wednesday 10:15 - 11:45 MA 750

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


Office Hours

Dr. Matteo Petrera Monday 10:00 - 12:00 MA 819
Dr. Raphael Boll by appointment MA 817

Contents

Initial value problems, dynamical systems, stability theory, bifurcation theory, Lagrangian mechanics, Hamiltonian mechanics


News

  • 20.10.15. Look here for the question about local Lipschitz continuity
  • 02.11.15. Interesting video about recurrence (Poincarè) theorem. Also this one is nice
  • 04.11.15. Here you can download a Maple routine to produce phase portraits of two-dimensional IVPs
  • 19.11.15. Until 13.12.15 Mats Vermeeren will be responsible for the tutorial and the homework. His office hours: Tuesdays, 10:30 - 11:30 (MA 818)
  • 02.12.15. Interesting video about Henon-Heiles system. Here you can download a Maple routine to produce Henon-Heiles orbits
  • 14.12.15. No Lectures and Tutorial in the week 11-15 of January 2016
  • 08.02.16. No tutorial on 10.02.16
  • 15.02.16. The certificates for the tutorial (necessary for the oral exam) are ready! Please take your one by Frau J. Downes, MA 701

Lectures

  • Lecture 1. Introduction and motivations, ODEs and IVPs, solutions, existence and uniqueness theorems for IVPs (Picard-Lindelöf theorem, Peano theorem)

  • Lecture 2. Well-posed IVPs, dependence of solutions on initial data and parameters

  • Lecture 3. Prolongations of solutions, definition of dynamical system (continuous and discrete), orbits, fixed points, cycles, phase portrait

  • Lecture 4. Invariant sets, invariant functions, Lyapunov stability, autonomous IVPs as continuous dynamical systems, geometric method, flows, Lie theorem

  • Lecture 5. Lie series, invariant functions and their characterizations, examples

  • Lecture 6. Lie bracket, Lie algebra of smooth vector fields, commutation of flows, evolution of phase space volumes, Liouville theorem

  • Lecture 7. Recurrence (Poincarè) theorem, stability of fixed points, Lyapunov functions, linear systems

  • Lecture 8. Stability theorems for linear systems

  • Lecture 9. Lyapunov functions for linear systems, linearized IVPs, Poincarè-Lyapunov theorem

  • Lecture 10. Topological equivalence of dynamical systems, Hartman-Grobman theorem, stable and unstable manifolds

  • Lecture 11. Center manifolds, non-autonomous IVPs, principal matrix solutions, fundamental solutions

  • Lecture 12. Wronskian of IVPs, Abel-Liouville theorem, periodic IVPs, Floquet theorem, definition of bifurcation

  • Lecture 13. Bifurcations theory, normal forms of bifurcations, saddle-node bifurcations, Hopf bifurcations

  • Lecture 14. Introduction to classical mechanics, Newton equations, conservative systems, Lagrangian systems

  • Lecture 15. Euler-Lagrange equations, principle of least action, conservative Lagrangian systems, Dirichlet theorem, linearization of Euler-Lagrange equations

  • Lecture 16. Symmetries of Lagrangian systems, Noether theorem in Lagrangian form, Legendre transformation, canonical Hamiltonian systems

  • Lecture 17. Canonical Hamilton equations, Hamiltonian vector fields, symplectic Lie group and algebra, symplectic structure of the canonical Hamiltonian phase space

  • Lecture 18. Poisson brackets, Noether theorem in Hamiltonian form, canonical and symplectic transformations

  • Lecture 19. Canonical and symplectic transformations, preservation of the form of Hamilton eqs., preservation of Poisson brackets

  • Lecture 20. Algebraic and differential forms, Lie condition for canonical transformation, Liouville 1-form

  • Lecture 21. Symplectic transformations and preservation of the symplectic 2-form, generating functions

  • Lecture 22. Introduction to mechanics on smooth manifolds, basic facts on smooth manifolds

  • Lecture 23. k-forms, k-vector fields (definitions and examples)

  • Lecture 24. Lie derivatives, Poisson manifolds, Hamiltonian mechanics on Poisson manifolds

  • Lecture 25. Distributions and foliations of manifolds, rank of a Poisson manifold, examples of Poisson manifolds

  • Lecture 26. Symplectic manifolds, Hamiltonian mechanics on symplectic manifolds, symplectic foliation of a Poisson manifold

  • Lecture 27. Arnold-Liouville integrable systems on symplectic manifolds (examples)


Literature

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

Exercise sheets


Homework policy

  • To get a certificate for the tutorial you need to satisfactorily complete 60% of the homework assignments.
  • Homework assignments are due weekly. They may be turned in directly to Dr. Boll at the beginning of the Wednesdays's Tutorial (10:15) or left in the letter box of Dr. Boll (MA 701, Frau J. Downes) before 10:15. Late homeworks will not be accepted!

Matteo Petrera . 15.02.2016.