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Institut für Mathematik

# Arbeitsgruppe Geometrie Geometry Group

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## Mathematical Physics 1: Dynamical Systems and Classical Mechanics (WS 2015/2016) Lectures Tutorials Dr. Matteo Petrera Tuesday 12:15 - 13:45 MA 749 Thursday 12:15 - 13:45 MA 749 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 Dr. Raphael Boll Monday 10:00 - 12:00 MA 819 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.

### 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.