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Mathematical Physics 1: Dynamical Systems and
Classical Mechanics (WS 2019/2020)
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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
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17.02.20: You can pick up your Übungsschein in MA 701, Frau J. Downes
Literature
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M. Petrera, Mathematical Physics 1. Dynamical systems and
classical mechanics. Lecture Notes, Logos, 2013.
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V. Arnold, Mathematical methods of classical mechanics, Springer, 1989.
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C. Chicone, Ordinary differential equations with applications, Springer, 2006.
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M.W. Hirsch, S. Smale, Differential equations, dynamical systems and
linear algebra, Academic Press, 1974.
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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
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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.
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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.
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