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Mathematical Physics 1: Dynamical Systems and
Classical Mechanics (WS 2015/2016)
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This is a course of the
Berlin Mathematical School
held in English.
Office Hours
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
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Lecture 1. Introduction and motivations, ODEs and IVPs, solutions, existence and uniqueness theorems for IVPs
(Picard-Lindelöf theorem, Peano theorem)
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Lecture 2. Well-posed
IVPs, dependence of solutions on initial data and parameters
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Lecture 3. Prolongations of solutions,
definition of dynamical system (continuous and discrete), orbits, fixed points,
cycles, phase portrait
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Lecture 4. Invariant sets, invariant
functions, Lyapunov stability, autonomous IVPs as continuous dynamical systems,
geometric method, flows, Lie theorem
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Lecture 5. Lie series, invariant functions and their
characterizations, examples
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Lecture 6. Lie bracket, Lie algebra of smooth
vector fields, commutation of flows, evolution of phase space volumes, Liouville theorem
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Lecture 7.
Recurrence (Poincarè) theorem, stability of fixed points, Lyapunov functions,
linear systems
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Lecture 8.
Stability theorems for linear systems
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Lecture 9.
Lyapunov functions for linear systems, linearized IVPs, Poincarè-Lyapunov theorem
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Lecture 10.
Topological equivalence of dynamical systems, Hartman-Grobman theorem, stable and unstable
manifolds
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Lecture 11.
Center manifolds, non-autonomous IVPs, principal matrix solutions, fundamental solutions
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Lecture 12.
Wronskian of IVPs, Abel-Liouville theorem, periodic IVPs, Floquet theorem, definition of bifurcation
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Lecture 13.
Bifurcations theory, normal forms of bifurcations, saddle-node bifurcations, Hopf bifurcations
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Lecture 14.
Introduction to classical mechanics, Newton equations, conservative systems, Lagrangian systems
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Lecture 15.
Euler-Lagrange equations, principle of least action,
conservative Lagrangian systems, Dirichlet theorem,
linearization of Euler-Lagrange equations
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Lecture 16.
Symmetries of Lagrangian systems, Noether theorem in Lagrangian form, Legendre transformation,
canonical Hamiltonian systems
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Lecture 17.
Canonical Hamilton equations, Hamiltonian vector fields,
symplectic Lie group and algebra,
symplectic structure
of the canonical Hamiltonian phase space
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Lecture 18.
Poisson brackets, Noether theorem in Hamiltonian form, canonical and symplectic transformations
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Lecture 19.
Canonical and symplectic transformations, preservation of the form of Hamilton eqs.,
preservation of Poisson brackets
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Lecture 20.
Algebraic and differential forms, Lie condition for canonical transformation,
Liouville 1-form
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Lecture 21.
Symplectic transformations and preservation of the symplectic 2-form, generating functions
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Lecture 22.
Introduction to mechanics on smooth manifolds, basic facts on smooth manifolds
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Lecture 23.
k-forms, k-vector fields (definitions and examples)
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Lecture 24.
Lie derivatives, Poisson manifolds, Hamiltonian mechanics on Poisson manifolds
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Lecture 25.
Distributions and foliations of manifolds, rank of a Poisson manifold, examples
of Poisson manifolds
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Lecture 26.
Symplectic manifolds, Hamiltonian mechanics on symplectic manifolds,
symplectic foliation of a Poisson manifold
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Lecture 27.
Arnold-Liouville integrable systems on symplectic manifolds
(examples)
Literature
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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.
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M. Petrera, Mathematical Physics 1. Dynamical systems and
classical mechanics. Lecture Notes, Logos, 2013.
Exercise sheets
Homework policy
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To get a certificate for the tutorial you need to satisfactorily complete
60% of the homework assignments.
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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!
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