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


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 twodimensional 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 HenonHeiles system.
Here you can download a Maple routine to produce HenonHeiles orbits
 14.12.15. No Lectures and Tutorial in the week 1115 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
(PicardLindelöf theorem, Peano theorem)

Lecture 2. Wellposed
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, PoincarèLyapunov theorem

Lecture 10.
Topological equivalence of dynamical systems, HartmanGrobman theorem, stable and unstable
manifolds

Lecture 11.
Center manifolds, nonautonomous IVPs, principal matrix solutions, fundamental solutions

Lecture 12.
Wronskian of IVPs, AbelLiouville theorem, periodic IVPs, Floquet theorem, definition of bifurcation

Lecture 13.
Bifurcations theory, normal forms of bifurcations, saddlenode bifurcations, Hopf bifurcations

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

Lecture 15.
EulerLagrange equations, principle of least action,
conservative Lagrangian systems, Dirichlet theorem,
linearization of EulerLagrange 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 1form

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

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

Lecture 23.
kforms, kvector 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.
ArnoldLiouville 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!
