Videos, and Games
Mathematical Physics 1: Dynamical Systems and
Classical Mechanics (WS 2015/2016)
This is a course of the
Berlin Mathematical School
held in English.
Initial value problems, dynamical systems, stability theory, bifurcation theory,
Lagrangian mechanics, Hamiltonian mechanics
- 20.10.15. Look
here for the question about local Lipschitz continuity
- 02.11.15. Interesting
video about recurrence (Poincarè) theorem. Also
Here you can download a Maple routine to produce phase portraits
of two-dimensional IVPs
- 19.11.15. Until 13.12.15
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
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
Lecture 6. Lie bracket, Lie algebra of smooth
vector fields, commutation of flows, evolution of phase space volumes, Liouville theorem
Recurrence (Poincarè) theorem, stability of fixed points, Lyapunov functions,
Stability theorems for linear systems
Lyapunov functions for linear systems, linearized IVPs, Poincarè-Lyapunov theorem
Topological equivalence of dynamical systems, Hartman-Grobman theorem, stable and unstable
Center manifolds, non-autonomous IVPs, principal matrix solutions, fundamental solutions
Wronskian of IVPs, Abel-Liouville theorem, periodic IVPs, Floquet theorem, definition of bifurcation
Bifurcations theory, normal forms of bifurcations, saddle-node bifurcations, Hopf bifurcations
Introduction to classical mechanics, Newton equations, conservative systems, Lagrangian systems
Euler-Lagrange equations, principle of least action,
conservative Lagrangian systems, Dirichlet theorem,
linearization of Euler-Lagrange equations
Symmetries of Lagrangian systems, Noether theorem in Lagrangian form, Legendre transformation,
canonical Hamiltonian systems
Canonical Hamilton equations, Hamiltonian vector fields,
symplectic Lie group and algebra,
of the canonical Hamiltonian phase space
Poisson brackets, Noether theorem in Hamiltonian form, canonical and symplectic transformations
Canonical and symplectic transformations, preservation of the form of Hamilton eqs.,
preservation of Poisson brackets
Algebraic and differential forms, Lie condition for canonical transformation,
Symplectic transformations and preservation of the symplectic 2-form, generating functions
Introduction to mechanics on smooth manifolds, basic facts on smooth manifolds
k-forms, k-vector fields (definitions and examples)
Lie derivatives, Poisson manifolds, Hamiltonian mechanics on Poisson manifolds
Distributions and foliations of manifolds, rank of a Poisson manifold, examples
of Poisson manifolds
Symplectic manifolds, Hamiltonian mechanics on symplectic manifolds,
symplectic foliation of a Poisson manifold
Arnold-Liouville integrable systems on symplectic manifolds
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.
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!