Conservative Dynamical Systems

Konstantinos Efstathiou, Heinz Hanßmann

spring	time	place
lectures	tuesday 10:15 - 13:00	BBL 276

ECTS : 8 credit points

Course material: V.I. Arnol'd, Mathematical Methods of Classical Mechanics (2nd edition), GTM 60, Springer-Verlag (1989)

Description

Classical Mechanics deals with dynamical systems without damping or friction. Here think of ideal pendula, springs or tops, the Solar system, etc. We deal with the Newtonian, the Lagrangian (or variational) and the Hamiltonian theory of such systems.

Important issues are the Noether Theorem, that links conservation laws to symmetry and the Liouville Theorem stating the conservation of volume by the phase flow. As a direct consequence of the latter, Hamiltonian systems can't have attractors.

Main part of the theory deals with symplectic manifolds, that also has applications in other parts of physics, like in optics. The language introduced in this course is needed for all reading of modern Dynamical Systems and Mathematical Physics.

Assumed knowledge

Ordinary Differential Equations,
Manifolds (a bit),
Introductory Dynamical Systems.

Examination

Presentation and home work excercises.

Subjects for presentation

1. "Noether's theorem" (from chapter 4): Yudistira Lesmana
2. "Small oscillations" (from chapter 5): Max Potters
3. "Parametric resonance" (from chapter 5): Nick Reinerink
4. "Inertial frames / Coriolis force" (from chapter 6): Alexander Ly
5. "The free rigid body" (from chapter 6): Giannis Moutsinas
6. "The Lagrange top" (from chapter 6): Leonie van den Berge
7. "Extremal principles" (from chapter 9): Panagiotis Sarridis
8. You may have your own suggestion: please discuss with us

Literature

R. Abraham and J.E. Marsden

Foundations of Mechanics (2nd ed.)

Benjamin (1978)

V.I. Arnol'd

Mathematical Methods of Classical Mechanics (2nd ed.)

GTM 60, Springer (1989)

V.I. Arnol'd, V.V. Kozlov and A.I. Neishtadt

Mathematical Aspects of Classical and Celestial Mechanics

in Dynamical Systems III

Springer (1988)

R.H. Cushman and L.M. Bates

Global Aspects of Classical Integrable Systems

Birkhäuser (1997)

V. Guillemin and S. Sternberg

Symplectic techniques in physics

Cambridge University Press (1984)

J.E. Marsden

Lectures on mechanics

LMS Lecture Notes Series 174, Cambridge University Press (1992)

J.E. Marsden and T.S. Ratiu

Introduction to Mechanics and Symmetry

Springer (1994)

K.R. Meyer and G.R. Hall

Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem

Applied Mathematical Sciences 90, Springer (1992)

J. Montaldi and T. Ratiu

Geometric Mechanics and Symmetry: the Peyresq Lectures

LMS Lecture Notes Series 306, Cambridge University Press (2005)

Contents

Tuesday 3 february. Introduction and overview, systems with 1 degree of freedom, harmonic oscillator, mathematical pendulum. Systems with 2 degrees of freedom, resonant oscillators, central force field. Exercises (.pdf, .ps).

Tuesday 10 february. Variational principle, Euler-Lagrange equations, Mathematical pendulum - Lagrangean viewpoint. Exercises (.pdf, .ps).

Tuesday 17 february. 2-body-problem, effective potential, Legendre transformation. Exercises (.pdf, .ps).

Tuesday 24 february. Conservation of phase space volume, theorem of Liouville, Poincaré Recurrence Theorem, coupled oscillators. Exercises (.pdf, .ps).

Tuesday 3 march. Lagrangean mechanics on manifolds, tangent bundle. Noether's theorem. Exercises (.pdf, .ps).

Tuesday 10 march. Natural mechanical systems, principle of d'Alambert, Lagrange multiplier. Exterior algebra. Small oscillations. Exercises (.pdf, .ps).

Tuesday 17 march. Exterior product, differential form, exterior derivative, Stokes theorem, canonical 1-form, symplectic form, symplectic manifold, Hamiltonian vector field, Poisson bracket. Exercises (.pdf, .ps).

Tuesday 24 march. Noether's theorem revisited, Lie derivative, Hamiltonian flow preserves symplectic structure (i.e. is canonical), theorem of Liouville, locally Hamiltonian vector fields, Poincaré mappings are volume-preserving. Exercises (.pdf, .ps).

Tuesday 31 march. Linear symplectic mappings, generating functions, the Hamilton-Jacobi method. Exercises (.pdf, .ps).

Tuesday 7 april. Action angle variables, Liouville's theorem. Exercises (.pdf, .ps).

Tuesday 21 april. The Lagrange top, systems with symmetries, reduction of circle symmetries. Exercises (.pdf, .ps).

Tuesday 28 april. Inertial frames / Coriolis force, previous exercises revisited.

Tuesday 12 may. Averaging, adiabatic invariants. Exercises (.pdf, .ps).

Tuesday 19 may. Birkhoff normal form, Lie series, the free rigid body. Exercises (.pdf, .ps).

Tuesday 26 may. The 1:2 Resonance. Exercises (.pdf, .ps).

Tuesday 2 june. Resonant normal form, Lie triangle, extremal principles.

Tuesday 9 june. Parametric resonance.