WISM439 : Geometric Mechanics

Heinz Hanßmann

spring	time	place
lectures	thursday 15:15 - 17:00	MIN 0.11

ECTS : 7.5 credit points

In this course we study integrable mechanical systems from a geometric point of view, using concepts and techniques that yield insight on small perturbations away from integrability.

Description

Hamiltonian systems naturally occur in frictionless mechanics, but are also used to answer questions that arise in optics or when studying certain partial differential equations. After a short introduction we present the necessary theory motivated by examples like the N-body problem or the geodesic flow.

A central result is the existence of so-called action angle variables of an integrable Hamiltonian system; the action variables are conserved quantities in involution. Expressing a given system in action angle variables not only simplifies the study of the dynamical properties of the system itself, but also makes perturbations of the system accessible to both quantitative and qualitative investigations.

Assumed knowledge

Ordinary Differential Equations
Manifolds (a bit: `tangent space')
Introductory Dynamical Systems

Examination

A combination of worked out home work exercises and a final exam.

Literature

R. Abraham and J.E. Marsden

Foundations of Mechanics (2nd ed.)

Benjamin (1978)

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)

K. Efstathiou

Metamorphoses of Hamiltonian systems with symmetries

LNM 1864, Springer (2005)

G. Gallavotti

The elements of mechanics

Springer (1983)

V. Guillemin and S. Sternberg

Symplectic techniques in physics

Cambridge University Press (1984)

P. Liberman and C.-M. Marle

Symplectic geometry and analytical mechanics

D. Reidel (1987)

A.J. Lichtenberg and M.A. Lieberman

Regular and stochastic motion/chaotic dynamics

Springer (1983/1992)

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)

W. Thirring

A course in mathematical physics

Vol.1. Classical dynamical systems

Springer (1978)

Contents

For dates in the future: what is planned; those days where the date still has to be corrected are included to give an impression of what is possible.

Thursday 5 february. Introduction, N-body problem, tangent bundle. Exercises (pdf, ps).

Thursday 12 february. Co-tangent bundle, symmetry group, Lie bracket. Exercises (pdf, ps).

Thursday 19 february. Lie groups, rigid body, exterior algebra. Exercises (pdf, ps).

Thursday 26 february. Differential forms, Stokes theorem, symplectic forms and manifolds. Exercises (pdf, ps).

Thursday 5 march. Theorem of Darboux, symplectic and Poisson structures in local co-ordinates, theorem of Liouville. Exercises (pdf, ps).

Thursday 12 march. Regular symmetry reduction, reduced system, momentum mapping. Exercises (pdf, ps).

Thursday 19 march. Reduced dynamics, geodesic flow, free rigid body. Exercises (pdf, ps).

Thursday 26 march. Integrable systems, invariant tori, action angle variables. Exercises (pdf, ps).

Thursday 2 april. No lecture and no exercises.

Thursday 9 april. Generalized action angle variables, Lagrange top, Andoyer variables. Exercises (pdf, ps).

Thursday 16 april. Non-resonant torus, non-degenerate frequency mapping, perturbed system. Exercises (pdf, ps).

Thursday 23 april. Diophantine conditions, KAM theory, perturbed quasi-periodicity.

Thursday 30 april. Long time stability, elliptic equilibria, normal forms.

Thursday 7 may. Normal form algorithm, perturbation analysis, Lie-Deprit triangle.

Thursday 21 may. Spectrum, fully resonant oscillators, high order resonances.

Thursday 28 may. Low order resonances, reduction of the 1:2 resonance, singular points and their isotropy groups.

Thursday 4 june. Dynamics of the 1:2 resonance, bifurcation analysis in one degree of freedom, period doubling.

Thursday 11 june. Genuine 1st order resonances, non-integrability, the 1:2:4 resonance.

Thursday 18 june. Reduced phase space in two degrees of freedom, dynamics on the singular part, equilibria on the regular part.

Thursday 25 july. Presentations