WISM439 : Geometric Mechanics

Heinz Hanßmann, Arthemy Kiselev, Bob Rink and Holger Waalkens

spring	time	place
lectures	monday 14:00 - 16:45	BBL 272

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 and allow for generalization towards infinite dimensions.

Description

Lagrangian and 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 geodesic flow or normal form approximations of non-integrable systems. The course ends with a bi-Hamiltonian (algebraic) approach to infinite-dimensional systems.

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, which becomes particularly important in the infinite-dimensional case. 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),
Introductory Dynamical Systems.

Examination

A combination of presentation of (shorter) weekly exercises and worked out (longer) home work excercises.

Literature

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

Mathematical Aspects of Classical and Celestial Mechanics

in Dynamical Systems III

Springer (1988)

D. Bambusi

Normal forms and semi-classical approximation

Universitá di Milano (2005)

A.V. Bocharov, V.N. Chetverikov, S.V. Duzhin, N.G. Khor'kova, I.S. Krasil'shchik, A.V. Samokhin, Yu.N. Torkhov, A.M. Verbovetsky and A.M. Vinogradov

Symmetries and conservation laws for differential equations of mathematical physics

Translations of Mathematical Monographs 182, AMS (1999)

R.H. Cushman and L.M. Bates

Global Aspects of Classical Integrable Systems

Birkhäuser (1997)

T. Kappeler and J. Pöschel

KdV and KAM

Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge) 45, Springer (2003)

A.V. Kiselev

Methods of the geometry of differential equations in the analysis of integrable models of field theory

J. Math. Sci. 136 (2006) 4295-4377

A.V. Kiselev and J.W. van de Leur

Involutive distributions of operator-valued evolutionary vector fields

Preprint arXiv:math-ph/0703082 (70 pages) sections 2.1 and 2.2

J. Krasil'shchik and A. Verbovetsky

Homological methods in equations of mathematical physics

Lectures given in August 1998 at the International Summer School in Levoca, Slovakia

Preprint arXiv:math/9808130 (150 pages)

J.E. Marsden and T.S. Ratiu

Introduction to Mechanics and Symmetry

Springer (1994)

T. Miwa, M. Jimbo and E. Date

Solitons. Differential equations, symmetries and infinite-dimensional algebras

Cambridge Tracts in Mathematics 135, Cambridge University Press (2000)

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)

P.J. Olver

Applications of Lie groups to differential equations

Graduate Texts in Mathematics 107, Springer (1993)

P.H. Richter, H.R. Dullin, H. Waalkens and J. Wiersig

Spherical Pendulum, Actions, and Spin

J. Phys. Chem. 100 (1996) 19124-19135

B. Rink

Lecture Notes on Geometric Mechanics

Vrije Universiteit Amsterdam (2009)

Contents

Monday 2 february. Mechanical systems. Lagrangean mechanics, Lagrange's theorem 2.1. Homework: exercises 1.2, 1.4 and 1.5.

Monday 9 february. Euler-Lagrange equations, natural mechanical systems, Lagrangean equations for continua. Homework: exercises 2.1, 2.2, and 2.4.

Monday 16 february. Hamiltonian systems, the geodesic flow. Exercise 3.2 to be handed in on 2 march.

Monday 23 february. Canonical transformations, Poisson bracket, Poisson mapping. Exercises (.pdf, .ps).

Monday 2 march. Coupled harmonic oscillators, normal form theory, non-resonant frequencies. Exercises (.pdf, .ps).

Monday 9 march. Resonant normal form, semiclassical normal form. Exercise (.pdf, .ps) to be handed in on 23 march.

Monday 16 march. The nonlinear wave equation (a fairy tale). Euler-Lagrange equations, Hamiltonian on space of Fourier coefficients, Poisson bracket, desired smoothness properties for normal form procedure, formal computation in non-resonant case, adaptions for fully resonant case.

Monday 23 march. Mechanics on Lie groups. Exercise 4.4 to be handed in on 6 april.

Monday 30 march. Action angle variables, pendulae, statement of the Liouville-Arnol'd theorem, examples, necessity of conditions, symplectic form, canonical formalism.

Monday 6 april. Poincaré-Cartan invariant, symplectic transformations. Exercises (.pdf, .ps) to be handed in on 20 april.

Monday 20 april. Principle of least action, generating functions, Hamilton-Jacobi method. Exercises (.pdf, .ps).

Monday 27 april. Proof of the Liouville-Arnol'd theorem, spherical pendulum. Exercise (.pdf, .ps) to be handed in on 11 may.

Monday 4 may. Noether symmetries of Euler-Lagrange PDE, invariant solutions.

Monday 11 may. Conservation laws and the Noether theorem. Exercises (.pdf, .ps) to be handed in on 25 may.

Monday 18 may. Hamiltonian differential operators, Poisson brackets.

Monday 25 may. Integrability of nonlinear evolutionary systems: Magri scheme and Lax representations. Direct and inverse scattering problems.