WISM439 : Geometric Mechanics

Heinz Hanßmann




spring time place
lectures thursday 11:00 - 13:00 BBG 005

ECTS : 7.5 credit points




The lectures take place in BBG 005, except for the first lecture in MIN 207 and the second lecture in BBG 061.




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

Lagrangean 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.

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: `tangent space')
Introductory Dynamical Systems




Examination

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




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)

T. Kappeler and J. Pöschel
KdV and KAM
Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge) 45, Springer (2003)

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)

Jesús Palacián, Flora Sayas and Patricia Yanguas
Regular and Singular Reductions in the Spatial Three-Body Problem
Qualitative Theory of Dynamical Systems 12 (2013) 143-182

Jesús Palacián, Flora Sayas and Patricia Yanguas
Flow reconstruction and invariant tori in the spatial three-body problem
Journal of Differential Equations 258 (2015) 2114-2159

W. Thirring
A course in mathematical physics
Vol.1. Classical dynamical systems
Springer (1978)




Contents

Thursday 5 february. Introduction, N-body problem.

Thursday 12 february. Co-tangent bundle, differential forms. Two exercises (pdf, ps) for presentation next thursday.

Thursday 19 february. Symplectic forms and manifolds, Lie bracket. Two exercises (pdf, ps) for presentation next thursday.

Thursday 26 february. Lie groups, momentum mapping. Two exercises (pdf, ps) for presentation next thursday.

Thursday 5 march. Regular symmetry reduction, reduced system. Two exercises (pdf, ps) for presentation next thursday.

Thursday 12 march. Geodesic flow, free rigid body. First homework exercise (pdf, ps), to be handed in april 2.

Thursday 19 march. Extended exercise session, dynamics of the free rigid body.

Thursday 2 april. Integrable systems, generalized action angle variables. Exercise (pdf, ps) for presentation next thursday.

Thursday 9 april. Andoyer variables, invariant tori. Exercise (pdf, ps) for presentation next thursday.

Thursday 16 april. KdV-equation, elliptic equilibria. Exercise (pdf, ps) for presentation next thursday.

Thursday 23 april. Normal forms for equilibria, normal form algorithm. Exercise (pdf, ps) for presentation next thursday.

Thursday 30 april. Normal forms for periodic orbits and invariant tori, perturbation analysis. Second homework exercise (pdf, ps), to be handed in may 21.

Thursday 7 may. KAM theory, lunar problem.

Thursday 21 may. Nonlinear wave equation, spatial Kepler system. Exercise (pdf, ps) for presentation.

Thursday 4 june. 3-body problem, normalization, symmetry reduction. Third homework exercise (pdf, ps), to be handed in june 25.

Thursday 11 june. Reduction of the discrete symmetry, singular points and their isotropy groups.

Thursday 25 june. Bifurcation analysis in one degree of freedom, elliptic equilibria.

Thursday 2 july. Invariant tori in the 3-body problem, open problems.