Dynamical Systems (Hamiltonian Mechanics)

Henk Broer, Heinz Hanßmann

spring	time	place
lectures	monday 10:15 - 13:00	BBL 106

ECTS : 8 credit points

Course material: V.I. Arnold, 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. "Parametric resonance" (section 25): Martin Potocny, 14 may
2. "The Lagrange top" (section 30): Jeroen van der Holst, 14 may
3. "The Principle of Least Action", "The Principle of Maupertuis" or "Huygens's Principle" (sections 45, 46): Frits Veerman, 14 may
4. "Fluid dynamics" (appendix 2): Carl Shneider, 21 may
5. "Symmetry reduction" (appendix 5): Charlotte Vlek, 21 may
6. "Poisson structures" (appendix 14): Sweitse van Leeuwen, 21 may
7. "The Korteweg-de Vries equation" (appendix 13): Ton van Boxtel, 30 may
8. "The planar restricted three body problem": Arnold Jansen, 30 may
9. "Birkhoff normal form theory" (appendix 7)
10. 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. Arnold

Mathematical Methods of Classical Mechanics (2nd ed.)

GTM 60, Springer (1989)

V.I. Arnold, 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

Monday 5 february. Introduction and overview, Newton's 2nd law, harmonic oscillator, systems with 1 degree of freedom, mathematical pendulum, central force field. Exercises (.pdf, .ps).

Monday 12 february. Anharmonic oscillator, Hamiltonian pitchfork bifurcation, centre-saddle bifurcation, variational principle, Lagrangian, Euler-Lagrange equations, Legendre transformation. Exercises (.pdf, .ps).

Monday 19 february. Mathematical pendulum - Lagrangian viewpoint, 2-body-problem, effective potential, Laplace vector. Exercises (.pdf, .ps).

Monday 26 february. Conservation of phase space volume, theorem of Liouville, Poincaré Recurrence Theorem, Lagrangian mechanics on manifolds, principle of d'Alambert, Lagrange multiplier, tangent bundle. Exercises (.pdf, .ps).

Monday 5 march. Exterior algebra, differential form, exterior derivative, Stokes theorem, canonical 1-form, symplectic form, symplectic manifold, Hamiltonian vector field, examples, Hamiltonian flow preserves symplectic structure (i.e. is canonical), theorem of Liouville. Exercises (.pdf, .ps).

Monday 12 march. Mathematical pendulum - Hamiltonian viewpoint, small oscillations, 2 coupled pendulums, absolute and relative integral invariants, time dependent Hamiltonian vector fields (in 3 dimensions and in 2n+1 dimensions), integral invariant of Poincaré-Cartan. Exercises (.pdf, .ps).

Monday 19 march. The rigid body, rotation group SO(3) and its (co)tangent bundle, tensor of inertia, momentum mapping, symmetry reduction. Exercises (.pdf, .ps).

Monday 26 march. Canonical equations remain canonical under canonical transformations, principle of Maupertuis, action angle variables in one degree of freedom. Exercises (.pdf, .ps).

Monday 2 april. Energy-momentum mapping and action-angle variables of the free rigid body, invariant tori in the geodesic flow on a surface of revolution, action angle variables for integrable systems, the spherical pendulum. Exercises (.pdf, .ps).

Monday 16 april. Fundamental problem of classical mechanics, Diophantine conditions. Exercises (.pdf, .ps).

Monday 7 may. KAM theory and its consequences, averaging, adiabatic invariants. Exercises (.pdf, .ps).

Monday 14 may. Parametric resonance, the Lagrange top, extremal principles.

Monday 21 may. Symmetry reduction, fluid dynamics, Poisson structures.

Wednesday 30 may. The planar restricted three body problem, the Korteweg-de Vries equation.