Conservative Dynamical Systems

Heinz Hanßmann, Holger Waalkens


fall time place
lectures tuesday 14:00 - 17:00 MI 611

ECTS : 8 credit points

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


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.


Presentation and home work excercises.

Examples of subjects for presentation

  1. "Noether's theorem": Deborah Cabib and Ralph Langendam
  2. "Parametric resonance": Weronika Siwek
  3. "The free rigid body": Ondrej Budac
  4. "The Lagrange top": Emmanuel Ursella
  5. "Fluid dynamics": Bas van Opheusden
  6. "Poisson structures": Brian van de Camp and Vincent Knibbeler
  7. "The Korteweg-de Vries equation": Stefanie Postma and Lotte Sewalt
  8. "Birkhoff normal form theory": Oscar Heslinga and Thomas de Jong
  9. "The double pendulum": Frank Plomp
  10. "Huygen's priciple and geometric optics": Dirk van Kekem
  11. "Symmetry reduction": Ralph Klaasse
  12. You may have your own suggestion: please discuss with us


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)

P.J. Olver
Chapter 6 of Applications of Lie groups to differential equations
Springer (1986)


Tuesday 7 september. Introduction and overview, systems with 1 degree of freedom, harmonic oscillator, systems with 2 degrees of freedom, energy surface, accessible region in configuration space. Exercises (.pdf, .ps).

Tuesday 14 september. Central force field, effective potential, 2-body-problem, variational principle, Euler-Lagrange equations. Exercises (.pdf, .ps).

Tuesday 21 september. Mathematical pendulum - Lagrangean viewpoint, Legendre transformation, Hamiltonian equations, mathematical pendulum - Hamiltonian viewpoint, conservation of phase space volume, theorem of Liouville, Poincaré Recurrence Theorem. Exercises (.pdf, .ps).

Tuesday 28 september. Small oscillations, coupled harmonic oscillators, Lagrangean mechanics on manifolds, tangent bundle, Noether's theorem. Exercises (.pdf, .ps).

Tuesday 5 october. Natural mechanical systems, principle of d'Alambert, Lagrange multiplier, parametric resonance, exterior algebra. Exercises (.pdf, .ps).

Tuesday 12 october. Exterior product, differential form, path integral, exterior derivative, Stokes theorem, free rigid body, symplectic manifold, symplectic form, canonical 1-form, Hamiltonian vector field. Exercises (.pdf, .ps).

Tuesday 19 october. Hamiltonian flow preserves the Hamiltonian, Poisson bracket, Noether's theorem revisited, Lagrange top, Lie bracket, Lie derivative. Exercises (.pdf, .ps).

Tuesday 26 october. Cartan's magic formula, Hamiltonian flow preserves the symplectic structure (is canonical), theorem of Liouville, Poincaré mappings preserve volume, fluid dynamics, linear symplectic structures, symplectic geometry. Exercises (.pdf, .ps).

Tuesday 2 november. Linear symplectic mappings, spectrum, Darboux's theorem, Hamiltonian systems on S^2, Euler's equations. Exercises (.pdf, .ps).

Tuesday 9 november. Integral invariants, geometric description based on vortex lines, principle of least action. Exercises (.pdf, .ps).

Tuesday 16 november. Generating functions, the Hamilton-Jacobi method, Poisson structures, action angle variables, integrable systems, Liouville's theorem. Exercises (.pdf, .ps).

Tuesday 23 november. Proof of Liouville's theorem, the Korteweg-de Vries equation, averaging. Exercises (.pdf, .ps).

Tuesday 30 november. Adiabatic invariants, Birkhoff normal form theory. Exercises (.pdf, .ps).

Tuesday 7 december. Fundamental problem of classical dynamics, Diophantine conditions, perturbation theory, the double pendulum, the geometry of reaction dynamics, transition state theory in two degrees of freedom. Exercises (.pdf, .ps).

Tuesday 14 december. Transition state theory in more degrees of freedom, Huygen's priciple and geometric optics, systems with symmetries.