Henk Broer > DSMP > MATHS > JBI > FWN > RUG


Seniorenacademie Groningen en Drenthe (HOVO)

Romeins dodecaeder gevonden bij Hartwerd (Fr.)

Cursus 17WG32 (voorjaar 2017):

Nieuwe ontwikkelingen in de wiskunde (2): De wetenschappelijke revolutie van de 17e eeuw

coordinatie Drs M.C. van Hoorn en Prof dr H.W. Broer. Vanaf de 17e eeuw vond een wetenschappelijke revolutie plaats, waarbij op vrijwel hetzelfde moment nieuwe methoden in de wiskunde, mechanica en optica opkwamen. De centrale figuur hierbij is Newton, maar ook Fermat, Huygens, Leibniz en de Bernoulli's zijn belangrijk. Later komen Euler, Lagrange, Gauss, Hamilton, Riemann en vele anderen. In het laatste college maken we de stap naar het heden voor de behandeling van een actueel thema uit de dagelijkse (les-) praktijk.

Hieronder volgen handouts van de achtereenvolgende colleges zoals die van 7 Maart tot en met 11 April 2015 wekelijks op dinsdagen tussen 10:15 en 12:00 uur zullen worden gegeven in het Logegebouw "l'Union Provinciale", Turfsingel 46, 9712 KR, Groningen (op de hoek met de W.A. Schoftenstraat). Voor meer informatie klik hier.

In de colleges gaat het om hoofdlijnen, maar telkens zullen verdiepingsmogelijkheden worden aangereikt. Het zal ook zeker gaan om het plezier dat zovelen in wiskunde hebben.


Wetenschappelijke revolutie als basis van de moderne wetenschap, colleges 1 en 5 door Prof dr H.W. Broer:

De ruimte in de loop van de tijd, college 2 door Prof dr G. Vegter:

Meer dan reele getallen, college 3 door Prof dr J. Top:

Klassieke meetkunde van de oude Grieken tot heden, colleges 4 en 6 door Drs M.C. van Hoorn:

Hieronder volgen de drie onderwijstaken waarbij ik betrokken ben. Voor dag, uur en plaats, zie de FWN roosters.

Hamiltonian Mechanics (master course)

Eise Eisinga's ceiling (Franeker, the Netherlands)


Aim is to develop mathematical aspects of classical mechanics via Newtonian and Lagrangian systems to the world of Hamiltonian systems, which most naturally live on symplectic manifolds. The entire theory, including the benefits of the symplectic formalism will be illustrated with many examples, eventually touching on current research.

Lectures and instruction

- Introduction, one and two degrees of freedom; - The central force field, Keplers second law; - The variational principle, Euler-Lagrange; - The Legendre transformation to Hamilton-Jacobi, Liouville and Noether; - Poincare recurrence, symmetry; Bottema - Applications to small oscillations; Bottema again - The symplectic formalism; - Time dependent systems; - Integral invariants; - Applications to mechanics and optics; - Averaging methods and adiabatic invariants; - Miscellaneous applications, perturbation theory.

Assumed knowledge

Ordinary Differential Equations and some Advanced Calculus, Mechanics useful but not required.

Capita selecta Hamiltonian Mechanics

Exercises, homework and exam

Please check with Exercises Hamiltonian Mechanics and Dynamical Systems

- Homework I: from the exercises 1.4, 2.1, 3.1 and 6.1,

  to be handed in on Friday May 20, noon 
  as a hard copy in my mailbox 
  (near the secretaries at the 4th floor of the BB) 

- Homework II: from the exercises 2.2, 6.2, 7.5 and 9.4,

  to be handed in on Friday June 10, noon
  as a hard copy in my mailbox

 !!! Take-Home Exam 2014

 !!! Take-Home Exam 2015

 !!! Take-Home Exam 2016

 !!! Take-Home Exam 2017

Written take-home exam in the form of a quiz: to be handed in with Henk Broer (email: broerhw@gmail.com). By handing in the exam the student declares that he / she has done all the work by him- / herself.

Optional is an essay that replaces the exam: a solid piece of work between 10 and 20 pages, clearly expressing one or more mathematical ideas. The subject has to be discussed with Henk Broer first.

Homework counts for rounding off purposes.


V.I. Arnold (1989) Mathematical Methods of Classical Mechanics. Springer, 1989.   bib

Lectures (schedule of MASTERMATH video course 2013)

  • Lecture 1, Newtonian Mechanics: 1a 1b 1c
  • Lecture 2, Newtonian Mechanics: 2a 2b 2c
  • Lecture 3, Lagrangian Mechanics: 3a 3b
  • Lecture 4, Lagrangian Mechanics: 4a 4b 4c
  • Lecture 5, Lagrangian Mechanics (Oscillations): 5a 5b 5c
  • Lecture 6, Hamiltonian Mechanics: 6a 6b 6c
  • Lecture 7, Hamiltonian Mechanics: 7a 7b 7c
  • Lecture 8, Hamiltonian Mechanics (Poincare-Cartan): 8a 8b
  • Lecture 9, Hamiltonian Mechanics (Darboux): 9a 9b 9c
  • Lecture 10, Hamiltonian Mechanics (Adiabatic Invariants): 10a 10b
  • Lecture 11, Hamiltonian Mechanics (Resonance): 11a 11b 11c
  • Lecture 12, Hamiltonian Mechanics (Kolmogorov-Arnold-Moser): 12a 12b
  • Lecture 13, Hamiltonian Mechanics (Optics): 13a 13b
  • Lecture 14, Hamiltonian Mechanics (Liouville-Arnold-Duistermaat): 14a 14b

Chaos Theory (bachelor course)


This Minor course (optional) gives an overview of Chaos Theory as this is a part of the mathematical discipline of Nonlinaer Dynamical Systems. The discipline is concerned with everything that moves. Think of mathematical descriptions of the motion of mechanical, optical or electronic devices, like the solar system, pendula and swings, LRC networks, etc. Also it is very useful in meteorology, economics, biology, but even in developmental psychology. Sometimes it is appropriate to work with a discrete time set (think of the population dynamics of one day flies). In that case the dynamics is given by iteration of mappings. Also many dynamical systems are governed by ordinary differential equations. In the course we deal with subjects like (un) predictability and with fractal sets, illustrated by examples. We also will discuss historical elements. It is remarkable that a large initiative to Chaos Theory came from the biologist Robert May, the meteorologist Edward Lorenz, the astronomer Michel Hénon and from the physicist Mitchell Feigenbaum. Apart from these also mathematicians like Henri Poincaré, Steven Smale, René Thom and many others played a role.

Presentations by me

. Determinism, Chaos and Chance. Henk Broer, 2011.   url
. On Poincare’s legacy in dynamical systems. Henk Broer, 2012.   url
. Determinisme, Chaos en Toeval. Henk Broer, 2011.   url
. Kepler’s Derde Wet en de Stabiliteit van het Zonnestelsel. Henk Broer, 2011.   url
. Dimensie en Dispersie, het ‘meten’ van chaos. Henk Broer, 2011.   url
. Huygens and Bernoulli’s brachistochrone. Henk Broer, 2011.   url
. Resonance and Fractal Geometry. Henk Broer, 2011.   url
. Multi-periodic dynamics: overview and some recent results. Henk Broer, 2011.   url


Written presentation in the form of an essay (between 10 and 20 pages)

First assignments to students

   1. Send an email to broerhw@gmail.com with your nam, S-number and discipline of study
   2. Look for 2 to 3 fellow students to write an essay together

Possible essay subjects, after discussion with me:

  • The chaotic pendulum
  • The chaotic magnetic pendulum
  • Julia and Mandelbrot sets
  • Kepler and Newton, circles and ellipses
  • Poincare and the three body problem
  • Multiperiodicity and stability
  • Moreover you can choose subjects from the literature as indicated below,

both examples and theoretical excusrsions


H.W. Broer and F. Takens (2011) Dynamical Systems and Chaos. Applied Mathematical Sciences 172, Springer, 2011.   url
M.W. Hirsch, S. Smale, and R.L. Devaney (2013) Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, 2013.   bib
J. Barrow-Green (1997) Poincare and the Three Body Problem. American Mathematical Society - London Mathematical Society, 1997.   bib
H.W. Broer. De chaotische schommel. Pythagoras, 35(5):11–15, 1997.   bib
H.W. Broer, J. van de Craats, and F. Verhulst (2003) Het einde van de voorspelbaarheid? Chaostheorie, ideeën en toepassingen. Aramith Uitgevers – Epsilon Uitgaven 35, 2003.   bib
R.L. Devaney (2003) An Introduction to Chaotic Dynamical Systems, 2nd Ed.. Westview Press, 2003.   bib
F. Diacu and P. Holmes (1996) Celestial Encounters. Princeton University Press, 1996.   bib
E. Lorenz (1993) The Essence of Chaos. University of Washington Press, 1993.   bib
B.B. Mandelbrot (1977) The Fractal Geometry of Nature. Freeman, 1977.   bib
H.O. Peitgen, H. Juergens, and D. Saupe (1992) Chaos and Fractals, New Frontiers of Science. Springer, 1992.   bib
D. Ruelle (1991) Chance and Chaos. Princeton University Press, 1991.   bib
I. Stewart (1989) Does God play Dice? Penguin, 1989.   bib

Dynamical Systems and Chaos (master course)

Eise Eisinga's ceiling (Franeker, the Netherlands)


This is an advanced course based on the bachelor courses on nonlinear dynamical systems. The books contain many topics, which will partly be dealt with in an eclectic way. The background of the course is a large phenomenology of periodic, quasi-periodic and chaotic dynamics as this occurs in a wide range of dynamical systems, which often are more or less realistic models. The theory behind this involves the idea of persistence of properties under variations of initial state or of parameters and, among other things, the notion of dispersion exponent. In later chapters and the appendices material is included that is closer related to the Groningen research. One item concerns the persistence of quasi-periodicity under small perturbations and another the reconstruction of dynamics from time series.


Most of the lectures will be held by the teacher and have a course like nature. Other lectures are seminar like talks held by the students, after self-study and coaching by the teacher. The subjects will be chosen (or assigned) from the books or from adjacent material.

Assumed knowledge

Ordinary Differential Equations, Manifolds, Metric Spaces, Introductory Dynamical Systems.


Seminar talks, where each seminar talk will be accompanied by an essay. Talk and essay together will be followed by an oral examination.


H.W. Broer and F. Takens (2011) Dynamical Systems and Chaos. Applied Mathematical Sciences 172, Springer, 2011.   url
R.C. Robinson (1995) Dynamical Systems. CRC Press, 1995.   bib