Teaching
Seniorenacademie Groningen en Drenthe (HOVO)
Romeins dodecaeder gevonden bij Hartwerd (Fr.)
Cursus 23WV-1G24 (voorjaar 2023)
Lange lijnen in de wiskunde
Coördinatie Drs M.C. van Hoorn en Prof dr H.W. Broer.
De hedendaagse wiskunde is zeer uitgebreid. Ook in ’moderne’ wiskunde komt echter vaak ‘oude’ wiskunde voor, die soms al vanaf de Grieken of Arabieren werd ontwikkeld. Bovendien kunnen nieuwe onderwerpen zich ontwikkelen tot terugkerende thema’s. In deze cursus gaat het om zulke terugkerende thema’s, oud of nieuw, die telkens actueel blijken. Zoals ook in eerdere cursussen komen verschillende onderwerpen aan bod. Twee onderwerpen komen uit het raakvlak van wiskunde en mechanica. Een ander onderwerp betreft de meetkunde van het heelal. Ook zijn er voordrachten over statistiek en didactiek.
Alle voordrachten vinden plaats in de kleine zaal van het Logegebouw van de Vrijmetselaars Groningen op de hoek van de Turfsingel en de W.A. Scholtenstraat (ingang W.A. Scholtenstraat), vanaf 10:15 uur
Voordrachten
College 1 door Prof dr H.W. Broer 7 maart 2023
Hemelmechanica voor en na Eise Eisinga
College 2 door Prof dr G. Vegter 14 maart 2021
Klassieke Meetkunde en Hyperbolische Meetkunde
Toegift: Topologie en Meetkunde van het Heelal
College 3 door Prof dr C.J. Albers 21 maart 2023
Coronastatistiek
Statistische Zonden
College 4 door Drs M.C. van Hoorn 28 maart 2023
Paragrafen uit de elementaire meetkunde
College 5 door Prof dr H.W. Broer 4 april 2023
De Foucault-slinger en Kamerlingh Onnes
Achtergrondmateriaal:
Henk Broer, Marcello Seri and Floris Takens, Oscillations: swings and vibrations from a mathematical point of view
College 6 door Prof dr J. Top 11 April 2023
Getaltheorie door de eeuwen heen
College 7 door Drs M.C. van Hoorn 18 April 2023
Rekenvaardigheid in de verdrukking
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Cursus 20WG28 (voorjaar 2020):
Wiskunde in meer dimensies
Vanaf de 17e eeuw vond een wetenschappelijke revolutie plaats, waarbij op vrijwel hetzelfde moment nieuwe methoden in de wiskunde, mechanica en optica opkwamen. Vanaf de Grieken speelde alles zich af in het 2-dimensionale vlak en de 3-dimensionale ruimte af, maar in de tweede helft van de 19e eeuw werden ook hoger dimensies geïntroduceerd, hetgeen een belangrijke rol heeft gespeeld bij de ontwikkeling van de relativiteitstheorie. In deze ontwikkeling heeft de Nederlandse wiskundige P.H. Schoute een belangrijke rol gespeeld.
Hieronder volgen handouts van de achtereenvolgende colleges zoals die van 25 februari tot en met 7 April 2020 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.
Voordrachten
Bernoulli in Groningen (rond 1700)
college 1 door Prof dr H.W. Broer:
- Johann Bernoulli in Groningen: triomf van de calculus
- Bernoulli's lightray solution for the brachistochrone problem from Hamilton's viewpoint
Achtergrondmateriaal:
- Bernoulli’s lichtstraal-oplossing van het brachistochrone probleem door de ogen van Hamilton
- Huygens en de brachistochroon van Bernoulli
Wat niet kan is nog nooit gebeurd
college 2 door Prof dr R.H. Koning:
Stereometrie, klassiek en niet meer klassiek
college 3 door Drs M.C. van Hoorn
Galileïsche dansen bij Jupiter (vanaf 1800 tot nu)
college 4 door Prof dr H.W. Broer
met als inleiding
De meetkunde van het heelal
college 5 door Prof dr G. Vegter:
- Slides
- Videos
Meer dan reele getallen
college 6 door Prof dr J. Top:
Meerdimensionale meetkunde
college 7 door Drs M.C. van Hoorn:
Hamiltonian Mechanics (master course)
Eise Eisinga's ceiling (Franeker, the Netherlands)
Description
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.
Literature
- Arnold, V.I. (1989) Mathematical Methods of Classical Mechanics. Springer. (BibTeX)
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)
Description
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
- (2011) Determinism, Chaos and Chance. Henk Broer. (URL) (BibTeX)
- (2012) On Poincare's legacy in dynamical systems. Henk Broer. (URL) (BibTeX)
- (2011) Determinisme, Chaos en Toeval. Henk Broer. (URL) (BibTeX)
- (2011) Kepler's Derde Wet en de Stabiliteit van het Zonnestelsel. Henk Broer. (URL) (BibTeX)
- (2011) Dimensie en Dispersie, het `meten' van chaos. Henk Broer. (URL) (BibTeX)
- (2011) Huygens and Bernoulli's brachistochrone. Henk Broer. (URL) (BibTeX)
- (2011) Resonance and Fractal Geometry. Henk Broer. (URL) (BibTeX)
- (2011) Multi-periodic dynamics: overview and some recent results. Henk Broer. (URL) (BibTeX)
Examination
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
Literature
- Broer, H.W. and Takens, F. (2011) Dynamical Systems and Chaos. Applied Mathematical Sciences 172, Springer. (URL) (BibTeX)
- Hirsch, M.W., Smale, S. and Devaney, R.L. (2013) Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press. (BibTeX)
- Barrow-Green, J. (1997) Poincare and the Three Body Problem. American Mathematical Society - London Mathematical Society. (BibTeX)
- Broer, H.W. (1997) De chaotische schommel.. . (BibTeX)
- Broer, H.W., van de Craats, J. and Verhulst, F. (2003) Het einde van de voorspelbaarheid? Chaostheorie, ideeën en toepassingen. Aramith Uitgevers -- Epsilon Uitgaven 35. (BibTeX)
- Devaney, R.L. (2003) An Introduction to Chaotic Dynamical Systems, 2nd Ed.. Westview Press. (BibTeX)
- Diacu, F. and Holmes, P. (1996) Celestial Encounters. Princeton University Press. (BibTeX)
- Lorenz, E. (1993) The Essence of Chaos. University of Washington Press. (BibTeX)
- Mandelbrot, B.B. (1977) The Fractal Geometry of Nature. Freeman. (BibTeX)
- Peitgen, H.O., Juergens, H. and Saupe, D. (1992) Chaos and Fractals, New Frontiers of Science. Springer. (BibTeX)
- Ruelle, D. (1991) Chance and Chaos. Princeton University Press. (BibTeX)
- Stewart, I. (1989) Does God play Dice? Penguin. (BibTeX)
Dynamical Systems and Chaos (master course)
Eise Eisinga's ceiling (Franeker, the Netherlands)
Description
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.
Lectures
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.
Examination
Seminar talks, where each seminar talk will be accompanied by an essay. Talk and essay together will be followed by an oral examination.