From Newton to Poincaré: two centuries of PDEs ´ Ad ´ am Besenyei [email protected]Dept. of Applied Analysis, E ¨ otv ¨ os Lor ´ and University, Budapest Conference Simon L ´ aszl ´ o 70, Budapest, May 20, 2010 – p. 1/10
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Adam Besenyei
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 1/10
Outline two centuries
two physical phenomena
and some fun
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 2/10
PDEs are part of physics
Mathematics is a part of physics. Physics is an experimental
science, a part of natural science. Mathematics is the part of
physics where experiments are cheap. (Arnold)
Mathematics is an experimental science, and definitions do not come
first, but later on. (Heaviside)
Geometry is a part of physics. (Hilbert)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 3/10
PDEs are part of physics
classical mechanics Newton, Euler, Lagrange, Hamilton, Jacobi,
Liouville, Darboux, Noether Newton’s laws of motions calculus of
variations symplectic topology
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 3/10
PDEs are part of physics
classical mechanics
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 3/10
PDEs are part of physics
classical mechanics
celestial mechanics
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 3/10
PDEs are part of physics
classical mechanics
celestial mechanics
electricity, magnetism
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 3/10
PDEs are part of physics
classical mechanics
celestial mechanics
electricity, magnetism
vibration D’Alembert, Euler, D. Bernoulli, Lagrange, Chladni,
Germain, Navier, Poisson, Kirchhoff
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 3/10
PDEs are part of physics
classical mechanics
celestial mechanics
electricity, magnetism
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 3/10
PDEs are part of physics
classical mechanics
celestial mechanics
electricity, magnetism
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 3/10
Vibration Vibration of a string before Newton
Pythagoras (c. 550 BC)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 4/10
Vibration Vibration of a string before Newton
Pythagoras (c. 550 BC) theory of music relations between musical
notes (length, tension, mass) simple ratios between harmonics
general principle in celestial mechanics (“music of spheres”)
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Vibration Vibration of a string before Newton
Pythagoras (c. 550 BC)
f ∼
1
L
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Vibration Vibration of a string after Newton
Brook Taylor (1715) Methodus Incrementorum Directa et Inversa used
mechanical principles acceleration (∂2
t u) proportional to curvature
(∼ ∂2
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 5/10
Vibration Vibration of a string after Newton
Brook Taylor (1715)
Jean le Rond D’Alembert (1747) Recherches sur la courbe que forme
une corde tenduë mise en vibration D’Alembert’s solution
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Vibration Vibration of a string after Newton
Brook Taylor (1715)
Leonhard Euler (1749) same formula
special solutions
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 5/10
Vibration Vibration of a string after Newton
Brook Taylor (1715)
Leonhard Euler (1749)
Daniel Bernoulli (1753) Réflexions et éclaircissements sur les
nouvelles vibrations des cordes all functions are representable by
the series of Euler debate between Euler and Bernoulli
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 5/10
Vibration Vibration of a string after Newton
Brook Taylor (1715)
Leonhard Euler (1749)
Daniel Bernoulli (1753)
Joseph-Louis Lagrange (1759) Recherches sur la nature et la
propagation du son limiting case of n masses
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 5/10
Vibration Vibration of a string after Newton
Brook Taylor (1715)
Leonhard Euler (1749)
Daniel Bernoulli (1753)
Joseph-Louis Lagrange (1759)
TO BE CONTINUED...(Fourier)
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Vibration Vibration of a plate
Leonhard Euler (1767) De motu vibratorio tympanorum
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
Vibration Vibration of a plate
Leonhard Euler (1767)
Jacob Bernoulli (1789) Essai théorethique sur les vibrations des
plaques élastiques, rectangles et libres
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Vibration Vibration of a plate
Leonhard Euler (1767)
Jacob Bernoulli (1789)
Ernst Chladni (1787) [Robert Hooke (1680)] metal plate excited by a
violin bow sand forms nodal patterns
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Vibration Vibration of a plate
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chladni.mp4
Chladni’s book on acoustics (1802)
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Vibration Vibration of a plate
Chladni’s book on acoustics (1802)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
Vibration Vibration of a plate
Chladni’s book on acoustics (1802)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
Vibration Vibration of a plate
Chladni’s book on acoustics (1802)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
Vibration Vibration of a plate
Chladni’s book on acoustics (1802)
Chladni meets Napoléon (1809)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
Vibration Vibration of a plate
Chladni’s book on acoustics (1802)
Chladni meets Napoléon (1809) french edition of Chladni’s book
Academy Prize 3000 francs Sophie Germain and Poisson begin to work
submissions refused by Lagrange, Poisson third attempt (1816) prix
extraordinaire later essays on elasticity at own expense
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Vibration Vibration of a plate
Sophie Germain (1816)
Claude-Louis Navier (1820)
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Vibration Vibration of a plate
Sophie Germain (1816)
Claude-Louis Navier (1820)
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Heat transfer Isaac Newton
anonymous paper
Newton’s law of cooling: rate of temperature change is proportional
to the temperature difference between the object and its
surroundings
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Heat transfer Joseph Fourier (1768–1830)
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Heat transfer Joseph Fourier as a scientist
works in Grenoble (from 1804)
Mémoire sur la propagation de la chaleur dans les corps solides
(1807)
Academy Prize (1811)
greenhouse effect, solar radiation (1824)
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Heat transfer Joseph Fourier as egyptologist & prefect
secretary of Cairo Institute (from 1798)
governor of Lower Egypt
monograph on Egypt (1810)
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Heat transfer Théorie analytique de la chaleur
Fourier’s great mathematical poem (Lord Kelvin)
Bible of the mathematical physicists (Sommerfeld)
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Heat transfer Théorie analytique de la chaleur
heat equation
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Heat transfer Théorie analytique de la chaleur
heat equation
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Heat transfer Théorie analytique de la chaleur
heat equation
Fourier series
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
Heat transfer Théorie analytique de la chaleur
heat equation
Fourier series
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
Heat transfer Théorie analytique de la chaleur
heat equation
Fourier series
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
Listen to Fourier series Wave: Amplitude spectrum:
400Hz sine wave
400Hz square wave Conference Simon Laszlo 70, Budapest, May 20,
2010 – p. 8/10
sine.wav
400Hz sawtooth wave
400Hz triangle wave Conference Simon Laszlo 70, Budapest, May 20,
2010 – p. 8/10
saw.wav
400Hz oboe tone
400Hz clarinet tone Conference Simon Laszlo 70, Budapest, May 20,
2010 – p. 8/10
oboe.wav
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The End
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Outline
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Vibration
Happy Birthday!
The End