From Newton to Poincaré - ELTE
46
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
Transcript of From Newton to Poincaré - ELTE
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”)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 4/10
Vibration Vibration of a string before Newton
Pythagoras (c. 550 BC)
f ∼
1
L
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 4/10
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
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 5/10
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)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 5/10
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
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
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
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
Vibration Vibration of a plate
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
chladni.mp4
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)
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
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
Vibration Vibration of a plate
Sophie Germain (1816)
Claude-Louis Navier (1820)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
Vibration Vibration of a plate
Sophie Germain (1816)
Claude-Louis Navier (1820)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
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
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
Heat transfer Joseph Fourier (1768–1830)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
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)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
Heat transfer Joseph Fourier as egyptologist & prefect
secretary of Cairo Institute (from 1798)
governor of Lower Egypt
monograph on Egypt (1810)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
Heat transfer Théorie analytique de la chaleur
Fourier’s great mathematical poem (Lord Kelvin)
Bible of the mathematical physicists (Sommerfeld)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
Heat transfer Théorie analytique de la chaleur
heat equation
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
Heat transfer Théorie analytique de la chaleur
heat equation
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
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
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 9/10
The End
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 10/10
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
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”)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 4/10
Vibration Vibration of a string before Newton
Pythagoras (c. 550 BC)
f ∼
1
L
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 4/10
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
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 5/10
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)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 5/10
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
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
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
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
Vibration Vibration of a plate
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
chladni.mp4
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)
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
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
Vibration Vibration of a plate
Sophie Germain (1816)
Claude-Louis Navier (1820)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
Vibration Vibration of a plate
Sophie Germain (1816)
Claude-Louis Navier (1820)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 6/10
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
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
Heat transfer Joseph Fourier (1768–1830)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
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)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
Heat transfer Joseph Fourier as egyptologist & prefect
secretary of Cairo Institute (from 1798)
governor of Lower Egypt
monograph on Egypt (1810)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
Heat transfer Théorie analytique de la chaleur
Fourier’s great mathematical poem (Lord Kelvin)
Bible of the mathematical physicists (Sommerfeld)
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
Heat transfer Théorie analytique de la chaleur
heat equation
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 7/10
Heat transfer Théorie analytique de la chaleur
heat equation
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
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
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 9/10
The End
Conference Simon Laszlo 70, Budapest, May 20, 2010 – p. 10/10
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
