MMAT5520. Lau Chi Hin The Chinese University of Hong Kong

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1 MMAT550 Lau Chi Hin The Chinese Univesit of Hong Kong

2 Isaac Newton ( )

3 Keple s Laws of planeta motion 1. The obit is an ellipse with the sun at one of the foci.. A line joining a planet and the sun sweeps out equal aeas in equal time. 3. The squaes of the obital peiods ae diectl popotional to the cubes of the semi-majo axes.

4 Keple s Laws of planeta motion Invese squae law Consevation of momentum Diffeential equation 3 T R Equal aea Elliptic obit

5 Keple s Laws of planeta motion Centipetal foce: Assume invese squae law T R T R R m R mv F 1 R F Then 3 1 R T T R R

6 Keple s Laws of planeta motion da dt 1 d dt is constant Angula momentum L m v m is constant

7 Keple s Laws of planeta motion Newton second Law: a m F e e e GM ˆ ) ( )ˆ ( ˆ 0 GM

8 Keple s Laws of planeta motion d dt l l In fact, this is known alead fom consevation of angula momentum.

9 Keple s Laws of planeta motion GM l GM Theefoe we need to solve 3 GM l

10 Keple s Laws of planeta motion GM l a Let and a u 3 " ' ' ' ' a u u l u d d dt d a l a lu dt d a lu u u a a lu u a d d dt d u a dt d

11 Keple s Laws of planeta motion l l u a 3 GM u" l u 3 a 3 3 l u 3 a The equation is simplified to u" u 1

12 Keple s Laws of planeta motion The geneal solution is u" u u 1 1 cos a 1 cos Recall: a u which epesents a conic cuve with focus at the oigin.

13 Edmond Halle ( ) Claim that the comet sightings of 1456, 1531, 1607 and 168 elated to the same comet. Pedicted that the comet would etun in The Halle s comet was seen again on 5th Dec 1758.

14 Electomagnetism Gauss Law E da q S 0

15 Electomagnetism Gauss Law fo magnetism S B da 0

16 Electomagnetism Faada s Law S E dl whee B t B da S B

17 Electomagnetism Ampee s Law S whee Bdl 0I 0 0 E E da S t E

18 Maxwell s Equations Name Integal fom Diffeential fom Gauss Law Gauss Law Faada s Law Ampee s Law S S S S Bdl E da B da 0 E dl q 0 t 0I 0 0 B t E E B E 0 0 B t B j E t

19 Electomagnetic wave In vacuum, Maxwell s equations become E 0 B 0 B E t E B 0 0 t

20 Electomagnetic wave Using the identit We have A A A t E E E t E t E B t E t B E E E

21 Electomagnetic wave E 0 0 E t The above equation shows the existence of wave of oscillating electic and magnetic fields which tavel at a speed ,000 kms which is ve close to the speed of light. Maxwell then claimed that light is in fact electomagnetic wave.

22 Electomagnetic wave

23 whee Special Relativit ae the electomagnetic tenso and the 4-cuent. 0 4,,,, F F F J c F x z x z z x z x B B E B B E B B E E E E F z x J J J c J Maxwell s equation in tenso fom and

24 Lotka-Voltea Equation Also known as the pedato-pe equations. It is used to descibe the dnamics of biological sstems. whee x dx dt d dt : numbe of pedato : numbe of pe x x

25 Single soliton solution KdV Equation Koteweg-de Vies equation x x t a ct x c c t x cosh ), (

26 Soliton

27 Soliton

28 Minimal Suface Equation div 1 u u 0 Mean cuvatue fee 1 H 1 0

29 Soap Bubble Constant Mean cuvatue 1 H 1 constant

30 Geneal Relativit Accoding to Einstein field equation, gavit is descibed as a cuved space time caused b matte and eneg. 1 8G R Rg 4 T c R R g T : Ricci tenso : scala cuvatue : metic tenso : eneg-momentum-stess tenso

31 Schwazschild Black Hole A black hole with no change o angula momentum. Schwazschild metic: ds 1 GM GM 1 dt 1 d d

32 Expanding Univese Robetson-Walke metic ) ( d S d t R dt c ds 0 cuvatue, sinh 0 cuvatue, 0 cuvatue, sin S

33 Schödinge equation In quantum mechanics, paticles ae descibed b wave function satisfing whee h H i h d dt : Planck s constant : wave function H : Hamiltonian opeato

34 Schödinge equation Hamonic Oscillato Electon obitals

35 Navie-Stokes Equation Navie-Stokes Equation descibe the motion of viscous fluid. whee v t v p f vv : velocit : densit : pessue : extenal foce The continuit equation eads p v f v 0

36 Black-Scholes equation Black-Scholes model the pice of an option b V t 1 S V S S V S V 0 whee V : pice of the option S : pice of the undeling instument : volatilit : constant inteest ate

37 Calabi s Conjectue Let (M, g ij ) be a compact Kähle manifold. An closed (1,1)-fom which epesents the fist Chen class of M is the Ricci fom of a metic detemines the same cohomolog class as. g ij

38 Calabi s Conjectue Equivalent to the existence of solution of the following complex Monge-Ampèe equation det g whee M ij exp( F) z z det Vol( M ) Poved b Yau Shing Tung in i j g ij 1 exp( F)

39 Poincaé s Conjectue Eve compact simpl-connected 3 dimensional manifold is homeomophic to the 3 dimensional sphee.

40 Genealized Poincaé s Conjectue If a compact n dimensional manifold is homotopic to the n dimensional sphee, then it is homeomophic to the n dimensional sphee.

41 Genealized Poincaé s Conjectue Dimension Solve Yea Field s Medal 1 o Classical 5 o above Stephen Smale Michael Feeman Gigoi Peelman

42 Ricci flow Poved b Peelman b using Ricci flow defined b Hamilton. g t ij R ij Peelman declined both the Fields medal and the Cla Millennium Pize.

43 Can Antson each the othe end? u Can I each the othe end? Rubbe band 1m 1ms 1 What is the minimum value of u fo Antson to each the othe end?

44 Definition An Odina Diffeential Equation of ode n is an equation of the fom F( x, ', ",, ( n) ) 0 whee of. (n) denotes the nth deivative

45 Definition If thee ae moe than one independent vaiable and the equation involves patial deivatives, then it is called Patial Diffeential Equation.

46 Examples Fist ode ODE: i) Linea equations a) b) d 4 0 dx d x cos dx x ii) Benoulli equation ' p( x) q( x) n

47 Examples Second ode ODE: i) Linea equations ii) Non-linea equations x e x b a x sin ' " ) ' " ) 3 x e b a ' " ) " )

48 Examples PDE: i) Elliptic u u xx 0 ii) Paabolic u t u xx u ii) Hpebolic u xx u u tt 0

49 Solution Diffeential Equation Solution x ' Ce d dx x x 3x x 3 x C " 3' 4 5e x 4x C1e C e x xe x u 4u 0 u cosx t xx tt * Paticula solution

50 IVP and BVP Initial Value Poblem: " 3' 0 sin x, 0, '(0) x 1 0, Bounda Value Poblem: " 3 ' 0 0, sin x, ( ) x 0,

51 Can Antson each the othe end? 1cms 1 Can I each the othe end? Rubbe band 1m 1ms 1 Can Antson each the othe end if he uns at 1cm pe second?

52 Antson can alwas each the othe end when u > 0. When u = 0.01 and v = 1 t 0.01ms 1 1 x 1ms 1 dx dt Sol: x x x(0) t 1 t 1 1 lnt x ln t t t e It takes about eas

53 What we ae inteested in? 1. Exact Solutions. Existence 3. Uniqueness 4. Numeical Solutions Futhe poblems: 5. Regulait 6. Well-posedness

54 Fist Ode Equation The fist ode ODE d dx f ( x, ) can be intepeted as a diection field. The integal cuves ae solutions of the equation.

55 Diection Field d dx sin x sin x dx cos x C

56 Diection Field d dx 10 5

57 Diection Field d dx e x 5

58 Diection Field d dx Ce x 5

59 Diection Field d dx x

60 Diection Field d dx x Ce x x 1

Lau Chi Hin The Chinese University of Hong Kong

Lau Chi Hin The Chinese University of Hong Kong Lau Chi Hin The Chinese Univesit of Hong Kong Can Antson each the othe end? 1cms 1 Can I each the othe end? Rubbe band 1m 1ms 1 Can Antson each the othe end? Gottfied Wilhelm Leibniz (1646-1716) Isaac

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