Mathematical Methods and Classical Mechanics Review

PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 35 Review 1. Comments on presentation and exam 2. Mathematical methods 3. Classical mechanics concepts 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 1

11/25/2019 PHY 711 Fall 2019 -- Lecture 35 2 2

Comments on the Computational Project. Presumably you have all completed or nearly completed your project. Nevertheless, just to remind The purpose of this assignment is to provide an opportunity for you to study a topic of your choice in greater depth. The general guideline for your choice of project is that it should have something to do with classical mechanics, and there should be some degree of analytical or numerical computation associated with the project. The completed project will include a presentation to the class (~20 min + 5 min for questions). After completing your presentation, please turn in a copy of your presentation slides plus any notes and references. If you have not already signed up for your presentation time, please do so as soon as possible. 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 3

Comments on presentation schedule: 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 4

Comments on take-home final Similar to mid-term in form Available: Fri. Dec. 6, 2019 Due before: Mon. Dec. 16, 2019 before 11 AM 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 5

Review of mathematical methods 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 6

11/25/2019 PHY 711 Fall 2019 -- Lecture 35 7

https://dlmf.nist.gov/ 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 8

Example special functions 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 9

Complex numbers = + 2 1 Define 1 i i = z x iy ( )( ) 2 = = + = + 2 2 * z zz x iy x i y x y Polar representation sin cos i + ( ) = = i z e Functions of complex variables ( ) f z = ( ) ( ) ( ) f z ( ) f z + + ( , ) u x y ( , ) iv x y i Derivatives: Cauchy-Riemann equations ( ) f z x ( ) u z x ( ) v z x f z x ( ) f z i y ( ) u z i y u z x ( ) v z i y ( ) v z y ( ) u z y = + = + = i i i ( ) ( ) f z i y ( ) ( ) v z y ( ) v z x ( ) u z y df dz = = = Argue that = and 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 10

Analytic function ( ) is analytic if it is: continuous single valued its first derivative satisfies Cauchy-Rieman conditions A closed integral of an analytic function is zero. f z However: 1 z = = Behavior of ( ) about the point 0: f z z n For an integer , consider n = 2 2 0 1 1 n n i 1 z e i d = = = 1 (1 ) n i n dz e id n n in 2 i e 0 0 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 11

Contour integration methods -- ( ) = ( ) z 2 Res ( ) f dz i f z p ( ) z = p y C No contribution z p ( ) ( ) f z z Res ( ) ( ) f z p ( ) f z 2 Res i z p z p p ( ) z = x z p z p C 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 12

General formula for determining residue: ( ) Res ( z ) f z ( ) g z p Suppose that in the neighborhood of , ( ) z f z ( ) p m z z z z z p p p Since ( ) is analytic near , we ca n make a Taylor exansi n about ( ( 1)! m dz o : g z z z p p ) 1 m z z 1 m ( ) ( ) dg z d g z ( ) p p p + + + + ( ) g z ( ) ... .... g z z z p p 1 m dz ) ( ( ) m 1 m ( ) d z z f z 1 ( ) p lim z = Res ( ) f z p 1 m ( 1 )! m dz z p 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 13

Fourier transforms -- Definition of Fourier Transform function a for ( : ) t f = i t ( ) ( ) f t d F e Backward transform : 1 = i t ) F( dt f(t) e 2 Check : 1 = ' i t i t ( ) ' ' f t d dt f(t ) e e 2 1 ( ) = = ' i t t ( ) ' ' ' ' ' ( ) f t dt f(t ) d e dt f(t ) t t 2 Note: The location of the 2 factor varies among texts. 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 14

Properties of Fourier transforms -- Parseval's theorem: ( ) ( ) * * = ( ) ( ) 2 dt f(t) f(t) d F F * ( ) ( ) ( ) * = ' i t i t Check: ' ' dt f(t) f(t ) dt d F e d F e ( ) ( ) ( ) ' i t * = ' ' d F d F dt e ( ) ( ) ( ) * = ' ' 2 d F d F ( ) ( ) F * = 2 d F 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 15

Doubly discrete Fourier Transforms Doubly periodic functions 2 T + t 2 1 T N 1 N ( ) + 2 / 2 1 i N = f F e + 2 1 N = N N ( ) + 2 / 2 1 i N = F f e = N Fast Fourier Transforms (FFT) 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 16

Mechanics topics Scattering theory Lagrangian mechanics Hamiltonian mechanics Liouville theorem Rigid body motion Normal modes of oscillation about equilibrium Wave motion Fluid mechanics (ideal or including viscosity; linear and nonlinear) Heat conduction Note: The following review slides are necessarily brief. Please refer to the original lecture slides for details. Please email: natalie@wfu.edu with any corrections/suggestions 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 17

Scattering theory Note: The notion of cross section is common to many areas of physics including classical mechanics, quantum mechanics, optics, etc. Only in the classical mechanics can we calculate it from a knowledge of the particle trajectory as it relates to the scattering geometry. d bdb b Figure from Marion & Thorton, Classical Dynamics d d d b db b db Note: We are assuming that the process is isotropic in = = d d d sin sin 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 18

Lagrangian mechanics Given the Lagrangian function: ( ) , q = , , L L q t T U The physical trajectories of the generalized coordinates ( ) q t ( ) , q = Are those which minimize the action: , S L q t dt Euler-Lagrange equations: d dt L q L q d dt q L L q = = 0 for each : 0 q For the case that there both mechanical and electromagnetic contributions in terms of electric and magnetic fields: , 1 , , t t c t q L T U q t c ( ) A r t ( ) ( ) ( ) ( ) = = E r r B r A r , , t t ( ) ( ) = + r r A r , , t mech 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 19

Recipe for constructing the Hamiltonian and analyzing the equations of motion ( , ) ) = Construct 1. Lagrangian function : ( ( , ) L L q t q t t L Compute . 2 generalize momenta d : p q = Construct . 3 Hamiltonia expression n : H q p L t ( , ) ) = motion Hamiltonia Form . 4 function n : ( ( , ) H of H q p t t : Analyze . 5 canonical equations dq dp H H = = dt p dt q 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 20

Liouvilles Theorem (1838) The density of representative points in phase space corresponding to the motion of a system of particles remains constant during the motion. ( , ) t ) = Denote density the q of particles phase in space : ( ( , ) t D D q p t dD D D D = + + q p dt p t dD = According Liouville' to theorem s : 0 dt 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 21

Rigid body motion Moment I In a reference frame attached to the object, there are 3 moments of inertia and 3 distinct principal axes of inertia p 1 tensor r : ( ) 2 r (dyad notation) m p r p p p Representation of rotational kinetic energy: ( 1 2 ) 1 1 1 ~ ~ ~ = + + 2 2 2 2 3 , , , , , T I I I 1 2 3 2 2 1 ( ) 2 = + sin cos sin I 1 2 1 ( ) 2 + + sin sin cos I 2 2 1 2 + + cos I 3 2 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 22

Eulers transformation between body fixed and inertial reference frames ~ = + + 0 3 ' 2 e e e 3 0 3 e ( ) ) + = + e sin cos sin 1 ( + e sin sin cos 2 + + e cos 3 e 3 x y y x ' 2 e 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 23

Normal modes of vibration -- potential in 2 and more dimensions 2 V x ( ) 2 + ( , ) V x y ( , ) V x y x x 1 2 eq eq eq 2 , x y eq eq 2 2 V y V ( ) ( )( ) 2 + + y y x x y y 1 2 eq eq eq 2 x y , , x y x y eq eq eq eq V(x,y) 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 24

Example normal modes of a system with the symmetry of an equilateral triangle -- continued 3 u3 Potential contribution for spring 13: 1 2 ( ) 2 = + u u V k 13 13 3 1 13 13 2 ( ) u u 1 2 13 3 1 k u2 13 2 1 2 1 2 1 2 3 ( ) ( ) + k u u u u u1 3 1 3 1 x x y y 2 1 2 3 = + x y 13 13 2 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 25

Example normal modes of a system with the symmetry of an equilateral triangle -- continued = + + Potential contributions: V V V V 12 13 23 2 2 ( ) ( ) u u u u 1 2 1 2 + 13 3 1 12 2 1 k k 12 13 2 ( ) u u 1 2 + 23 3 2 k 23 1 2 ( ) 2 k u u 2 1 x x 2 1 2 1 2 3 ( ) ( ) + + k u u u u 3 1 3 1 x x y y 2 2 1 2 1 2 3 ( ) ( ) + k u u u u 2 3 2 3 x x y y PHY 711 Fall 2019 -- Lecture 35 2 11/25/2019 26

Example normal modes of a system with the symmetry of an equilateral triangle -- continued u u u u u u u u 1 x 1 x 2 x 2 x k m 3 x 3 x = 2 1 y 1 y u u 2 y 2 y u u 3 y 3 y 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 27

Discrete particle interactions continuous media The wave equation Initial value solutions attributed to D'Alembert: to the wave equation; (x,t) 2 2 t = = ( ) and x = 2 0 where 0 0 ( ) x c (x, ) (x, ) 2 2 t x x ct + 1 2 1 2 ( ) ( , ) x t = x ct + + + ( ') x dx ( ) ( ) ' x ct c x ct 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 28

Mechanical motion of fluids Newton s equations for fluids Use Euler formulation; following particles of fluid Density : Variables (x,y,z,t) Pressure p(x,y,z,t) (x,y,z,t) v Velocity Navier-Stokes equation + v 1 1 1 3 ( ) ( ) = f + + + 2 v v v v p t Continuity condition + ( ) = v 0 Viscosity contributions t 11/25/2019 PHY 711 Fall 2019 -- Lecture 35 29

Fluid mechanics of incompressible fluid plus surface Non-linear effects in surface waves: p0 z h y z=0 x Dominant non-linear effects soliton solutions 3 x ct h ( , ) x t = = 2 sech constant 0 0 0 2 h gh = + where 1 0 h c gh PHY 711 Fall 2019 -- Lecture 35 0 1 / h 2 11/25/2019 30