PHY 711 Fall 2014 -- Lecture 39 112/01/2014

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 39

1. Brief introduction to the physics of elastic continua (Chap. 13 of F & W)

2. Brief review of topics covered this semester

3. Course evaluation forms

PHY 711 Fall 2014 -- Lecture 39 212/01/2014 2

Final grades due 12/17/2014

PHY 711 Fall 2014 -- Lecture 39 312/01/2014

PHY 711 Fall 2014 -- Lecture 39 412/01/2014

Brief introduction to elastic continua

reference deformation

r1 r1’

2 2

2 1 2

1 1 1 2

1 2 1 1

( ) (' '

' '

)

( )

ur r r r

r r r r r

r r

u rr

u

PHY 711 Fall 2014 -- Lecture 39 512/01/2014

Brief introduction to elastic continua

Deformation components:

1 1

2 2

i i i

j j j

ij ij

j j

i i

O

u uu u u

x x x x x

ò

V a b c ' ' ' 'V ba c (1 )' 1 TrV VV uò

TrdV d

V

uò

PHY 711 Fall 2014 -- Lecture 39 612/01/2014

Stress tensor3

1

:

Lame' coeffi

ˆ com

cient

ponent of

s :

or :

force acting on surface

Generalization of Hooke's

law, F

thij j

x

ii

j

jj ij

j

i

i

j

kx

uuT

x x

T dA i

T

u

n

Tr 2ij ij ò ò

2Note that: Tr = 3 Tr

3

1

23

32ij ij ij

T

T Tr T

ò

ò

bulk modulus=p

VK V

PHY 711 Fall 2014 -- Lecture 39 712/01/2014

Example -- hydrostatic pressure:

2 333

ij ij

ij ij ij

T dp

dp dp

K

p

VK V

ò

22 33

1 ij ij ijT Tr T

ò

1

Example -- uniaxial pressure: 0 otherwise

in terms of Young's modulus

9

3

zz zz

ij

dp ij zzT

K

T

E

E

K

ò

PHY 711 Fall 2014 -- Lecture 39 812/01/2014

Dynamical equations of elastic continuum

2

22

1

3K

t

uu u f

1/ 2

In the absence of external forces, this reduces

to two decoupled wave equations representing

longitudinal and transverse modes:

=

where 0 and 0

4

3 and

t

t

l t

l

l

Kc c

u uu

u u

1/ 2

PHY 711 Fall 2014 -- Lecture 39 9

Scattering theory:

12/01/2014

Brief review of topics covered in this course --

PHY 711 Fall 2014 -- Lecture 39 1012/01/2014

angleat

detector into scattered is that beamincident of Area

areaunit per particlesincident ofNumber

particle per target at particles detected ofNumber

section cross alDifferenti

d

d

bdbdb

min1/

2 20

Scattering angle equation for

central potential ( ):

12

(1 / )1

r

V r

b duV u

b uE

sin sin

d d b db b db

d d d d

PHY 711 Fall 2014 -- Lecture 39 1112/01/2014

Rotating reference frames

VωωVωV

ωVV

VωV

ωV

dt

d

dt

d

dt

d

dt

d

dt

d

dt

d

dt

d

dt

d

bodybodyinertial

bodybodyinertial

22

2

2

2

01e

02e

03e

Inertial frame Body frame

1e

2e

3e

ω

Centrifugal force

12/01/2014 PHY 711 Fall 2014 -- Lecture 39 12

Newton’s laws; Let r denote the position of particle of mass m:

rωωrωr

ωa

Fr

Fr

mdt

dm

dt

dm

dt

dm

dt

dm

dt

dm

bodyinertial

ext

body

ext

inertial

2 2

2

2

2

2

2

Coriolis force

rωωrωr

ωFrr

m

dt

dm

dt

dm

r

mGM

dt

dm

earth

e

earth

2 'ˆ22

2

Equation of motion on Earth’s surface

w

z (up)

x (south)

y (east)

q

PHY 711 Fall 2014 -- Lecture 39 1312/01/2014

Calculus of variation example for a pure integral functions

Find the function ( ) which extremizes ( ), ,

where ( ), , ( ), , .

Necessary condition: 0

f

i

x

x

dyy x I y x x

dx

dy dyI y x x f y x x dx

dx dx

I

, ,

Euler-Lagrange equations:

0 /dy

x x ydx

f d f

y dx dy dx

PHY 711 Fall 2014 -- Lecture 39 1412/01/2014

Application to particle dynamics

2

1

(time)

(generalized coordinate)

(Lagrangian)

(action)

Denote:

, ;t

t

x t

y q

f L

I S

dqq

dt

S L q q t dt

, ,

Euler-Lagrange equations:

0 /dq

t t qdt

L d L

q dt dq dt

PHY 711 Fall 2014 -- Lecture 39 1512/01/2014

The Lagrangian is given by:

( ), ,d

L t t T Udt

rr

Kinetic energy

Potential energy

0

0

Note: For a particle of charge , the potential function can have the form

, , ,

where , denotes non-electromagnetic field contributions and

,1 where , ,

q U

qU U t q t t

cU t

tt t

c

r r r A r

r

A rE r r

, ,t

t t

B r A r

PHY 711 Fall 2014 -- Lecture 39 1612/01/2014

Recipe for constructing the Hamiltonian and analyzing the equations of motion

q

H

dt

dp

p

H

dt

dq

ttptqHH

LpqH

q

Lp

ttqtqLL

:motion of equations canonical Analyze .5

,)(,)( :functionn Hamiltonia Form .4

:expressionn HamiltoniaConstruct .3

:momenta dgeneralize Compute .2

,)(,)( :function LagrangianConstruct 1.

PHY 711 Fall 2014 -- Lecture 39 1712/01/2014

x

p

Notion of phase space – example for one-dimensional motion due to constant force

2

0 0

200 0 0 0

( )

,2

1

2( ) i

i i i i

px p F x

mp

p t p t x t x tm

pH x p F

m

F t F

PHY 711 Fall 2014 -- Lecture 39 1812/01/2014

Liouville’s Theorem (1838) The density of representative points in phase space corresponding to the motion of a system of particles remains constant during the motion.

0 : theoremsLiouville' toAccording

,)(,)( :space phasein particles ofdensity theDenote

dt

dD

t

Dp

p

Dq

q

D

dt

dD

ttptqDD

Importance of Liouville’s theorem to statistical mechanical analysis: In statistical mechanics, we need to evaluate the probability of various configurations of particles. The fact that the density of particles in phase space is constant in time, implies that each point in phase space is equally probable and that the time average of the evolution of a system can be determined by an average of the system over phase space volume.

PHY 711 Fall 2014 -- Lecture 39 1912/01/2014

223232

1212

233

222

211

2

1

2

1

2

1

2

1

2

1

xxkxxk

xmxmxmL

Example – linear molecule

1x2x

3x

m1 m2 m3

Analysis of small oscillations near equilibrium

PHY 711 Fall 2014 -- Lecture 39 2012/01/2014

m1 m2 m3

m1 m2 m3

m1 m2 m3

01

Om

k2

CO m

k

m

k 23

Normal modes of oscillation for linear molecule example--

PHY 711 Fall 2014 -- Lecture 39 2112/01/2014

Longitudinal waves as the continuum limit of a linear mass-spring system --

1ixix 1ix

m m m

0

2

10

2

2

1

2

1

iii

ii xxkxmVTL

2 11 iiii xxxkxm

22

2

11

2

2

2

),()(

:system thisof version continuum theimagine Now

xx

xxx

txtxtx

iii

iii

PHY 711 Fall 2014 -- Lecture 39 2212/01/2014

/

:equation Continuum

2 :equation Discrete

2

2

2

2

2

22

2

2

11

xxm

xk

t

xxk

tm

xxxkxm iiii

system parameter with units of (velocity)2=c2

2 22

2 2

Wave equation for longitudinal or

transverse displacement :

0

μ(x,t)

ct x

PHY 711 Fall 2014 -- Lecture 39 2312/01/2014

Solution methods for wave equations and generalizations1. D’Alembert’s method2. Fourier transforms3. Laplace transforms; contour integration4. Eigenfunction expansions5. Green’s functions6. Variational methods

PHY 711 Fall 2014 -- Lecture 39 2412/01/2014

r

w

22

2

1 1Kinetic energy:

2 2

1

2

1

2

inertial

p p p pp p

p p pp

p p p pp

d

dt

T m v m

m

m

rω r

ω r

ω r ω r

ω ω r r r ω

Physics of rigid bodies rotating about a fixed origin

ω I ω�

PHY 711 Fall 2014 -- Lecture 39 2512/01/2014

2

Matrix notation:

xx xy xz

yx yy yz ij p ij p pi pjp

zx zy zz

I I I

I I I I m r r r

I I I

I�

Moment of inertia tensor

1 1 2 2 3 3

1For general coordinate system:

2

For (body fixed) coordinate system that diagonalizes

ˆ ˆ moment of inertia tensor: 1,2,3

1ˆ ˆ ˆ

2

ij i jij

i i i

T I

I i

T I

I e e

ω e e e

�

2i i

i

PHY 711 Fall 2014 -- Lecture 39 2612/01/2014

03e

y

xy

x '2e

3e

3'2

03 ˆ ˆ ˆ ~ eeeω

3

2

1

ˆcos

ˆcossinsin

ˆsincossin ~

e

e

eω

Euler angles for describing angular velocity in terms of body fixed contributions --

PHY 711 Fall 2014 -- Lecture 39 2712/01/2014

Equations describing fluid physics

Newton-Euler equation of motion:

Continuity condition: 0

applied

p

t

t

vv v f

v

PHY 711 Fall 2014 -- Lecture 39 2812/01/2014

Solution of equations in the linear approximation – (linear sound waves)

1 0p

t t

v

v v f v

0

20 0

=0

Assume: 0

p p p p c

v fv

2

00

0 0

20

0 0 0 00

22 0 0

20

Linearized equations: 0

Let

0

ˆ

i tit

c

e e

kc

t

c

c

t

k kr r

vv

v v

k

v v

v

k k

PHY 711 Fall 2014 -- Lecture 39 2912/01/2014

Wave analysis in fluid physics beyond linear regime1. Non-linear effects in sound waves – shock

phenomena2. Application of Euler-Newton equations to surface

waves in water3. Korteweg-de Vries soliton analysis

PHY 711 Fall 2014 -- Lecture 39 3012/01/2014

Analysis of heat conduction

a

T1

b

c

2,,

p

th

p

T t qT

c

tt c

k

rr

T0

PHY 711 Fall 2014 -- Lecture 39 3112/01/2014

Newton-Euler equations for viscous fluids

2

Navier-Stokes equation

1 1 1

3

Continuity condition

0

pt

t

vv v f v

v

v

PHY 711 Fall 2014 -- Lecture 39 3212/01/2014

Presentations – After your presentation, please send me your slides and notes. In order to encourage discussion after the talks, points will be awarded for peer questions.