Mechanics Continuous Systems, Lecture Notes - Physics, Study notes of Mechanics

Download Mechanics Continuous Systems, Lecture Notes - Physics and more Study notes Mechanics in PDF only on Docsity!

MechanicsPhysics 151^ Lecture 24Continuous Systems and Fields(Chapter 13)

What We Did Last Time „^ Built Lagrangian formalism for continuous system^ „^ Lagrangian

Æ

„^ Lagrange’s equation

Æ

„^ Derived simple wave equation „ Energy and momentum conservation given by theenergy-stress tensor „ Conservation lawstake the form of (time derivative) = (flux into volume) „ Ran out of time here

Æ^ See Goldstein 13.3 if interested „^ Today’s lecture doesn’t use it

L^ dxdydz =^ L∫∫∫

,

d dx η^ μμ

η ⎛^ ⎞∂^

⎜^ ⎟⎜^ ⎟∂^

L^ ⎝ ⎠

L

, , T^ μν

∂^ L^ η^ δ≡ −L^ μμνη∂ν

dT^ μν^0 = dx ν

The Rod Again „^ Consider the 1-dim elastic rod again^ „^ Lagrangian density

x

2

2 1^ d^^2

dK dt^

dx

⎡^ μ

⎛^ ⎞^

⎛^ ⎞

=^

⎢^

⎜^ ⎟^

⎜^ ⎟

⎝^ ⎠^

⎝^ ⎠

⎢^

⎣^

L

∂^ L^ ^ μη= =η∂

2

(^22) 1^ d^ 2 2 2 2

dK dt^

dx η K d dx

η

πη

⎡^ μ πη μ

⎛^ ⎞^

⎛^ ⎞

=^ −^

=^

⎢^

⎜^ ⎟^

⎜^ ⎟

⎝^ ⎠^

⎝^ ⎠

⎢^

⎣^

⎛^ ⎞

=^ +^

⎜^ ⎟ ⎝^ ⎠

H^

L

Wait!What am I going todo with this term?

Hamiltonian Formalism „^ Hamiltonian formalism treats time as special^ „^ Because of the way momentum is defined „^ Natural structure of classical field theory is symmetricbetween time and space^ „^ At least in Lagrangian formalism^ „^ Hamiltonian is not so usefulas in the case of discrete systems „^ Quantum field theory is built primarily on Lagrangian^ „^ c.f. Non-relativistic QM is almost all Hamiltonian

 η= ∂ ∂L 0 ,

d dx^ μρ μ

ρ η

η ⎛^ ⎞∂^

⎜^ ⎟⎜^ ⎟∂^

L^ ⎝ ⎠

L

Fourier Transformation „^ What happens to the Lagrangian?^ „^ Integrate with

x^ and use

etc.

2

2 1 2 1 sin

sin

cos^

cos

n^ 2 n^ m^

m^

n^ n^ n^

m^ m^ m

n^

m^

n^

m

d^

dK dt^

dx q k x q^ k x

K^ q k

k x^

q k^ k x

η

η ⎡^ μ μ

⎤ ⎛^ ⎞^

⎛^ ⎞ =^

−⎢

⎥ ⎜^ ⎟^

⎜^ ⎟ ⎝^ ⎠^

⎝^ ⎠ ⎢^

⎥ ⎣^

⎦ ⎡^

⎤

=^

− ⎢^

⎥

⎣^

⎦

∑^

∑^

∑^

∑

L

^

^0 sin^ sin

L n^

m^ L nm k x^ k xdx

∫ 2 2 2

2 2 2

L^

n^

n^ n^

n n n

n^

n^

n q^

k q^

Kk

L^

L

dx^

K^

q^

q

⎡^ μ

⎤^ ⎡

=^

−^

=^

⎢^

⎥^ ⎢

⎣^

⎦^ ⎣

∑^

∑^

∑

L^ ∫

^

( , )^0 

( ) sin n n n x t^ q

t^ k x ∞ η = ∑= What does this look like?

Harmonic Oscillators „^ The Lagrangian represents an infinite array ofindependent harmonic oscillators^ „^ Angular frequencies are „^ Vibration of continuous system can be decomposedinto a set of discrete oscillators^ „^ True for any linear system^ „^

Lagrangian density must be 2

nd^ order homogeneous

function of the field’s derivatives „ Small oscillation around equilibrium always OK

2 2 2 2 2

n n n 2 n

Kk L^ q

⎡^ μ^ q

⎣^

∑ (^2) n n^

n

Kk^ vk ω = =^ μ

Wave velocityWavenumber

Other Examples? „^ Linear fields

Æ^ Harmonic oscillators

Æ^ Particles

„^ We know an excellent example: Electromagnetic field „^ Corresponding particle = photon^ „^ Photoelectric effect tells us „ Is it possible that all particles are quantized field? „^ For a particle of mass

m , „^ Make correspondence with a harmonic oscillator „^ But first of all, the field must satisfy relativity

E^ ω=^ =^2 4

2 E^ m c

p c = +

2 2 2

(^4 2 2) m c p c

ω^ =^

2 4 m c^ 2 2 2 k c ω = +^2 =

Must satisfythis dispersionrelation

Relativistic Field Theory „^ We had difficulty with relativity and multi-particles^ „^ Each particle’s EoM looked like^ „^ When combined, we didn’t knowwhose time to use „^ With field like

η( x , t ), time is just another parameter

„^ Action integral and Lagrange’s equationslook symmetric for time and space „ Can we just call

0 x =^ ct^ and call it done?

„^ Almost…

dp^ sK = s d^ τ^ s^ Proper time of particle

s , I dxdydzdt = L∫∫∫

0 d dx^ μρ μ

ρ ⎛^ ⎞∂^ ∂−^ ηη

= ⎜^ ⎟⎜^ ⎟∂^

∂ L^ ⎝ ⎠ L

Field Equation „^ We derived Lagrange’s equation from Hamilton’sprinciple for continuous field in the last lecture^ „^ Derivation is unchanged

Æ^ Same equations hold „^ Note „^ Ready to look at an easy example…

,

d^ μ^ dx ρ μ

ρ

⎛^ ⎞∂^

⎜^ ⎟⎜^ ⎟∂^

L^ ⎝ ⎠

L

0 1 2

I^ dx dx dx dx δ δ=^

L∫^0

,^

d^ ( ) (^ )

d d ct^

dt

d^

d^

d

dx^

d ct^

ηη^ ρρ dt

⎛^ ⎞

⎛^ ⎞^

⎛^ ⎞

∂^

∂^

⎜^ ⎟= =

⎜^ ⎟^

⎜^ ⎟⎜^ ⎟

⎜^ ⎟^

⎜^ ⎟

∂^

∂^

⎝^ ⎠^

⎝^ ⎠

L^

L^

L^

Didn’tchangethis term

Scalar Field „^ The simplest field is a scalar field

φ

„^ Lagrangian density may be a function of „^ For free field,

L^ has no explicit dependence on

x^ μ

„^ Only a few scalar quantities can be formed „^ Try „^ What kind of field is this?

d^ φ^ μ dx ( , , ) x φ φL, μ μ

(^2 2) λ , , (^0) λ φ φ^ μ φ= −L

(^ )^

(^2) , 0 ,

d^

d dx^

ν dx ν

ν ν

φ^ μ φ

⎛^ ⎞∂^ ∂−^ φ^ φ

=^

+^ =

⎜^ ⎟⎜^ ⎟∂^

L^ ⎝ ⎠

L^

2 2 0

d φ^ μ φ+^ ν^ dx dx ν

Known asKlein-Gordon equation

The Field – What is It? „^

gives particles with mass

„^ OK, but what is the field

φ^ itself?

„^ Vibration of elastic material

Æ^ Phonons

„^ Vibration of electromagnetic field

Æ^ Photons

„^ The field

φ^ doesn’t have to be “physical” „^ It “exists” only in the sense that quantized excitation of

φ^ are

physical (particles) „ QM calls it wave function,whose (amplitude)

2 is interpreted as the probability of a particle being there^ „^ Still an indirect definition of “existence”

(^2 2) λ , , (^0) λ φ φ^ μ φ= −L

= m^ μ= 0 c

Mystical^ ether

anybody?

Vector Field „^ Field can be more complicated than scalar^ „^ How about a 4-vector, for example?^ „^ Such field represents particles with spins^ „^

4-vector field

Æ^ Particles with spin = 1

„^ Electromagnetic field is an obvious example^ „^ Corresponding particle is photon, with spin 1^ „^ Recall

is a 4-vector „^ Connectionwith^ E^

(^ and B

,^ )

A^ c φ=^ μ

A

0 0 x^ y^ z^00 x^

z^ y y^ z^

x z^ y^

x E^ c^ E^

c^ E^ c E^ c^

B^ B E^ c^ B

B E^ c^ B

B νμ A^ A μν F x x μν

−^ −^

− − − ⎡^ −

⎤

⎢^

⎥

∂^ ∂^

⎢^

⎥

=^ −^

= ⎢^

⎥

∂^ ∂^

⎢^

⎥

⎢^

⎥

⎣^

⎦

Electromagnetic Field „^ To build

L, we can use

μ^ μν^ A ,^ F

μand j

„^ Easy to find

L^ that works: „^ Field equation is

,^

F 2 , 2 0

d^

d^ F^

j

dx^ A^

A^ dx^

A d F F^ j dx dF^ λρ^ λρ^ μ j dx ν

ν μ νμ

μ ν μν νμ^ μ ν μν μ ν

⎛^ ⎞^

∂^ ∂−^

=^

⎜^ ⎟⎜^ ⎟∂^

∂^

⎝^ ⎠

−^ +=

= −^

−^ =

L^ L

λρ^ F F^ λρ^ λ^ j A^4

λ =^

L

A^ A = −^ ,^ ,^ ρ^ λ^ λ ρ^ What wewanted

Free EM Field „^ Does it satisfy the usual wave equation?^ „^ For free field (

μ^ j = 0), the field equation reduces to „^ This doesn’t give you the usual plane waves etc. „^ Problem: Given

E^ and^ B ,

μ^ A is not uniquely defined „^ Extra condition to fix this ambiguity „ Impose Lorentz gauge condition

λρ^ F F^ λρ^ λ^ j A^4

λ =^

L

μν dF^0 =ν^ dx

2

d^ A^

A^

A^

A

dx^ x^

x^ x^

x^ x^

x νμ

ν

μ

ν

ν

ν μν

μ ν ⎛^

∂^ ∂^

∂^

−^ =

⎜^

⎜^

∂^ ∂^

∂^ ∂^

∂^ ∂

⎝^

μ A ∂ 0 =μ x ∂

2

A^

d A^

A

x^ x^ c μμ^^ dt

μ ∂^ =^ ν ν

− ∇^ =

∂^ ∂

EM waveswith^ v^ =^ c