Special Relativity: Lagrangian Formulation and Practical Approach, Study notes of Mechanics

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Mechanics

Physics 151

Lecture 17

Special Relativity

(Chapter 7)

What We Did Last Time „^

Worked on relativistic kinematics^ „

Essential tool for experimental physics „ Basic techniques are easy:

„^

Define all 4 vectors „^

Calculate c-o-m energy and boost „^

Go about with business

„^

Examples:^ „

Particle creation „ Elastic scattering „ Particle decays

Lagrangian Formulation „^

Proper Approach^ „

Set up a covariant form of Hamilton’s principle „ Keep everything in clean tensor forms „^

Practical Approach^ „

Build a Lagrangian that reproduces 3-force in a frame „ May or may not be correct in other frames „ Works OK pretty often, but no guarantee

Practical Formalism „^

For a single particle of mass

m

„^

Let’s check if this works „^

Looks OK for the 3-d part… Try to push this path

2

2 1

L^

mc

V

−^

−^

x 2

2

i

i

i^

i

L^

mc

mc

p

v

∂^

∂^

=^

∂^

∂^

i^

i^

i

i^

i^

i

d^

L^

L^

V

p^

p^

F

dt^

v^

x^

x

∂^

∂^

⎛^

=^

+^

=^

−^

⎜^

∂^

∂^

⎝^

⎠^

^

3-d equationof motion

β^ = reduced velocity

Space componentgood. But no time

component

Energy Function „^

Energy function

h

is defined by

„^

This is total energy „^

It’s conserved if

V

is time-independent

„^

Proved this before – No changes by going relativistic

2

2

1

L^

mc

V

−^

−^

x

2

2

2

2

2

i^ i

i

L^ i

mc

v^

mc

h^

x^

L^

mc

V^

V

x

=^

−^

=^

+^

−^

+^

=^

∂^

−^

^

conservative

Simple Example „^

Particle accelerating under constant force^ „

Electron in an electric field „ Lagrange’s equation „ Integrate twice, assuming

x^

v^

= 0 at

t^

x

φ^

=^

V

φ^

e − E

2

2 1

L^

mc

eEx

−^

2

d^

L^

L^

d^

mc

eE

dt^

v^

x^

dt

⎛^

∂^

⎛^

⎞^

⎜^

−^

=^

−^

⎜^

⎟^

⎜^

∂^

⎝^

⎠^

⎝^

⎠^

2 1 d^

eE

dt

mc

⎛^

⎜^

⎜^

⎝^

2

(^

eEmc eEmc

t t

+^

(^

)

2

2

(^

)^

eEmc

mc x^

t

eE =^

+^

Simple Example

„^

Low-velocity limit

Æ

„^

t^ Æ

limit

Æ

„^

Look at it in terms of energy

2

(^

eEmc eEmc

t t

+^

(^

)

2

2

(^

)^

eEmc

mc x^

t

eE =^

+^

2 (^

eEx

mc

=^

LHS

( ) V x = −

0

2

RHS

p^

c^

mc

T

=^

−^

=^

Energy conservation

eE v^

t m =^

2 eE 2 x^

t m =

β^

x^

ct →

All as expected

mc x^

eE

=^

Relativistic Oscillator „^

Consider a 1-dim. harmonic oscillator^ „

Let’s use energy conservation this time „ Solution exists only when „ Oscillation between twopoints expected

2

2 1

L^

mc

V

−^

m

x

2 2

const

mc 1

E^

V

=^

+^

2 1 2 V^

kx

2

4

2

2

(^

m cE^

V

β^

=^

−^

2

E^

V^

mc

−^

x ( ) V x

E^

2

E^

mc −

b

b −

What’s thefrequency?

Semi-Relativistic Oscillator

„^

Period is longer than non-relativistic oscillator „^

Relativistic solution slower than the non-relativistic one „^

Difference depends on the amplitude of oscillation

2

2

2

3

2

4

2

2

2

(^

(^

b^

b^

x^

m^

kb

dx

b

c^

k^

mc

c

b^

x

κ^

⎛^

+^

−^

⎛^

=^

=^

+^

=^

⎜^

⎜^

⎝^

−^

⎝^

∫

2

max

2

2

0

V 8

kb mc

mc

=^

=^

Wrong sign in textbook

/

Limitations of Practical Approach „^

gives correct relativistic answers

for many practical problems

„^

It is an ad-hoc technique^ „

Not Lorentz covariant by construction

„^

Time is treated separately from space

„^

Lorentz transformation of Lagrangian is not given^ „

Must redefine

L

in each inertial frame

„^

Truly relativistic theory should respect relativity fromthe principle all the way up^ „

Let’s see how well it works…

2

2 1

L^

mc

V

−^

−^

x

Truly Relativistic Formalism „^

Hamilton’s principle^ „

We want the action integral to be Lorentz scalar „ Integration should not be by

t , but by a Lorentz-invariant

variable

Æ

Proper time

τ^ could be a good choice?

„^

Lagrangian

L

must then be a Lorentz scalar

„^

Lagrange’s equation should look like

„^

Solution is not unique. None of them perfect^ „

Let’s look at one – Goldstein Section 7.10 for more

I^

Ldt

δ^

∫

L^

d^

L

x^

d^

u

μ

μ

∂^

⎛^

−^

⎜^

∂^

Symmetric for time andspace components

Free Lagrangian „^

We try a force-free Lagrangian^ „

Looks like the non-relativistic kinetic energy „ Lorentz scalar „^

Lagrange’s equation would be^ „

Conservation of 4-momentum „ Time component is conservation of energy „^

Energy function doesn’t give total energy, though

1 mu u 2

ν ν

(^

)^

d mu

d d^

u^

d

μ

μ

⎜^

2

h^

u^

mu u

mc

u μ

μ^

μ

μ ∂Λ =^

Conserved,but not energy

Limitations of Purist Approach „^

We don’t know 4-force for anything but EM^ „

Most real-world problems cannot be solved this way „^

What to do with multi-particle system^ „

Lagrangian formalism allows coordinate transformation

„^

Each coordinate does not correspond to a single particle

„^

Problem will be solved only when we give up theparticle picture

L^

d^

L

x^

d^

u

μ

μ

∂^

⎛^

−^

⎜^

∂^

⎠^ Proper time of what?

I^

Ld

δ^

δ^

=^

∫

Summary „^

Constructed Lagrangian formulation^ „

Practical approach provides useful tools

„^

Relativistic solutions can befound for many systems „^

Not really relativistic at heart

„^

Purist approach can be built only for limited cases^ „

E.g. single particle in EM field

„^

Done with special relativity^ „

Next: Hamiltonian formalism

2

2 1

L^

mc

V

−^

−^

x

1 mu u 2

qu A μ

μ

μ

μ