Introduction

The textbook

“Classical Mechanics” (3rd Edition)By H. Goldstein, C. P. Poole, J. L. SafkoAddison Wesley, ISBN: 0201657023

Herbert Goldstein(1922-2005)

Charles P. Poole John L. Safko

Misprints: http://astro.physics.sc.edu/goldstein/

World picture

• The world is imbedded in independent variables (dimensions) xn

• Effective description of the world includes fields (functions of variables):

• Only certain dependencies of the fields on the variables are observable – ηm(xn) – we call them

physical laws

?...3,2,1,0n

zxyxxxtxgE 3210 ,,, .,.

)( nm xη ?...3,2,1,0m

Systems

• Usually we consider only finite sets of objects: systems

• Complete description of a system is almost always impossible: need of approximations (models, reductions, truncations, etc.)

• Some systems can be approximated as closed, with no interaction with the rest of the world

• Some systems can not be adequately modeled as closed and have to be described as open, interacting with the environment

Example of modeling

To describe a mass on a spring as a harmonic oscillator we neglect:

• Mass of the spring• Nonlinearity of the spring • Air drag force• Non-inertial nature of reference frame• Relativistic effects• Quantum nature of motion• Etc.

Account of the neglected effects significantly complicates the solution

World picture

• How to find the rules that separate the observable dependencies from all the available ones?

• Approach that seems to work so far: use symmetries (structure) of the system

• Symmetry - property of a system to remain invariant (unchanged) relative to a certain operation on the system

Symmetries and physical laws (observable dependencies)

• Something we remember from the kindergarten:

For an object on the surface with a translational symmetry, the momentum is conserved in the direction of the symmetry:

p =

cons

t

p ≠ const

Symmetries and physical laws (observable dependencies)

• Observed dependencies (physical laws) should somehow comply with the structure (symmetries) of the systems considered

Structure

Physical Laws

Structure

Physical Laws

How?

Best F

it

Recipe

• 1. Bring together structure and fields

• 2. Relate this togetherness to the entire system

• 3. Make them fit best when the fields have observable dependencies:

Structure

FieldsFields

Structure

Physical Laws

Best F

it

mη mη

Algorithm

• 1. Construct a function of the fields and variables, containing structure of the system

• 2. Integrate this function over the entire system:

• 3. Assign a special value for I in the case of observable field dependencies:

Idxxx

x

System

nnin

nmi

S

,)(η

L

nin

nmi

S xx

x,

)(ηL ?...3,2,1,0i

Idxxx

xη

System

nnin

nmi

S

~,

)(

L

Some questions

• Why such an algorithm?Suggest anything better that works

• How difficult is it to construct an appropriate relationship between system structure and fields?It depends. You’ll see (here and in other physics courses)

• Is there a known universal relationship between symmetries and fields?Not yet

• How do we define the “best fit” value for I ? You’ll see

Evolution of a point object

• How about time evolution of a point object in a 3D space (trajectory)?

• At each moment of time there are three (Cartesian) coordinates of the point object

• Trajectory can be obtained as a reduction from the field formalism

)(

)(

)(

tzz

tyy

txx

Trajectory: reduction from the field formalism

• Let us introduce 3 fields R1(x’,y’,z’,t), R2(x’,y’,z’,t), and R3(x’,y’,z’,t)

• We can picture those three quantities as three components of a vector (vector field)

),',','(ˆ),',','(ˆ

),',','(ˆ),',','(

32

1

tzyxRktzyxRj

tzyxRitzyxR

Trajectory: reduction from the field formalism

• Different points (x’,y’,z’) are associated with different values of three time-dependent quantities

'x

'y

'z

0

1R

2R

3R

1R

2R

3R

1R

2R

3R

And they move!

Trajectory: reduction from the field formalism

• Here comes a reduction: the vector field iz zero everywhere except at the origin (or other fixed point)

'x

'y

'z

01R

2R

3R ),0,0,0(ˆ),0,0,0(ˆ

),0,0,0(ˆ),',','(

32

1

tRktRj

tRitzyxR

)(ˆ)(ˆ)(ˆ

)(

321 tRktRjtRi

tR

No (x’,y’,z’)dependence!

How about our algorithm?

• 1.

• 2.

System

nnin

nmi

S dxxx

xI ,

)(ηL

tdt

tdx

x

xim

i

Snin

nmi

S ,)(

,)( Rη

LL3,2,1

?...3,2,1,0

m

i

dtdzdydxt

dt

tdim

i

S ''',)(R

L

''',

)(dzdydxdtt

dt

tdim

i

S

RL ',

)(

Vdtt

dt

tdim

i

S

RL

dtt

dt

tdLI

im

i

S ,)(R

)'( SS LdV L

How about our algorithm?

• 3.

• Let’s change notation

• Not bad so far!!!

dtt

dt

tRdLI

im

i

S ,)(~

dtt

dt

tdLI

im

i

S ,)(R

dtt

dt

trdLI

im

i

S ,)(~

Questions?