Introduction. The textbook “Classical Mechanics” (3rd Edition) By H. Goldstein, C. P. Poole, J....
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Transcript of Introduction. The textbook “Classical Mechanics” (3rd Edition) By H. Goldstein, C. P. Poole, J....
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?