Introduction
The textbook
“Classical Mechanics” (3rd Edition)
By H. Goldstein, C. P. Poole, J. L. Safko
Addison 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
n  0,1,2,3... ?
E.g., x0  t , x1  x, x2  y, x3  z
• Effective description of the world includes fields
(functions of variables):
ηm ( xn )
m  0,1,2,3... ?
• Only certain dependencies of the fields on the
variables are observable – ηm(xn) – we call them
physical laws
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:
Symmetries and physical laws
(observable dependencies)
• Observed dependencies (physical laws) should
somehow comply with the structure (symmetries) of
the systems considered
How?
Physical Laws
Structure
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:
ηm
ηm
Physical Laws
Structure
Algorithm
• 1. Construct a function of the fields and variables,
containing structure of the system
  i η m ( xn )

LS 
, xn 
i
 xn

i  0,1,2,3... ?
• 2. Integrate this function over the entire system:
  i η m ( xn )

LS 
, xn dxn  I
i

System
 xn

• 3. Assign a special value for I in the case of
observable field dependencies:
  i ηm ( xn )

~


L
,
x
dx

I
S
n
n
i

 x

System
n


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
x  x (t )
y  y (t )
z  z (t )
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)

R( x' , y ' , z ' , t )  iˆR1 ( x' , y ' , z ' , t ) 
 ˆjR ( x' , y ' , z ' , t )  kˆR ( x' , y ' , z ' , t )
2
3
Trajectory: reduction from the field
formalism
• Different points (x’,y’,z’) are associated with
different values of three time-dependent quantities
R3
z'
R1
R3
R1 R2
x' R1
0
R2
R3
R2
y'
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'
R1

z'
R( x' , y ' , z ' , t )  iˆR1 (0,0,0, t ) 
ˆR (0,0,0, t )
ˆ

j
R
(
0
,
0
,
0
,
t
)

k
2
3
R3

 R (t )
0
 iˆR (t )  ˆjR (t )  kˆR (t )
R2
1
y'
2
No (x’,y’,z’)
dependence!
3
How about our algorithm?
  i η m ( xn )

 d i R m (t )  i  0,1,2,3... ?
  LS 
• 1.LS 
,
x
n
i
 dt i , t  m  1,2,3
 x



n


  i η m ( xn )

• 2. I 


L
,
x
dx
S
n
i

 x
 n
System
n


 d i R m (t ) 
  LS 
, t dx' dy' dz ' dt
i
dt


i
i
 d R m (t ) 
 d R m (t ) 
  LS 
, t dt  dx' dy' dz '   LS 
, t dt V '
i
i
dt
 dt



i
 d R m (t ) 
I   LS 
, t dt ( L dV '  L )
i
S
S

dt


• 3.
How about our algorithm?
i
 d R m (t ) 
I   LS 
, t dt
i
 dt

 d Rm (t ) 
~
I   LS 
, t dt
i
 dt

i
• Let’s change notation
 d rm (t ) 
~
I   LS 
, t dt
i
 dt

i
• Not bad so far!!! 
Questions?