1. Introduction

2. Misprints: http://astro.physics.sc.edu/goldstein/
Herbert Goldstein
( )
The textbook
“Classical Mechanics” (3rd Edition)
By H. Goldstein, C. P. Poole, J. L. Safko
Addison Wesley, ISBN:
Charles P. Poole
John L. Safko
Misprints:

3. 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

4. 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

5. 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

6. 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

7. 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 = const
p ≠ const

8. Symmetries and physical laws (observable dependencies)
Observed dependencies (physical laws) should somehow comply with the structure (symmetries) of the systems considered
Structure
Physical Laws
Physical Laws
How?
Structure
Best Fit

9. Recipe Best Fit 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
Physical Laws
Best Fit
Structure
Fields

10. 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:

11. 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

12. 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

13. 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)

14. Trajectory: reduction from the field formalism
Different points (x’,y’,z’) are associated with different values of three time-dependent quantities
And they move!

15. Trajectory: reduction from the field formalism
Here comes a reduction: the vector field iz zero everywhere except at the origin (or other fixed point)
No (x’,y’,z’)
dependence!

16. How about our algorithm?1. 2.

17. How about our algorithm?3.
Let’s change notation
Not bad so far!!! 

18. Questions?