Introduction

The textbook “Classical Mechanics” (3rd Edition) By H. Goldstein, C. P. Poole, J. L. Safko Addison Wesley, ISBN: Herbert Goldstein ( ) Charles P. PooleJohn L. Safko Misprints:

World picture The world is imbedded in independent variables (dimensions) x n Effective description of the world includes fields (functions of variables): Only certain dependencies of the fields on the variables are observable – η m (x n ) – 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: p = const 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 Fit

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 Fields Structure Physical Laws Best Fit

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:

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

Trajectory: reduction from the field formalism Let us introduce 3 fields R 1 (x’,y’,z’,t), R 2 (x’,y’,z’,t), and R 3 (x’,y’,z’,t) We can picture those three quantities as three components of a vector (vector field)

Trajectory: reduction from the field formalism Different points (x’,y’,z’) are associated with different values of three time-dependent quantities 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) No (x’,y’,z’) dependence!

How about our algorithm? 1. 2.

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

Questions?