PHYS3510 ADVANCED MECHANICS, FIELDS AND CHAOS

Not offered in 2009

This course is highly recommended for students planning to do Honours in Physics and for students interested in theoretical physics.

The course has several aims.


1. The first aim is to develop efficient approaches for description of complex mechanical systems. In principle all physics in such systems is described by Newton's laws, but systems are so complex that direct application of the laws is practically impossible. This is why the more powerful techniques are necessary. The approaches developed in advanced mechanics give also a general view on conservation laws. For example it is demonstrated that conservation of energy is related to homogeneity of time and conservation of momentum is related to homogeneity of space.


2. The next aim is the most important one. This is to develop techniques for other fields of physics. The techniques are the Hamiltonian and the Lagrangian formalisms. Quantum mechanics is essentially based on Hamiltonian formalism. This is why study of advanced mechanics is very helpful for the understanding of quantum mechanics. General relativity, quantum field theory, standard model, and even advanced electrodynamics are fully based on Lagrangian formalism. Any serious study of these subjects is hardly possible without knowledge of advanced mechanics.


3. The introduction in field theories is the third aim of the course. Such topics as gauge invariance and functional derivative are discussed. The concept of the gauge invariance is the central one in modern physics. The present course demonstrates how entire electrodynamics (i.e. all Maxwell equations) can be derived from this concept.


4. The forth aim of the course is introduction in classical chaos. This is a relatively new field developed in the second half of 20th century. The development of the field was to large extent stimulated by the development of large particle accelerators. The problem of classical chaos is closely related to foundations of statistical mechanics and to such fundamental problem as irreversibility of time.

Assessment
2 hour written examination 70%
Two assignments 15%
Mid session test 15%

Syllabus of the course

  • Generalized coordinates
  • Lagrangian, Action
  • Principle of least action
  • Lagrange equations
  • Generalized momenta
  • Symmetries and conservation laws, Noether's theorem
  • Systems with constraints, Holonomic and nonholonomic constraints
  • Hamilton's equations
  • Poisson brackets
  • Phase space
  • Canonical transformations
  • Generating functions
  • Liouville's theorem
  • Adiabatic invariants, motion of charged particles in magnetic traps.
  • Action angle variables
  • Functional derivatives
  • Scalar field theory (elastic waves)
  • Lorentz invariant vector gauge theory (electrodynamics)
  • Linear stability
  • Tent map, Modulo map
  • Quadratic map, Lyapunov exponents
  • Quadratic map, bifurcations, windows
  • Dimension, Cantor set
  • Baker map, symbolic dynamics, periodic points
  • Lorentz gas, stable and unstable manifolds


Textbooks

L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon Press)
H. Goldstein, Classical Mechanics (Addison Wesley)
F. Scheck, Mechanics (Springer-Verlag)

 


Further Information

For more information about PHYS3510 contact:

Associate Professor Oleg P. Sushkov

last updated 1st February 2011