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 (SpringerVerlag)