PHYS 3510
Advanced Mechanics, Fields & Chaos
Tutorial Problems 4
1). Find the generic behaviour of an orbit (with arbitrary initial condition) for the system of
equations
x = x + y x(x 2 + y2 )
y = x + y y(x 2 + y 2 )
a) For a particular initial condition
PHYS 3510
Advanced Mechanics, Fields & Chaos
Tutorial Problems 2
1). Show that the following transformations are canonical.
sin p
Q = ln
,
q
P = qcot p
p
Q = cot 1
mq ,
P=
p 2 + m 2 2q 2
2m
Q = q cos p ,
P = q sin p
Q = tan1 (q p) ,
P=
1
2
q
Q =
The Constrained Particle (1)
(Ref. Pars Ch 1)
Let us suppose that the particle is acted on by a given force ( Fx , F y ,F z ) but is confined to a
given smooth surface. Let the equation for the surface be (x, y, z) = 0 . Then the coordinates
x, y, z of th
Hamilton-Jacobi Theory
Canonical transformations may be used as a general procedure for solving mechanical
problems. If the Hamiltonian is conserved, a solution can be sought by transforming to new
set of canonical coordinates which are all cyclic. The in
HAMILTONS EQUATIONS of MOTION
In the first two chapters we developed the Lagrangian formulation of classical mechanics;
now we turn to an alternative statement known as the Hamiltonian formulation - a yet more
powerful method. We will assume that the syst
Variational Principle & Lagranges Equations
2.1 Hamiltons Principle
We derived Lagranges equations starting from an instantaneous state of the system and
considering small virtual displacements from it: ie using a differential principle such as
DAlemberts
ADVANCED MECHANICS, FIELDS and CHAOS
Introduction
The subject of classical mechanics concerns the application of Newtons laws to mechanical
systems such as particles or rigid bodies. We learn some powerful mathematical formulations
of the theory - the Lag
PHYS 3510
Advanced Mechanics, Fields & Chaos
Tutorial Problems 3
1). Solve the problem of the motion of a point particle in the xy -plane under the influence of
a uniform gravitational field in the y direction, using the Hamilton-Jacobi method. Find both
Piecewise Linear One-dimensional Maps
To gain some understanding of chaotic systems it is simpler if we abandon differential
equations and look instead at simple mappings. Many of the ideas of dynamical systems can
be understood more easily using such map
Qu_1.
a) Q = p + iaq and P = ( p iaq) 2ia so
[Q,P ] = ia
1
1
1 = 1
2ia 2
b) Find F2 (q,P) . Need p = p(q,P) and Q = Q(q,P) .
Q = p + iaq
2iaP = p iaq
(1)-(2) gives
Q 2iaP = 2iaq Q = 2ia(P + q)
then
p = Q iaq = 2ia(P + q) iaq = ia(2P + q)
p(q,P)dq + f
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF PHYSICS
PHYS 3510 ADVANCED MECHANICS, FIELDS AND CHAOS
MID-SESSION TEST - 11 SEPTEMBER 2008
Do both questions.
Both questions of equal marks.
FORMULA SHEET
Euler-Lagrange equations
m
d L L
= l alk = Qk
dt qk q
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF PHYSICS
FINAL EXAMINATION
NOVEMBER 2008
PHYS3510
Advanced Mechanics, Fields and Chaos
Time Allowed 2 hours
Total number of questions - 4
Answer ALL questions
All questions ARE of equal value
Candidates may not b
The Lagrangian and Hamiltonian formulation for Continuum
systems and Fields
So far we have discussed systems with a finite number of degrees of freedom. Many
mechanical systems are continuous however, for example, a vibrating elastic solid. The
complete m