PHYS2120
Mechanics and Computational Physics
Students should only enrol into PHYS2010 if they have already completed PHYS2020.
All others should enrol into PHYS2120 Mechanics and Computational Physics in 2013.
Mechanics (also see Computational Physics lower down the page)
Lecture
notes and Past Exams
Information
for Session 1, 2013
Brief
Syllabus:
Coordinate
systems; Newton's Laws; Lagrange's equations; Harmonic oscillator,
damped and forced motion, resonance; Central forces, inverse
square law orbits; Many particle systems; Hamilton's equations;
Coupled oscillators.
Assumed
Knowledge:
The
course assumes familiarity with first year physics, e.g. PHYS1002
or PHYS1221 or PHYS1231; and first year mathematics, e.g.
MATH1231 or MATH1241. Corequisites: 2^{nd} year mathematics,
MATH 2011 or MATH2110 or MATH2100.
Course
Goals:
Mechanics
is the most basic and fundamental branch of physics, and has
been of central importance since the days of Galileo and Newton.
This course aims to provide students with further insight
into the principles and behaviour of mechanical systems, plus
an introduction into new mathematical formalisms which have
become the basis of much of modern physics, e.g in quantum
field theory, or the treatment of nonlinear systems. Specific
topics include:
 A
brief discussion of various coordinate systems which may
be useful in different problems;
 The
integration of Newton's law of motion for some standard
cases;
 An
advanced discussion of the harmonic oscillator, treating
the effects of a damping force and a harmonic driving force,
and the phenomenon of "resonance", which appears
in virtually every field of physics;
 A
general discussion of motion in a central field of force,
including the fundamental conservation laws. We cover the
special case of an inverse square force law, and the appearance
of elliptic orbits, providing some of the basic tools for
an understanding of astronomy;
 A
review of the general properties of manyparticle systems;
 A
first discussion of the Lagrangian formulation of mechanics,
and Hamilton's principle, leading to the Lagrange equations
of motion. This formalism is a basic tool used in later
years in discussing advanced mechanics, and relativistic
quantum mechanics and field theory;
 A
first introduction to the alternative Hamiltonian formulation
of mechanics, and Hamilton's equations of motion;
 A
brief discussion of two coupled oscillators, providing a
first example of a system with nontrivial interactions
between different degrees of freedom.
Learning
Objectives
 Students
will gain a physical understanding of the important phenomena
of resonance in oscillating systems, of the motion of planets
and central force motion in general, and of coupled oscillators.
 Students
will learn to formulate and to solve differential equations
describing physical systems
 Students
will learn to understand and apply the sophisticated formalisms
of Lagrange and Hamilton, which are basic to advanced theoretical
physics.
Why
is mechanics important?
A
knowledge of mechanics is clearly vital in understanding the
world around us. The principles of mechanics are used in everything
from designing Moon rockets or building bridges, right down
to playing a game of billiards. Furthermore, the mathematical
techniques and formalisms learnt in mechanics are used as
basic tools in all other areas of physics. It is vital to
understand the 'classical' material in this course, before
one can understand or appreciate the principles and techniques
used in 'modern' physics.
The
course is strongly recommended as groundwork for a number
of 3^{rd} year courses, e.g. PHYS3020 Statistical
Physics, PHYS3230 Electromagnetism, PHYS3410 Biophysics and
PHYS3510 Advanced Mechanics, Fields and Chaos, as well as
the 4^{th} year Honours course in Quantum Field Theory.
The
material covered is not by any means new. Inverse square law
orbits were already understood by Isaac Newton; and the discussions
of Lagrange and Hamilton date back to the 1800s. But despite
its antiquity, mechanics remains a living and thriving field
of research. In third year PHYS3510, Advanced Mechanics, Fields
and Chaos, students will get a first glimpse of some fascinating
topics at the frontiers, such as:
 Nonlinear
systems, and 'solitary waves';
 The
road to chaos;
 Etc.

Space
Rocket changing from a circular orbit to an elliptical
one 
How
to succeed  Strategies for Learning
Some
students find this subject confusing, and have difficulty
telling the wood from the trees. It is important to keep an
eye on the basic principles that emerge, particularly old
favourites like the conservation of energy and momentum. Like
most subjects, the key to success is hard work. Rather than
waiting for it all to be presented on a plate, it is vital
that students sit down and work out most of the tutorial
problems, by themselves or with a group. Only in this
way can they get the practice and experience necessary to
understand the physical phenomena and mathematical techniques
presented here. Practice, practice, practice! Approximately
one in four of the class periods will be devoted to tutorials,
where solutions to selected problems will be discussed.
It
is also very useful, as in any course, for each student to
prepare a concise summary of the material presented
in lectures, to fix the main ideas in his/her memory.
For rules regarding academic
honesty, etc, see here.
Resources
Textbook
The
material presented in this subject is covered by many good
textbooks. Students are recommended to obtain one of two
texts, depending on their needs.
For
a more elementary treatment, one could use:
G.R.
Fowles and G.L. Cassaday, Analytical Mechanics, 5^{th}
or 6^{th} edition; or
G.R. Fowles, Analytical Mechanics, 3^{rd} or 4^{th}
edition.
Students
with a stronger theoretical background or students intending
to pursue future studies in advanced mechanics could use:
H. Goldstein, C. Poole & J. Safko, Classical Mechanics,
3rd edition; or
H. Goldstein, Classical Mechanics, 1st or 2nd edition.
Note:
Goldstein is the recommended text for PHYS3510 Advanced
Mechanics, Fields and Chaos.
Additional
References
 L.
D. Landau and E. M. Lifshitz, Mechanics (Pergamon Press)
 LN
Hand & JD Finch, Analytical Mechanics (Cambridge University
Press)
 F.
Scheck, Mechanics (SpringerVerlag)

JV Jose & EJ Saletan, Classical Dynamics A Contemporary
Approach (Cambridge University Press)

C. Lanczos, The Variational Principles of Mechanics (Dover)
M.R. Spiegel, Theoretical Mrechanics
(Schaum outline);
 B.P.
Cowan, Classical mechanics
 K.
Rossberg, A First Course in Analytical Mechanics
 K.R.
Symon, Mechanics
 A.P.
Arya, Introduction to Classical Mechanics
Those
students having difficulties should consult the lecturer for
help. Further information on student support services may
be found on the School website here.
Detailed
Syllabus
TOPIC 
TEXT
REFERENCE
4^{th}
Edition 
TEXT
REFERENCE
6^{th}
Edition 
Vectors:
coordinate systems, derivatives of vectors. Kinematics:
velocity and acceleration, angular velocity, relative
motion. 
Chapter
1 
Chapter
1 
Mechanics:
Newton's laws, consequences 
4.1,4.2 
Chapter
2 
Conservative
forces, solution methods 
3.13.5 
2.3,4.1 
Oscillatory
Motion 


Harmonic
oscillator, damped and forced oscillations 
3.13.5 
3.13.4 
Resonance,
power supplied, harmonic motion in 2 and 3 dimensions 
4.4 
3.6,4.4 
Central
Forces 


Conserved
quantities, equation of motion, energy equation, Kepler's
laws, gravitational fields and forces, inverse square
law orbits and energies, stability and symmetry. 
Chapter
6 
Chapter
6 
Many
Particle Systems 


Internal
forces and torques, conservation laws, CM coordinates,
twobody problem, collisions, rotational inertia,
laminar motion of a rigid body. 
7.17.4
8.28.6 
7.17.3,7.5
8.18.3,8.58.6 
Lagrange's
Equations 


Constraints,
generalized coordinates and forces, kinetic energy,
Lagrangian, generalized momenta. 
10.110.5 
10.110.5 
Hamilton's
equations, symmetries and conservation laws, examples. 
10.7 
10.9 
Potential
function, equations of motion, normal frequencies
and modes, general solution, weighted and continuous
strings. 
Chapter
11 
Chapter
11 
.
For more
information about PHYS2120 contact:
last updated 1st February 2011
Computational Physics
Brief
Syllabus:
Introduction to computer software
used in Physics; the C programming language; integers/floatingpoint
numbers; the Mandelbrot set; generating and using
random numbers; plotting; numerical integration techniques;
manybody gravitation; cellular automata; Wolfram’s A New Kind of Science; Conway’s Game
of Life; subtleties in apparently simple algorithms;
analysis of experimental data. Numerical techniques
for the solution of nonlinear equations, for polynomial
interpolation, numerical integration, ordinary differential
equations and for waves.
Assumed Knowledge:
From the 2003 Undergraduate Handbook,
the prerequisites are: PHYS1002 or PHYS1022 or PHYS1221
or PHYS1231 or PHYS1241 and MATH1021 or MATH1231 or
MATH1241 or MATH1031. Excluded: PHYS2001.
No computer programming knowledge
in C is assumed. We will assume that you have a knowledge
of Maple and Linux, as taught in the Mathematics prerequisites.

Course
Goals:
The main aim of the course is to
teach you how to program a computer in order to be
able to solve a wide range of problems in Physics.
The first part of the course will
be used to teach you the elements of the C programming
language. C has been chosen since (1) it is freely
available, (2) it is commonly used, and (3) the basic
concepts used in C are common to many other programming
languages.
The remaining lectures will show
how your programming knowledge can be used to attack
a variety of interesting physics problems.

Learning Objectives
You will learn techniques required
to write reliable, efficient, software. This is not
easy, and you will learn that even simple problems,
such as finding the roots of a quadratic equation,
or calculating the standard deviation of a data set,
can be surprisingly subtle.
You will have a strong foundation
for picking up other programming languages and packages
during your career.
Why is computational physics
important?
Computers are one of the most powerful
tools available to the physicist. They are used
in all areas of physics: ranging from theoretical
calculations using supercomputers, analysing terabytes
of data, to controlling instrumentation. There
has even been speculation that the universe itself
is effectively a computer.
A knowledge of how to use computers
is invaluable to all scientists.

How
to succeed  Strategies for Learning
The only way to learn a computer
programming language is to practice programming, at
least weekly. PHYS2020 is one course where you can
not cram the day before the exam and expect to
pass. To assist with this learning process, you will
be expected to complete a series of six assignments.
Additionally a number of tutorial questions will also
be set.
It is the nature of programming
that the time required to solve a problem is difficult
to predict. You may find that you can complete
one assignment in 30 minutes, whereas the following
week you may spend three hours trying to track down
a problem that turns out to be a missing semicolon.
As you gain more experience, this unpredictability
will be reduced, and your enjoyment will be increased.
While some course notes will be
available on the web, it is vital to turn up to the
lectures. 
If
you find yourself completely lost in lectures, seek help
immediately from the lecturer or tutor. Most importantly…
For rules regarding academic honesty, etc, see the
School website here.
Resources
Textbook
There are no compulsory textbooks, but the book Computational Physics 2nd edition by Giordano Nakanishi is recommended, and is available in the UNSW bookshop.
There are notes on the C language available on the course website; however, a basic book on C is also very useful. There are many of these available, including some reasonably priced C books in the UNSW bookshop. The C language is well established now, and so second hand books are fine. The book "The C programming Language" by Kernigan and Ritchie remains the classic reference for C, although you should note that it doesn't deal with prototypes.
Additional References

Burden and Faires, Numerical
Analysis

Stephen Wolfram, A New Kind of
Science

Press, Teukolsky, Vetterling &
Flannery, Numerical Recipes in C, the art of scientific
computing
Those students having difficulties
should consult the lecturer and/or the tutor for help.
Further information on student support services may be
found on the School website at Second
Year Course Information.
Detailed Syllabus
TOPIC 
Introduction to the C programming
language 
Representation of data types: integers,
floatingpoint, binary, hexadecimal 
A short course on C: functions,
loops, arrays, pointers, input/output, printf/scanf,
the GNU C debugger, memory allocation, the math
library 
The Mandelbrot set; programming
with complex numbers 
Random numbers: generating them
and using them 
Solution of nonlinear equations 
Polynomial interpolation 
Numerical integration 
Ordinary differential equations 
Waves 
Analysis of experimental data 
Other possible examples: Euler and
midpoint approximations; nbody gravitation;
1D cellular automata; Stephen Wolfram's A
New Kind of Science; 2D cellular automata;
John Horton Conway's Game of Life; artificial
intelligence; Simulations of particles in a
gas 
Subtleties in simple algorithms:
the roots of a quadratic equation; calculating
the mean and standard deviation of a data set 
Further Information
For more information
about PHYS2120 contact:
last updated 21st February 2013
