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

• Level 2 Physics course
• Offered every year, Session 1
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: 2nd 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 many-particle 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 non-trivial 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 3rd year courses, e.g. PHYS3020 Statistical Physics, PHYS3230 Electromagnetism, PHYS3410 Biophysics and PHYS3510 Advanced Mechanics, Fields and Chaos, as well as the 4th 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';
• 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, 5th or 6th edition; or
G.R. Fowles, Analytical Mechanics, 3rd or 4th 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.

• L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon Press)
• LN Hand & JD Finch, Analytical Mechanics (Cambridge University Press)
• F. Scheck, Mechanics (Springer-Verlag)
• 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 4th Edition TEXT REFERENCE 6th 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 3.1-3.5 2.3,4.1 Oscillatory Motion Harmonic oscillator, damped and forced oscillations 3.1-3.5 3.1-3.4 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, two-body problem, collisions, rotational inertia, laminar motion of a rigid body. 7.1-7.4 8.2-8.6 7.1-7.3,7.5 8.1-8.3,8.5-8.6 Lagrange's Equations Constraints, generalized coordinates and forces, kinetic energy, Lagrangian, generalized momenta. 10.1-10.5 10.1-10.5 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

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Further Information

last updated 1st February 2011

Computational Physics
• Level 2 Physics course
• Offered every year, Session 1
• more course information is available on the PHYS2120 Moodle website

Brief Syllabus:

Introduction to computer software used in Physics; the C programming language; integers/floating-point numbers; the Mandelbrot set; generating and using random numbers; plotting; numerical integration techniques; many-body 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 non-linear 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.

• 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, floating-point, 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 non-linear equations Polynomial interpolation Numerical integration Ordinary differential equations Waves Analysis of experimental data Other possible examples: Euler and midpoint approximations; n-body 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