• Level 2 Physics course
• 3UOC
• Offered every year, Session 1

• Lecturer: Prof. Paul Curmi
• Lecture times: Wednesday 9-10 and Thursday 1-2
  
• Consultation time: Fri 3-4

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 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.

• 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.

Assessment

• 2 hour written examination 60%
• Two assignments 20%
• Mid session test 20%

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 (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

For more information about PHYS2010 contact: