PHY2007 Relativity II and Mechanics

2010-2011

Code: PHY2007
Level: 2
Title: Relativity II and Mechanics
Instructors: Prof. C.P. Winlove
CATS Credit Value: 10
ECTS Credit Value: 5
Pre-requisites: N/A
Co-requisites: N/A
Duration: L1-L11
Availability: unrestricted
Background Assumed: Relativity I and Vectors (PHY1105)
Directed Study Time: 22 lectures
Private Study Time: 66 hours
Assessment Tasks Time: 12 hours

Aims

The level 1 module PHY1105 analysed the motion of particle-like objects. However, to understand the motion of more realistic "rigid bodies" both translational and rotational aspects of their motion must be considered. In complicated cases (e.g. motion in central fields) it is convenient to use a non-inertial rotating frame of reference, and to use Euler-Lagrange equations. These are a powerful reformulation of classical mechanics and will provide the key to the solution to new range of problems.

Problems in nuclear and high-energy physics motivate extending the description of space-time given in PHY1105 by the development of a matrix description of the Lorentz transformations, and this will reveal the intimate relationship between electromagnetism, as discussed in PHY2006 and special relativity.

Intended Learning Outcomes

Students will be able to:

• solve problems involving the translational and rotational motion of bodies using rectangular, cylindrical and spherical polar coordinates;
• calculate moments of inertia and products of inertia of rigid bodies;
• use the conservation laws to solve mechanical problems;
• use the Euler-Lagrange equations to solve more complex problems;
• describe the basic concepts of Chaos theory and state how Chaos theory may be used in different disciplines;
• apply the theory of Special Relativity to solve problems in High Energy physics;
• demonstrate how Special Relativity theory unifies Optics, Mechanics and Electromagnetism.

Transferable Skills

Ability to think logically, analyse and solve problems of both a qualitative and numerical nature. Knowledge of chaos theory applicable to other disciplines.

Learning / Teaching Methods

Lectures, tutorials and problems classes

Assignments

Problems classes.

Assessment

Problems-class assignments (10%), 30-minute mid-term test in Week L6 (20%) and one 90-minute examination (70%).

Syllabus Plan and Content

1. Rotational Dynamics
   1. Coordinate systems; cylindrical and spherical polar coordinates
   2. Single particle (in a central field); angular velocity, circular orbits
   3. N-particle system; conservation of total angular momentum
   4. Solid bodies: rotational forms of Newton's laws, kinetic energy of a rotating body, moment of inertia, product of inertia, the parallel-axis theorem, the perpendicular-axis theorem
   5. Examples, including rolling motion, the compound pendulum, and gyroscopic motion
2. Rotating Reference Frames and Orbits
   1. Central fields: noncircular orbits
   2. Rotating reference frames: centrifugal force, Coriolis force
   3. Euler-Lagrange equations - simple examples
3. Nonlinear Dynamical Systems
   1. Chaos and its relevance to mechanics
   2. The stability of non-linear equations
   3. The non-linear oscillator
   4. Phase-Space Methods
   5. The pendulum revisited
   6. Mappings
   7. Characterisation of chaotic systems
4. Special Relativity
   1. Review of Fundamental Concepts
   2. Four Vectors
   3. Relativistic Mechanics
   4. Relativistic Optics
   5. Relativity and Electromagnetism
   6. Experimental Basis of Special Relativity

Core Text

Goldstein H., Poole C. and Safko J. (2002), Classical Mechanics (3^rd edition), Addison Wesley, ISBN 0-201-65702-3 (UL: 531 GOL)
Marion J.B. and Thornton S.T. (1995), Classical Dynamics of Particles and Systems (4^th edition), Harcourt Brace and Co (UL: 531.11 MAR)

Supplementary Text(s)

French A.P. (1975), Special Relativity, M.I.T Introductory Physics Series (UL: 530.11 FRE)
Kaplan D. and Glass L. (1998), Understanding Nonlinear Dynamics, Springer-Verlag (UL: 515.352 KAP)
Landau L.D. and Lifshitz E.M. (1976), Mechanics, Pergamon Press (UL: 531 LAN)
Rindler W. (1977), Essential Relativity, Springer-Verlag (UL: 530.11 RIN)

Formative Mechanisms

Students are able to monitor their own learning by attempting the set problems and attending classes arranged around those problems where points of difficulty may be discussed with the lecturers and postgraduate demonstrators.

Evaluation Mechanisms

The module will be evaluated using information gathered via the student representation mechanisms, the staff peer appraisal scheme, and measures of student attainment based on summative assessment.