Course Code    : AMAT 42463

Title                   : Advanced Topics in Mechanics

Pre-requisites   :  AMAT 21272

Learning Outcomes:

After the completion of this course unit, the student will be able to:
1. apply Lagrange’s equations to solve motion involving impulse
2. solve dynamic problems using Hamilton’s equations
3. verify canonical variables, determine generating functions and use canonical transformations
4. use Poisson brackets and their properties for finding constants of motion, determining canonical
transformations
5. approximate problems for small oscillations and determine normal frequencies and normal modes
6. solve motion of a heavy symmetric top by Hamiltonian formulation
7. determine stability of fixed points of a nonlinear dynamical system

 Course Contents:

Impulsive motion: Equations of motion, body acted on by a given impulse, body acted on by an impulsive
couple, impact of inelastic bodies, elastic bodies, applications of principle of virtual work, Lagrange's equations
for impulsive motion
Hamiltonian dynamics: Generalized Momentum, Hamilton's equations of motion, Liouville's Theorem, Poisson
brackets, Canonical transformation, Hamilton-Jacobi equation
Small oscillations: Lagrange's method, Normal modes, Roots of the Lagrangian determinant, Oscillations under
constraint, Stationary property of the normal modes
The Motion of a Top: Euler's Equations, Free Tops, Euler's Angles, The Heavy Symmetric Top
Chaos theory: Fixed points and their linear stability, Elements of bifurcation theory, Limit cycles,
Synchronization and phase dynamics, Chaos and strange attractors, Cellular automata

 Method of Teaching and Learning: A combination of lectures, tutorial discussions and presentations.

 Assessment: Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading:

1. Rao, A.V. (2011). Dynamics of Particles and Rigid Bodies: A Systematic approach, Cambridge
University Press.
2. Chorlton, F. (2 nd Ed., 2019). Textbook of Dynamics, D. Van Nostrand.
3. Gignoux C & Silvestre-Brac, B. (2014). Solved Problems in Lagrangian and Hamiltonian Mechanics,
Springer Netherlands.
4. Kelley, J.D. & Leventhal, J.J. (2016). Problems in Classical and Quantum Mechanics: Extracting the
Underlying Concepts. Springer.
5. Strauch, D. (2009). Classical Mechanics, An Introduction, Springer.
6. Goldstein, H., Poole, C. P. & Safko, J. (3 rd Ed., 2011). Classical Mechanics, Pearson.
7. Mondal, C.R. (2005). Classical Mechanics, Prentice Hall of India.
8. Ramsey, A.S. (1975). Dynamics, Parts I & II, Cambridge University Press.

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Course Unit Code     : AMAT  41423

Course Title               : Linear programming and Nonlinear Programming

Pre-requisites            : PMAT 21263

Learning outcomes:

Upon successful completion of the course unit the student will be able to:

• develop a linear programming model from problem description
• solve linear programming problems using graphical method, simplex method, revised simplex method,
  big M method, two-phase method
• obtain the solution of the dual of a linear programming using the solution of the primal problem
• analyze sensitivity of the model parameters
• solve transportation problems, assignment problems and transshipment problems
• solve problems using non-linear programming methods
• obtain the solution of various applications of linear programming problems using Excel, TORA and other
  appropriate software and analyze the solutions for decision making

 Course Content:

Introduction: optimization, types of optimization problems

Linear Programming: formulation of linear programming problem, extreme points, basic feasible solutions,
solutions using graphical methods, simplex method, revised simplex method, big Method, two-phase method,
solution behavior, duality theory, primal and dual problems, sensitivity analysis, reduction of linear inequalities.

Applications of Linear Programming Problems: transportation problem, assignment problem and
transshipments problems

Non-Linear programming: graphical illustration of nonlinear programming problems, types of nonlinear
programming problems, one-variable unconstrained optimization, multivariable unconstrained optimization, the
Karush-Kuhn-Tucker (KKT) conditions for constrained optimization, applications of nonlinear programming

Software: Solve Linear programming problems using Excel, TORA computer packages and other appropriate
computer software.

Method of Teaching and Learning:  A combination of lectures, group projects, case studies, tutorial discussions and presentations.

Assessment: A combination of lectures, group projects, case studies, tutorial discussions and
presentations.

Recommended Textbook:

1. Luenberger, D.G. (4th Ed., 2016). Linear and Nonlinear Programming, Springer.
2. Matousek, J. & Gärtner, B. (2009). Understanding and Using Linear Programming. Springer Berlin
   Heidelberg.
3. Hillier, F.S. & Lieberman, G. (11th Ed., 2021). Introduction to Operations Research, McGraw-Hill.
4. Taha, H.A. (10th Ed., 2017). Operations Research: An Introduction, Pearson.

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Course Code    : AMAT 42843

Title                   : Quantum Field Theory

Pre-requisites   : AMAT 41823

 Learning Outcomes:  

At the end of this course, the student should be able to demonstrate knowledge of basic properties of relativistic local field theory and the quantization of spin zero and spin half fields.

 Course Contents:

Relativistic wave equation, Review of mechanics of a finite system, Quantisation, General Theorems, Quantisation of spin zero fields and spin ½ fields, Momentum and angular momentum operators, Phase factor, Conventions between the spinners, Two - component theory.

 Method of Teaching and Learning: A combination of lectures, tutorial discussions and presentations. 

Assessment: Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading      :

1. Schiff, L.I., Quantum Mechanics (4e), McGraw-Hill India. (2014).
2. Prugovecki, E., Quantum Mechanics in Hilbert Space (2e), Courier Corporation, 2013
3. Lee, T. D., Particle Physics and Introduction to Field Theory, Taylor and Francis. (1981).

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Course Code  :           AMAT 43976

Title                :           Research/Study Project

 Learning Outcomes:

At the end of this course, the student should be able to 

1. plan a research project in the field of Mathematics

2. investigate a research problem

3. develop a theoretical/practical/conceptual framework towards achieving the research objectives

4. document a research project proposal, and progress in detail

5. solve a research problem using theoretical principles and practical knowledge

6. analyze research results critically

7. communicate research findings, information, and solutions to a specialized audience in a form of dissertation and oral presentations

8. practice research ethics, technical skills.

 Course Contents:

An undergraduate research project is an inquiry, investigation, or creation produced by a final year honours degree undergraduate that contributes to the discipline and reaches beyond the traditional curriculum. An undergraduate research project is designed to provide students with the opportunity to develop and practice advanced discipline-specific projects in collaboration with senior academics in the department.

 Method of Teaching and Learning: self-studies, discussions and student presentations, seminars and colloquiums

Assessment: A combination of self-study, seminars, presentations, reports and dissertation

Recommended Reading      : 

1. Robson, C. (2nd Ed., 2016). How to do a research project - A guide for undergraduate students, Wiley.

2. Reading list and material relevant for each selected topic to be provided at the beginning of the academic year by the supervisor