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**AMA2001 - Classical Mechanics (1**^{st} semester)

*Pre-requisite:* AMA1020 and PMA1020

*Lecturer:* Dr C Ramsbottom

**Introduction**

Mechanics has always been a source of inspiration for mathematics and mathematicians. Calculus, as well as the famous Newton's laws, were “invented” by Newton largely because he wanted to solve a single but very important mechanical problem, the problem of planetary motion. Mechanics was at the beginning of such branches of mathematics as theory of functions, calculus of variations, differential equations and more recently, theory of chaos. Mechanics and its mathematical methods are important in optics, electromagnetic theory, statistical mechanics, quantum mechanics, theory of relativity, quantum field theory and many ‘non-physical’ applications such as theory of optimisation and control.

For students, besides giving a comprehensive picture of mechanical phenomena and teaching how to solve a wide variety of problems, Classical Mechanics offers a unique opportunity to see the mathematical methods they have learned at work and to practice their mathematical skills. Vectors, partial derivatives, single and multiple integrals, differential equations, stationary points, complex numbers, as well as matrices and determinants are all among the tools used in the course.

**Contents**

*Introduction:* Basic revision of Newtonian mechanics: conservation of mechanical energy and angular momentum. Revision of key concepts in linear algebra (matrices): eigenvalues and eigenvectors. Revision of basic multivariable calculus (partial differentiation and multiple integration).
*Motion of a single particle in a central potential:* Attractive forces: planetary motion and transfer orbits. Repulsive forces: Rutherford scattering.
*Rotating frames of reference:* Angular velocity, effect of the Earth's rotation; apparent gravity, cyclones and anticyclones, Foucault's pendulum, Larmor precession.
*Conservation laws for a system of particles:* internal and external forces, conservation of energy, centre of mass as the origin.
*Rigid body motion:* Moments and products of inertia, parallel axis theorem, Euler's equations of motion, Euler angles and rotation matrices.
*Lagrange's equations:* Constraints, equations for holonomic constraints; examples.
*Motion of a top:* Precession and nutation; stability of a sleeping top; gyroscopes and the gyrocompass.
*Lagrange's equations for impulses:* Applications to systems of rods.
*Small oscillations:* Normal modes of oscillation.
*Lagrange's equations with non-holonomic constraints:* Constrained optimisation; Lagrange multipliers.

**Assessment**

Exam 60% Class test 20% Reports 20%

**AMA2004 - Numerical Analysis (2**^{nd} semester)

*Pre-requisite:* AMA1020 and PMA1020

*Lecturer:* Dr M Gruening

**Introduction**

Numerical Analysis is concerned with devising methods for finding approximate, numerical solutions to mathematically expressed problems. The methods are analysed for their accuracy, efficiency and robustness. As a simple example, consider solving the equation *f*(*x*) = 0, where *f*(*x*) is a specified function. If *f*(*x*) is of even moderate complexity, we will not be able to solve this equation analytically. In Numerical Analysis we develop different procedures, or algorithms, to solve this problem. Finding the most suitable requires an appreciation of the methods. For example, some will guarantee convergence to a solution, but may require much effort, while other methods may converge quickly with less effort, but may also diverge, depending on the function and initial guess. Faced with such differing behaviour of the methods, what is the ‘best’ strategy to adopt? In AMA2004 we cover the basic introductory material of Numerical Analysis. We investigate the solution of equations, interpolation, function approximation, differentiation, integration and the solution of ordinary differential equations.

An important element of this module are the practical classes, in which algorithms developed during the lectures are implemented using MATLAB, which is an interactive and programmable software package. The practical work culminates in a numerical project, in which a MATLAB program is written to implement a particular algorithm, or to investigate the behaviour of a method. Although prior familiarity with computers is helpful, it is not essential.

A third element of the module is the oral presentation. You are required to give a 5-minute presentation to a small group of students and staff.

**Contents**

*Introduction and basic properties of errors:* Introduction; Review of basic calculus; Taylor's theorem and truncation error; Storage of non-integers; Round-off error; Machine accuracy; Absolute and relative errors; Richardson's extrapolation.
*Solution of equations in one variable:* Bisection method; False-position method; Secant method; Newton-Raphson method; Fixed point and one-point iteration; Aitken's δ^{2} process; Roots of polynomials.
*Solution of linear equations:* Gaussian elimination; Pivoting strategies; Calculating the inverse; LU decomposition; Norms; Condition number; Ill-conditioned linear equations; Iterative refinement; Iterative methods.
*Interpolation and polynomial approximation:* Why use polynomials? Lagrangian interpolation; Neville's algorithm; Other methods.
*Approximation theory:* Norms; Least-squares approximation; Linear least-squares; Orthogonal polynomials; Error term; Discrete least-squares; Generating orthogonal polynomials.
*Numerical quadrature:* Newton-Cotes formulae; Composite quadrature; Romberg integration; Adaptive quadrature.
*Numerical solution of ordinary differential equations:* Boundary-value problems; Finite-difference formulae for first and second derivatives; Initial-value problems; Errors; Taylor-series methods; Runge-Kutta methods.

**Assessment**

Exam 50% Project (long) 25% Project (short) 15% Presentation 10%

All assessment components are compulsory

**AMA2005 - Fluid Mechanics (2**^{nd} semester)

*Pre-requisite:* AMA1020 and PMA1020

*Lecturer:* Dr F Pausinger

**Introduction**

This course provides an elementary introduction to the mechanics of non-viscous and viscous fluids. The course uses the methods of vector field theory that have been introduced at level 1. It also provides a useful link to several level-3 courses, such as AMA3001 Electromagnetic Theory and AMA3003 Tensor Field Theory.

The course starts with a revision of the main aspects of vector field theory that are needed later in the course. The main body of the course is concerned with fluid mechanics. The fundamental equations are derived. The theory is then applied to situations in which the equations simplify to the extent that analytic solutions are possible. This course does not consider computational solutions. It is hoped that, in addition to some reasonably challenging mathematics, the course contains enough simple treatments of every-day phenomena that all level-2 students will find manageable, stimulating and rewarding.

**Contents**

I. Review of Vector Field Theory

Vector algebra: scalar product; vector product; triple products. Co-ordinates: Cartesian; general orthogonal curvilinear coordinates; spherical polar; cylindrical polar. Scalar Fields: gradient; time-dependent fields. Vector fields: flux and divergence; Gauss divergence theorem; line-integrals and curl; Stokes theorem. Miscellaneous topics: ∇ notation; ∇^{2}; identities; Green's Theorem.

II. Fluid Mechanics

*1) General:* Basic definitions and properties: materials; solids, plastics and fluids; density; pressure; equation of state; body forces; surface forces; viscosity. Velocity: pathlines; streamlines; boundary conditions. Continuity equation; incompressible fluids; streamtubes. Euler's Momentum Equation.

*2) Applications:* Hydrostatics; Sound waves.

*3) Vorticity and circulation:* Definitions; Vortex lines and tubes; Vorticity equation; Kelvin's Circulation Theorem.

*4) Bernoulli's equation:* Derivation and conditions; Simple examples; Class demonstrations; Open channel flows.

*5) Two-dimensional flow of incompressible Fluid:* Stream function: properties; flux; vorticity; solid boundaries. Some model flows. Steady flow of an inviscid liquid past a cylinder.

*6) Irrotational flow in two dimensions:* Review of complex variables. Complex potential: definition; some model flows; Shifts and rotations; examples. Image theorems for walls and circles. Conformal mapping: general theory; particular mappings; the Joukowski transformation; the cambered aerofoil.

*7) Irrotational flow in three dimensions:* Method of separation of variables. Spherical polar solution; Cylindrical polar solution; Cartesian solution. Gravity surface-waves.

*8) Viscous fluids:* The stress tensor; surface forces revisited; deformations and the strain tensor; Newtonian fluids. Navier-Stokes equation: general derivation; incompressible fluids; boundary conditions. Examples. Laminar and turbulent flow; Reynolds and Froude numbers; dimensional analysis.

**Assessment**

Exam 70% Class teat 10% Project 20%

**AMA2008 Introduction to Partial Differential Equations ****(1**^{st} semester)

*Pre-requisite:* AMA1020 and PMA1020

*Lecturer:* Dr A Brown

**Contents**

*functions of a single variable (Revision)*
*even and odd functions*
*limits*
*differentiation*
*integration*
*Taylor series expansion*
*functions of two variables*
*partial differentiation*
*total differentials,*
*Taylor series expansion for two variables and stationary points*
*solving partial differential equations*
*the method of separation of variables*
*the heat conduction equation*
*the wave equation*
*Fourier series*
*Fourier analysis*

**Assessment**

Class test 40% Python computational project (teams) 60%