Aims

To introduce the mathematical methods required for the modelling and description of physical dynamic systems.

Module outline
In mathematics and physics, dynamics is the study of movement and change over time, in ways that can be described by mathematical equations or systems of equations. The aim is to explain and predict past and future patterns using basic principles of mathematics. Objects of interest might be tiny particles or huge stars, there might be a single object or very many.
Working from a mathematical point of view, we will formulate problems in terms of functions which can be differentiated or integrated, so essentially working with ordinary differential equations (ODEs). In modern mathematical usage, ‘dynamics’ describes the analysis of such ODEs. This will involve methods you’ve met (or are meeting) in other modules. We’ll focus on problems of idealised 'point particles' (simple bodies) and describe their motion when they are thrown or shot (ballistic), oscillating or orbiting something (circular and elliptical orbits).

Outline Of Syllabus

Particle dynamics: differentiation and integration of a vector-valued function; position, velocity and acceleration vectors in Cartesian and polar coordinates.
Newton's laws of motion and energetics: forces and linear momentum; angular momentum; kinetic and potential energies; motion under gravity; variable mass problems.
Spring oscillator and pendulum motion: small amplitude, simple harmonic motion; damped and forced oscillations; large amplitude motion and nonlinear oscillations.
Orbital motion: Newton's law of gravity; equations of orbital motion; Kepler's laws.
Multiple particles: two body system including reduced mass; introduction to N-body case; centre of mass.

Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

In the event of on-campus examinations not being possible, an on-line alternative assessment will be used for written examination 1.