Aims

To introduce the concept of spacetime, the theory of special relativity and some preliminary ideas from general relativity. To present basic ideas and techniques of variational calculus, including relevant applications.

Starting from situations such as GPS navigation, where the velocity of light plays an important role, we explore ideas on the fundamental nature of space and time which form the basis of the theory of relativity.

The calculus of variations answers questions like: What is the shortest route between two places on the Earth's surface? How do you maximise growth in an economy? It has wide applications to real-world problems. Most importantly, the ideas of variational calculus provide the basis for a reformulation of dynamics which underpins modern theoretical physics.

Outline Of Syllabus

Lorentz transformations: length contraction and time dilation; spacetime; 4-vectors and line elements.
Definitions of energy and momentum, E=mc^2.
Relativistic dynamics.
Introduction to General Relativity: curved space and geodesics.

Basics: finding extrema of functions; Lagrange multipliers.
Euler-Lagrange equations: theory and applications.
Variational problems in many dimensions and multiple variables.
Dynamics: kinetic and potential energy; Hamilton’s principle; generalised coordinates; dynamics of solid bodies; Hamiltonian dynamics.

Teaching Rationale And Relationship

Non-synchronous online materials are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on assessed work. Present-in-person and synchronous online sessions are used to help develop the students’ abilities at applying the theory to solving problems and to identify and resolve specific queries raised by students, and to allow students to receive individual feedback on marked work. Students who cannot attend a present-in-person session will be provided with an alternative activity allowing them to access the learning outcomes of that session. In addition, office hours/discussion board activity will provide an opportunity for more direct contact between individual students and the lecturer: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.
Alternatives will be offered to students unable to be present-in-person due to the prevailing C-19 circumstances.
Student’s should consult their individual timetable for up-to-date delivery information.

Assessment Rationale And Relationship

A substantial formal examination is appropriate for the assessment of the material in this module. The course assessments will 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.