MAS413 Analytical Dynamics and Classical Field Theory

Note: Information for future academic years is provisional. Timetable information and teaching staff are especially likely to change, but other details may also be altered, some courses may not run at all, and other courses may be added.

 Both semesters, 2022/23 20 Credits Lecturer: Dr Sam Dolan Timetable Reading List Aims Outcomes Assessment

Newton formulated his famous laws of mechanics in the late 17th century. Only later it became obvious through the work of mathematicians like Lagrange, Hamilton and Jacobi that underlying Newton's work are wonderful mathematical structures. In the first semester, the work of Lagrange, Hamilton and Jacobi will be discussed and how it has later affected the formulation of field theory. We will also discuss Noether's theorem, which relates symmetries of a system to the conservation law of certain quantities (such as energy and momentum). In the second semester, Einstein's theory of gravity, General Relativity, will be introduced. The physical principles of General Relativity and mathematical concepts from differential geometry presented. Some consequences of this theory, such as black holes and the expanding universe, will be discussed.

Prerequisites: MAS112 (Vectors and Mechanics); MAS314 (Introduction to Relativity)

The following modules have this module as a prerequisite:

 MAS415 Topics in Mathematical Physics

Outline syllabus

• Formulations of Lagrange and Hamilton;
• Calculus of variations and canonical transformations;
• Interpretation of relativistic systems in the formalisms of Lagrange and Hamilton;
• Fields;
• Noether's theorem;
• Physical Principles of General Relativity;
• Mathematical concepts from Differential Geometry;
• The Schwarzschild Spacetime;
• The Robertson-Walker-Friedmann Spacetime

Aims

• to introduce students to the formulations of mechanics by Lagrange and Hamilton;
• to show how mechanical problems can be formulated in much simpler ways;
• introduce new mathematical methods: calculus of variations and canonical transformations;
• show how relativistic systems can be described in the formalisms of Lagrange and Hamilton;
• introduce the concept of a field and how the ideas of Lagrange and Hamilton can be extended to describe fields such as the gravitational field and the electromagnetic field;
• show how Noether's theorem relates the conservation of quantities like energy and momentum to symmetries in Nature;
• to introduce students to the theory of General Relavity;
• to show how methods and concepts of differential geometry are used in General Relativity;
• to show some consequences of Einstein's theory (black holes and the expanding universe).

Learning outcomes

At the end of this module, students should be able to:
• Use Lagrange’s equations to study mechanical systems
• Apply the calculus of variations to systems with fixed end-points
• Apply Hamilton’s equations to mechanical systems
• Apply canonical transformations to Hamiltonian systems, including Hamilton-Jacobi theory
• Derive the field equations for scalar and electromagnetic fields from the Lagrangian
• Apply Noether’s theorem to systems of particles or classical fields
• Manipulate tensor quantities in both special and general relativity
• Understand the physical concepts of general relativity
• Compute mathematical quantities from differential geometry, such as curvature tensors
• Find geodesics for some simple space-times
• Understand the basic properties of the Schwarzschild space-time
• Understand the basic properties of the Friedmann-Robertson-Walker space-time

40 lectures, no tutorials

Assessment

One formal 3 hour written examination.