## MAS413 Analytical Dynamics and Classical Field Theory

 Both semesters, 2017/18 20 Credits Lecturer: Prof Carsten van de Bruck Timetable Reading List Aims 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)
No other modules have this module as a prerequisite.

## 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).

40 lectures, no tutorials

## Assessment

One formal 3 hour written examination.

Type Author(s) Title Library Blackwells Amazon
C Goldstein, Poole, Safko Classical Mechanics 531 (G) Blackwells Amazon
C Joel Franklin Advanced Mechanics and General Relativity Blackwells Amazon
C Morin Introduction to Classical Mechanics Blackwells Amazon
C Sean Carroll Spacetime and Geometry: An Introduction to General Relativity Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.

## Timetable (semester 1)

 Mon 11:00 - 11:50 lecture Hicks E39 Mon 12:00 - 12:50 lecture Hicks E39