# GEOMETRIC MECHANICS - 2017/8

Module code: MATM032

Module provider

Mathematics

BRIDGES TJ Prof (Maths)

Number of Credits

15

ECTS Credits

7.5

Framework

FHEQ Level 7

JACs code

G130

Module cap (Maximum number of students)

N/A

Module Availability

Semester 2

Independent Study Hours: 117

Lecture Hours: 33

Assessment pattern

Assessment type Unit of assessment Weighting
Examination EXAMINATION 75
School-timetabled exam/test CLASS TEST 1 (50 MINS) 10
School-timetabled exam/test CLASS TEST 2 (50 MINS) 15

Alternative Assessment

N/A

Prerequisites / Co-requisites

MAT3008/MAT3031 Lagrangian and Hamiltonian Dynamics

Module overview

This module applies Lagrangian and Hamiltonian dynamics to physical systems with symmetry.

Module aims

The module aims to extend students' knowledge of mechanics by considering systems with symmetry and their conservation laws.

Learning outcomes

Attributes Developed
1 Demonstrate understanding of mechanical systems on Lie groups, along with their symmetry properties. K
2 Interpret and apply variational principles in mechanics, and quote and apply the Euler-Poincare reduction theorem. KCT
3 Calculate momentum maps, and prove/disprove their conservation using symmetry arguments. KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

Module content

Topics covered will include some or all of:

Elements of multi-linear algebra, differential geometry and Lie group actions.

Euler-Poincaré variational principles (with and without symmetry breaking)

Legendre transform and symplectic spaces

Conservation laws: momentum maps and Noether's theorem

Lie-Poisson structures (with and without symmetry breaking)

Applications: rigid bodies, heavy tops, quantum dynamics, magnetic fields, etc.

Infinite dimensions: diffeomorphism groups and applications to fluids/plasmas

Methods of Teaching / Learning

Teaching is by lectures, 3 hours per week for 11 weeks. Extensive notes are provided.

Learning takes place through lectures, exercises and class tests.

Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

Understanding of fundamental concepts and ability to develop and apply them to a new context.

Subject knowledge through recall of key definitions, formulae and derivations.

Analytical ability through the solution of unseen problems in the test and examination.

Thus, the summative assessment for this module consists of:

One two hour examination at the end of the semester, worth 75% of the overall module mark

two fifty minute class tests, the first worth 10% and the second worth 15%

Formative assessment and feedback

Students receive written feedback via the marked class tests.  The solutions to the class tests are also reviewed in the lecture.  Un-assessed courseworks are also assigned to the students, and a sketch of solutions to these are provided.  Verbal feedback is provided during lectures and office hours.