# GEOMETRIC MECHANICS - 2025/6

Module code: MATM032

## Module Overview

This module develops the mathematics underpinning mechanical systems.  It builds on differential equations, symmetry and groups, geometry and classical dynamics.  It develops both fundamental theory of conservative (Hamiltonian and Lagrangian) systems with symmetry and gives detailed attention to examples such as rigid body motion (such as the dynamics of a spinning top), fluid mechanics (viewed as a dynamical system with many particles), robotics, magnetic field flow, and quantum mechanics.

### Module provider

Mathematics & Physics

BRIDGES Tom (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 69

Lecture Hours: 33

Guided Learning: 15

Captured Content: 33

Semester 2

None.

## Module content

Indicative content includes:

• 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.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-Semester Test (50 minutes) 20
Examination End-of Semester Examination (2 hours) 80

N/A

## 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 using the language of Geometric Mechanics.

• 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 in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1 to 2.

• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to Learning Outcomes 1 to 3.

Formative assessment
There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

Feedback
Students receive individual written feedback on the formative unassessed coursework and the in-semester test. The feedback is timed so that feedback from the first unassessed coursework assists students with preparation for the in-semester test. The feedback from both unassessed courseworks and the in-semester test assists students with preparation for the end-of-semester examination. This written feedback is complemented by verbal feedback given in lectures. Students also receive verbal feedback in office hours.

## Module aims

• The module aims to show how a wide range of mathematics is combined to develop a theory for mechanical systems. The mathematics (groups, differential equations, linear algebra, geometry of curves and surfaces) will be reviewed and then applied in sequence to the governing equations from mechanics.

## Learning outcomes

 Attributes Developed 001 Students will demonstrate understanding of mechanical systems on Lie groups, along with their symmetry properties. K 002 Students will interpret and apply variational principles in mechanics, and quote and apply the Euler-Poincare reduction theorem. KCT 003 Students will calculate momentum maps, and prove/disprove their conservation using symmetry arguments. KC 004 Students will delve into examples, to understand application of the theory. CP

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• A detailed introduction to Geometric Mechanics.

• Experience (through demonstration) of the methods used to interpret, understand and solve problems relating to physical systems with symmetry.

The learning and teaching methods include:

• Three one-hour lectures per week for eleven weeks, in which students will be encouraged to take lecture notes to facilitate their learning, and engagement with the module material. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.

• There are two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.

Lectures may be recorded. Lecture recordings are intended to give students the opportunity to review parts of the session that they might not have understood fully and should not be seen as an alternative to attendance at lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MATM032

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourcefulness and Resilience and Sustainability. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MATM032 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

Employability: Through the module, students cultivate advanced problem-solving skills applicable and valued across diverse industries, such as technology, engineering, and research.

Global and Cultural Capabilities: Student engagement in discussions during lectures naturally cultivates the sharing of the different cultures from which the students originate.

Resourcefulness and Resilience: MATM032 fosters resourcefulness and resilience by immersing students in intricate problem-solving scenarios. Dealing with problems involving rotations and energy conservation hones adaptability and perseverance.

Sustainability: The skills gained in addressing systems involving symmetry is vital in fast and efficient numerical integrators which in turn play a vital role in fields such as climate science and weather prediction, enabling innovative solutions for a more sustainable future.

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2025/6 academic year.