# LAGRANGIAN & HAMILTONIAN DYNAMICS - 2024/5

Module code: MAT3008

## Module Overview

This module introduces fundamental concepts in analytical dynamics and illustrates their application to real-world problems. The module covers the calculus of variations, Lagrangian and Hamiltonian formulations of dynamics, Poisson brackets, canonical transformations and symplectic manifolds. The module also leads to a deeper understanding of the role of symmetries and conservation laws in dynamical systems.

This module builds on material from MAT1036 Classical Dynamics and lays the foundations for MAT3039 Quantum Mechanics.

### Module provider

Mathematics & Physics

TRONCI Cesare (Maths & Phys)

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

Independent Learning Hours: 69

Lecture Hours: 33

Guided Learning: 15

Captured Content: 33

Semester 1

None

## Module content

Indicative content includes:

• Lagrangian Dynamics: Generalised coordinates. Calculus of variations and the principle of stationary action. The Euler-Lagrange equations of motion. Constrained motion. Integrals of motion. Motion in a central potential. Planetary motion.

• Hamiltonian Dynamics: Legendre transform. Phase-space variational principle and Hamilton’s equations of motion. Poisson brackets. Integrals of motion. Canonical transformations. Symplectic manifolds. Infinitesimal transformations. Noether’s Theorem.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 min) 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 subject knowledge, and recall of key definitions and results in Lagrangian and Hamiltonian dynamics.

• The ability to identify and use the appropriate methods to solve real-world problems in dynamical systems using the Lagrangian and Hamiltonian formalisms.

Thus, the summative assessment for this module consists of:

• One in-semester test corresponding to Learning Outcomes 1, 3 and 4.

• A synoptic examination corresponding to all Learning Outcomes 1 to 4.

Formative assessment

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

Feedback

Students will receive individual written feedback on both the formative unassessed courseworks and the in-semester test. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback in office hours.

## Module aims

• Introduce students to the Lagrangian and Hamiltonian formulations of dynamics.
• Enable students to understand the role of symmetries and conservation laws.
• Illustrate the application of various techniques for solving real-world problems in analytical dynamics.

## Learning outcomes

 Attributes Developed 001 Students will be able to choose an appropriate set of generalised coordinates to describe a dynamical system, and apply the Lagrangian formalism to determine the evolution of the system. KC 002 Students will be able to apply the Hamiltonian formalism to determine the evolution of a dynamical system. KC 003 Students will understand the role of symmetries and conservation laws in dynamical systems. KC 004 Students will be able to analyse and solve unseen problems relating to real-world examples of dynamical systems. KCT

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:

• Introduce students to the principle of least action, and Lagrangian and Hamiltonian dynamics.

• Provide students with experience of methods used to analyse real-world dynamical systems and solve for the dynamics using the Lagrangian and Hamiltonian formalisms.

The learning and teaching methods include:

• Three one-hour lectures for eleven weeks, with module notes provided to complement the lectures. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.

• Two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive feedback on these courseworks as guidance on their progress and understanding.

• Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending 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: MAT3008

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness 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 MAT3008 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: The module MAT3008 equips students with skills which significantly enhance their employability. The mathematical proficiency gained will hone their critical thinking and problem-solving abilities. Students will learn to interpret and evaluate physical real-world problems in dynamical systems, model these mathematically using ordinary differential equations, and hence deduce and interpret solutions. Mathematical modeling is a highly sought after skill in many professions.

Global and Cultural Capabilities: Students enrolled in MAT3008 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in lectures, which naturally facilitates the sharing of different cultures.

Resourcefulness and Resilience: MAT3008 is a module which demands the ability to analyse physical real-world problems in dynamical systems, formulate and solve these problems mathematically using the Lagrangian and Hamiltonian formalisms, and interpret the results. Students will gain skills in analysing unseen problems and lateral thinking, and will complete assessments which challenge them and build resilience.

Sustainability: The Lagrangian and Hamiltonian formalisms can be used to model dynamical systems relevant to sustainable practices. For instance, Lagrangian dynamics can be used to model the dispersal of pollutants and greenhouse gases in the atmosphere, which has an impact on climate change, and the dispersal of contaminants in bodies of water, which has an impact on water quality management. One or more case studies will be included in the module relating to uses of Lagrangian and Hamiltonian dynamics to model systems relating to real-world sustainable practices.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Physics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics with Astronomy BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics with Nuclear Astrophysics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics with Quantum Computing BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics with Nuclear Astrophysics MPhys 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics with Astronomy MPhys 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics MPhys 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics with Quantum Computing MPhys 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 1 Optional A weighted aggregate mark of 40% is required to pass the module

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 2024/5 academic year.