# LAGRANGIAN & HAMILTONIAN DYNAMICS - 2022/3

Module code: MAT3008

## Module Overview

This module introduces some fundamental concepts in analytical dynamics and illustrates their applications to relevant problems. The module covers the calculus of variations, Lagrangian and Hamiltonian formulations of dynamics, Poisson brackets, canonical transformations and Hamilton-Jacobi equations.

The module leads, among other things, to a deeper understanding of the role of symmetries  and conservation laws. This course lays the foundations for the Year 3 module  Quantum Mechanics  (MAT3039).

### Module provider

Mathematics & Physics

TRONCI Cesare (Maths & Phys)

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

Independent Learning Hours: 84

Lecture Hours: 33

Captured Content: 33

Semester 2

## Prerequisites / Co-requisites

Classical Dynamics  MAT1036

## Module content

Indicative content includes:

• Generalised coordinates, calculus of variations, Lagrangian and Euler-Lagrange equations.

• Planetary motion.

• Legendre transform, Hamiltonian and canonical equations of motion.

• Hamilton-Jacobi method.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 min) 20
Examination Exam (2 hrs) 80

N/A

## Assessment Strategy

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

·         Ability to choose  appropriate  generalised coordinates to describe a system

·         Ability to identify simple symmetries and conservation laws.

·         Subject knowledge through the recall of key definitions, theorems and their proofs.

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

Thus, the summative assessment for this module consists of:

·         One two hour examination  at the end of the semester ; worth 80% module mark.

·         One in-semester test; worth 20% module mark.

Formative assessment and feedback

Students receive written feedback via a number of marked coursework assignments over an 11 week period.

## 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 frequently encountered problems in analytical dynamics

## Learning outcomes

 Attributes Developed 1 Choose an appropriate set of generalised coordinates to describe a system CT 2 Apply the Lagrangian and Hamiltonian formulations of dynamics to determine the evolution of a system KCT 3 Use the Hamilton-Jacobi method to solve separable systems KC

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:

• Introduction to generalised coordinates, functionals, Lagrangian and Hamiltonian, principle of least action.

• Experience (through demonstration) of the methods used to determine the dynamics of a system by means of the Lagrangian and Hamiltonian formulation of dynamics.

The learning and teaching methods include:

• 3 x 1 hour lectures per week x 11 weeks. Blackboard lectures.  Q + A opportunities for students.

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

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics and Physics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 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 2022/3 academic year.