PX2677.5 Hamiltonian Mechanics
Introductory description
This module introduces the Hamiltonian formulation of classical mechanics. This elegant theory has provided the natural framework for several important developments in theoretical physics including quantum mechanics. The module starts by covering the general "spirit" of the theory and then goes on to introduce the details.
The module uses a lot of examples. Many of these should be familiar from earlier studies of mechanics while others, which would be much harder to deal with using traditional techniques, can be dealt with quite easily using the language and methods of Hamiltonian mechanics.
Module aims
To revise the key elements of Newtonian mechanics and use this to motivate and then develop Lagrangian and Hamiltonian mechanics
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Introduction. Analogy with optics and constructive interference; principle of least action; examples of L: TV, mc2/γ

Euler Lagrange Equations. 1d trajectory, TV case, worked examples; T+V as a constant of the motion; multiple coordinates with examples

Generalised coordinates and canonical momenta. Polar coordinates; angular momentum; moment of inertia of rigid bodies; treatment of constraints; examples

Symmetry and Conservation Laws

Hamiltonian formulation. Hamilton's equations, phase space, examples

Normal Modes and Small Oscillations. Inertial and stiffness matrices, diatomic and triatomic molecules
Learning outcomes
By the end of the module, students should be able to:
 Explain the significance of the Lagrangian and Hamiltonian
 Derive and solve the EulerLagrange equations for simple models
 Find the canonical momenta in a mechanical system and construct the Hamiltonian function
 Derive and solve Hamilton's equations for simple systems
 Explain the role of (and relations between) constraints, conserved quantities and generalised coordinates
Indicative reading list
A good text going well beyond the module is H Goldstein, Classical Mechanics;
A helpful reference for the beginning of the module is: Feynmann, Leighton & Sands, The Feynmann Lectures on Physics, Vol 2, Chapter 19
View reading list on Talis Aspire
Interdisciplinary
The Hamiltonian and Lagrangian formulations of mechanics use ideas from mathematics  variational methods and symplectic structures. These mathematical methods can lead to valuable conceptual understanding as well as more elegant methods for solving (and approximately solving) problems in mechanics. This module is taken by many mathematics and physics students.
Subject specific skills
Knowledge of mathematics and physics. Skills in modelling, reasoning, thinking
Transferable skills
Analytical, communication, problemsolving, selfstudy
Study time
Type  Required 

Lectures  20 sessions of 1 hour (27%) 
Private study  55 hours (73%) 
Total  75 hours 
Private study description
Working through lecture notes, solving problems, wider reading, discussing with others taking the module, revising for the exam, practising on past exam papers
Costs
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
Assessment group B1
Weighting  Study time  

2 hour online examination (Summer)  100%  
Answer 2 questions

Feedback on assessment
Personal tutor, group feedback
Courses
This module is Option list A for:
 Year 2 of UPXAFG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
 Year 2 of UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 2 of UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
 Year 2 of UPXAF304 Undergraduate Physics (BSc MPhys)
 Year 2 of UPXAF300 Undergraduate Physics (BSc)
 Year 2 of UPXAF303 Undergraduate Physics (MPhys)
 Year 2 of UPXAF3N1 Undergraduate Physics and Business Studies
 Year 2 of UPXAF3F5 Undergraduate Physics with Astrophysics (BSc)
 Year 2 of UPXAF3FA Undergraduate Physics with Astrophysics (MPhys)
This module is Option list B for:
 Year 2 of UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)
 Year 2 of UMAAG100 Undergraduate Mathematics (BSc)
 Year 2 of UMAAG103 Undergraduate Mathematics (MMath)
 Year 2 of UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 2 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 2 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 2 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 2 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)
 Year 2 of UMAAG101 Undergraduate Mathematics with Intercalated Year