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2019/20 Taught Postgraduate Module Catalogue

MATH5356M Advanced Hamiltonian Systems

20 creditsClass Size: 30

Module manager: Dr Vincent Caudrelier

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2019/20


MATH2650Calculus of Variations

This module is mutually exclusive with

MATH3355Hamiltonian Systems

This module is approved as an Elective

Module summary

The Hamiltonian formulation of dynamics is the mathematically most beautiful form of mechanics and (in fact) the stepping stone to quantum mechanics. Hamiltonian systems are conservative dynamical systems with a very interesting algebraic structure in the guise of the Poisson bracket. Hamilton's equations are invariant under a very wide class of transformation (the canonical transformations), and this leads to a number of powerful solution techniques, developed in the nineteenth century. The subject received a boost in the late twentieth century, with the development of integrable systems, which gave many new examples and techniques. This module can be thought of as a companion to the module, 'Discrete Systems and Integrability'.


1. Derive Lagrangian and Hamiltonian functions and write Hamilton's equations for simple mechanical systems.
2. Use phase portraits and Poincare maps to analyse simple Hamiltonian systems.
3. Calculate Poisson brackets and first integrals.
4. Use generating functions for canonical transformations and solve simple cases of the Hamilton-Jacobi equation.
5. To use Liouville's Theorem on complete integrability.
6. To use hidden symmetries to describe Lissajous figures in coupled oscillators and Kepler orbits in gravitation.

Learning outcomes
The aim of this module is to develop the theory of Hamiltonian systems, Poisson brackets and canonical transformations. After discussing some general algebraic and geometric properties, emphasis will be on complete integrability, developing a number of techniques for solving Hamilton's equations.


1. Review of 'pre-Hamiltonian' dynamics: Newtonian and Lagrangian dynamics. Hamilton's principle.
Legendre's transformation and the canonical equations of motion.
2. Introduction to Hamiltonian dynamics. Simple geometric properties. Phase portraits and Poincare maps.
Poisson brackets. First integrals and symmetries.
3. Canonical transformations and generating functions. The Hamilton-Jacobi equation. Separation of variables. Complete integrability.
4. Near integrable systems. Area preserving maps.
5. Hidden symmetries and super-integrability. Coupled Oscillators and Lissajous figures. The Kepler problem and the Runge-Lenz Vector.

Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Delivery typeNumberLength hoursStudent hours
Private study hours156.00
Total Contact hours44.00
Total hours (100hr per 10 credits)200.00

Private study

Studying and revising of course material.
Completing of assignments and assessments.

Opportunities for Formative Feedback

Regular problems sheets.

Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Exam typeExam duration% of formal assessment
Standard exam (closed essays, MCQs etc)3 hr 00 mins100.00
Total percentage (Assessment Exams)100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 30/09/2019


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