2023/24 Taught Postgraduate Module Catalogue

MATH5356M Advanced Hamiltonian Systems

20 creditsClass Size: 30

Module manager: Dr Vincent Caudrelier
Email: v.caudrelier@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2023/24

Pre-requisite qualifications

MATH2650 Calculus of variations or equivalent.

Pre-requisites

MATH2650

Calculus of Variations

This module is mutually exclusive with

MATH3355

Hamiltonian 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 and geometric 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, and emphasized the fundamental role of symmetries.

Objectives

1. Derive Lagrangian and Hamiltonian functions and write Hamilton's equations for simple mechanical systems.
2. Use symmetries and Noether’s theorem to derive constants of motion.
3. Calculate Poisson brackets and first integrals.
4. Use generating functions for canonical transformations and solve simple cases of the Hamilton-Jacobi equation.
5. Use Lie’s theorem on commuting vector fields and Liouville's Theorem on complete integrability.
6. Identify degenerate Lagrangian and calculate Dirac brackets. Use symmetries to reduce the phase space.

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.

Syllabus

1. Review of Lagrangian dynamics. Hamilton's principle. Noether’s theorem. Legendre's transformation and the canonical equations of motion.
2. Introduction to Hamiltonian dynamics. Simple geometric properties.
Poisson brackets. First integrals and symmetries. Noether’s theorem in Hamiltonian form.
3. Canonical transformations and generating functions. The Hamilton-Jacobi equation. Separation of variables. Complete integrability.
4. Hamiltonian formalism for non standard Lagrangians (higher order or degenerate). Dirac brackets. Introduction to moment maps and reduction of phase space.

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	44	1.00	44.00
Private study hours			156.00
Total Contact hours			44.00
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

Exams

Exam type	Exam duration	% of formal assessment
Open Book exam	3 hr 00 mins	100.00
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: 12/05/2023

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