## MATH3355 Hamiltonian Systems

### 15 creditsClass Size: 60

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

This module is not approved as a discovery module

### 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.

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.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 33 1.00 33.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Opportunities for Formative Feedback

Regular examples sheets

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Open Book exam 2 hr 30 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