# Mathematics / Applied Mathematics MSc

# 5E: Mathematical Physics MATHS5073

**Academic Session:**2019-20**School:**School of Mathematics and Statistics**Credits:**10**Level:**Level 5 (SCQF level 11)**Typically Offered:**Semester 2**Available to Visiting Students:**No**Available to Erasmus Students:**No

#### Short Description

This course introduces students to a geometric view of classical mechanics which draws together many problems in the constrained and unconstrained motions of systems of particles and continua as well as providing the underlying framework for modern quantum mechanics.

#### Timetable

17 x 1 hr lectures and 6 x 1 hr tutorials in a semester

#### Requirements of Entry

Mandatory Entry Requirements

3H: Mathematical Methods (MATHS4075)

3H: Mechanics of Rigid and Deformable Bodies (MATHS4078)

Recommended Entry Requirements

#### Excluded Courses

4H: Mathematical Physics (MATHS4107)

#### Assessment

Assessment

90% Examination, 10% Coursework.

Reassessment

MSc students will have an opportunity to resit

**Main Assessment In:** April/May

#### Course Aims

The main aim of this course is the study of the dynamical properties of systems consequent upon specific choices of Hamiltonian function. It will develop the theory from the Lagrangian approach familiar from previous courses, introducing the structures of symplectic geometry associated with phase spaces of particulate and rigid body motions. Hamiltonian symmetries will play a crucial role in the explicit description of the associated spaces of orbits. It will conclude with a discussion of some basic notions of Quantum Mechanics.

For the level 5 course students will be expected to obtain a deeper understanding of the material including proofs of some of the fundamental theorems of Hamiltonian mechanics and more advanced topics, such as Noether's Theorem, Liouville's Theorem, Hamilton-Jacobi theory, stability and chaos.

#### Intended Learning Outcomes of Course

By the end of this course students will be able to:

(a) Move seamlessly between the Lagrangian and Hamiltonian formulations of a given mechanical or optical system;

(b) Given an appropriate geometric situation, describe the associated Hamiltonian;

(c) Identify and employ integrals of motion arising from symmetries to reduce the order of a system;

(d) Implement symplectic transformations between equivalent Hamiltonian systems;

(e) Discuss separable coordinates in appropriate situations;

(f) Present treatments of very simple quantised systems.

(g) Be able to derive and apply some of the central theorems of Hamiltonian mechanics

(h) Discuss the stability of Hamiltonian systems and identify chaotic behaviour in certain simple systems

#### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.