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.

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

4H: Mathematical Physics (MATHS4107)

Assessment

90% Examination, 10% Coursework.

Reassessment

MSc students will have an opportunity to resit

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.

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

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