Unit name | Methods of Theoretical Physics |
---|---|

Unit code | PHYS38014 |

Credit points | 10 |

Level of study | H/6 |

Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |

Unit director | Dr. Furqaan Yusaf |

Open unit status | Not open |

Pre-requisites |
Level I/5 Mathematical Physics PHYS23020. |

Co-requisites |
None |

School/department | School of Physics |

Faculty | Faculty of Science |

The course introduces the mathematical foundations of modern theoretical physics. These are primarily the theory of Analytical Mechanics in the Lagrangian and Hamiltonian formulations, together with the necessary mathematical tools of the calculus of variations and the theory of complex analytic functions. This course introduces these fundamental ideas and so provides the foundations for understanding more advanced theoretical physics topics studied at Level 7 and beyond.

Aims:

The calculus of variations is introduced, providing the mathematical concept of a functional, while also physically motivated from Fermat's principle. The Brachistochrone Problem is used as an example of constrained variation and Lagrange multipliers are introduced. This foundation then allows the introduction of Lagrangian mechanics based upon the principle of least action. Virtual work is defined and the Euler Lagrange equations derived. 1st integrals and conservation laws are discussed Conservation laws and Poisson brackets are mentioned. Hamiltonian mechanics is introduced by Legendre transfomation, along with the key notions of phase space, canonical momentum and of generalized coordinates. Finally complex function theory is developed, concentrating on the notion of analyticity, with examples of 2 dimensional Laplace equation, and applications to contour integration. Multi-valued complex functions, branch points and branch cuts are introduced briefly.

Students will be able to:

- obtain the Euler-Lagrange equations for variational problems, and to solve them in simple physical cases.

- write a Lagrangian for a dynamical system, and to derive the corresponding equations of motion.

- derive the cannonical momentum and Hamiltonian for systems defined by a Lagrangian, and to solve the corresponding Hamilton's equations of motion.

- determine whether a given function of a complex variable is analytic, and locate poles and perform integrals using Cauchy's theorem of residues.

Lectures and problems classes

Formative assessment is provided through problems classes

Final assessment 2 hour unseen examination (100%)

- M.L. Boas, Mathematical Methods in the Physical Sciences (Wiley)

- Jerry B. Marion, Stephen T. Thornton,. Classical dynamics of particles & systems, (Brooks/Cole).