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## PH40073: Mathematical physics

Credits: 6
Level: Masters UG & PG (FHEQ level 7)
Period: Semester 2
Assessment: EX 100%
Supplementary Assessment: Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this unit you must take PH10004 and take PH20029
Description: Aims:
The aim of this unit is to develop students' understanding of some fundamental aspects of Physics, where a mathematical treatment is essential fully to appreciate the subject. For the section on phase transitions, the aim is for students to gain a quantitative understanding of the principles that govern first and second order phase transitions. For the section on classical mechanics, the aim is for students to understand and apply the Lagrangian formulation of classical mechanics.

Learning Outcomes:
After taking the section on phase transitions the student should be able to:
* perform mean field calculations of phase transitions;
* define critical exponents and discuss scaling relations and universality classes;
* describe in detail the principles of real-space renormalisation;
After taking the section on classical mechanics the student should be able to:
* show proficiency in using the Lagrangian and Hamiltonian formulations to solve problems in classical mechanics;
* use symmetries to derive conservation laws;
* formulate and analyse equations of motion for systems of oscillators;
* analyse nonlinear field models using methods of classical mechanics.

Skills:
Numeracy T/F A, Problem Solving T/F A.

Content:
Phase transitions: Phenomenology, classification of phase transitions. Mean field theories; Weiss theory, Landau theory, Van der Waals theory. Statistical mechanics of phase transitions; examples based on the Ising model. Introduction to scaling and the renormalisation group.
Classical mechanics: Calculus of variations. Hamilton's principle, Lagrangian formulation of classical mechanics, examples. Symmetry and conservation laws. Linear and non-linear dynamics. Classical field theory. Non-linear wave equations.
Programme availability:

#### PH40073 is Compulsory on the following programmes:

Department of Physics
• USPH-AFM02 : MPhys Physics (Full-time) - Year 4
• USPH-AFM04 : MPhys Physics with Research Placement (Full-time) - Year 4
• USPH-AAM03 : MPhys Physics with Year Abroad (Full-time with Study Year Abroad) - Year 4
• USXX-AFM01 : MSci (hons) Mathematics and Physics (Full-time) - Year 3

#### PH40073 is Optional on the following programmes:

Department of Mathematical Sciences
• TSMA-AFM08 : MSc Modern Applications of Mathematics (Full-time) - Year 1
• TSMA-AFL02 : PG Dip Modern Applications of Mathematics (Full-time) - Year 1
Department of Physics
• USXX-AFB03 : BSc (hons) Mathematics and Physics (Full-time) - Year 3
• USXX-AKB04 : BSc (hons) Mathematics and Physics with Placement (Full-time with Thick Sandwich Placement) - Year 4
• USXX-AAB04 : BSc (hons) Mathematics and Physics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4

NB. Programmes and units are subject to change at any time, in accordance with normal University procedures.