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Advanced Theoretical Techniques - PHY00068H

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  • Department: Physics
  • Module co-ordinator: Dr. Matt Hodgson
  • Credit value: 10 credits
  • Credit level: H
  • Academic year of delivery: 2022-23
    • See module specification for other years: 2021-22

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Module will run

Occurrence Teaching period
A Autumn Term 2022-23

Module aims

The Advanced Theoretical Techniques part of this module introduces mathematical ideas and tools which are essential to modern theoretical physics. Variational principles and Lagrangian mechanics have applications in many areas of physics, and are an essential to modern classical and fundamental quantum field theories. Mathematical ideas about reference frames and coordinate systems underpin special and general relativity, and are naturally expressed using tensors. This course will teach you how to use these mathematical tools to analyse challenging theoretical physics problems from a range of areas.

Section I: Integral Transforms & Variational Methods (9 lectures)

In this part, we look at Fourier transforms (first encountered in Maths III) in more detail, and develop a related concept, the Laplace transform. We will then see some applications of these transforms, including their use in solving differential equations. We then go on to look at a more advanced form of calculus, including functional differentiation and the calculus of variations. We will conclude by looking at the ubiquity of these ideas in many different areas of physics, often expressed as some form of “variational principle”, including applications in classical mechanics, optics, field theories and quantum mechanics.

Section II: Tensors (9 lectures)

A cornerstone of modern physics is the notion that no observer is more privileged than any other in terms of being able to deduce the laws of nature. This concept is manifested in the covariant nature of our mathematical description of the universe, and the properties of the algebraic quantities (which we call tensors) which represent physical quantities. In this second section of the module we will see why some matrices can represent physical quantities and others can’t, and demonstrate how various physical laws exhibit Galilean or Lorentzian invariance. We also look at non-orthogonal coordinate systems, which are relevant to general relativity (among other applications).

Module learning outcomes

  • calculate the Fourier transform (and inverse transform) of a given function
  • calculate the convolution of two functions
  • calculate the Laplace transform of a given function
  • use Fourier and Laplace transforms to solve partial differential equations
  • use the calculus of variations to solve problems in mechanics and find extremal solutions in other fields such as geometry (shortest path, geodesics) and statistics (maximum entropy).
  • use Lagrange multipliers in the calculus of variations to solve problems with constraints
  • understand the correspondence between symmetries and conservation laws in physical theories
  • identify symmetries in a functional, and find the corresponding conservation law
  • understand and describe how the calculus of variations extends to field theories
  • efficiently manipulate multidimensional mathematical objects
  • apply anisotropic physics models and interpret their physical consequences (through a variety of examples)
  • identify and apply spatial transformations, including three-dimensional rotations
  • distinguish between tensorial and non-tensorial objects
  • demonstrate invariance of various quantities in both Euclidean and Minkowski spaces
  • manipulate tensorial quantities (both covariant and contravariant) in non-orthogonal coordinate systems

Module content

Please note, students taking this module should either have taken the prerequisite module listed above (Mathematics II - PHY00030I) or the appropriate equivalent.

Syllabus

  • Concept of basis set

  • Integral Transforms: Motivation

  • Fourier transform & inverse transform

    • Dirac delta

    • Derivatives and solving differential equations

    • Convolution theorem

  • Laplace transform

    • Existence requirements

    • Techniques for finding inverse transforms

    • Derivatives and solving differential equations

    • Convolution theorem

  • Applications of Fourier and Laplace transforms

  • Variational Methods: Motivation

    • Functional differentiation

    • Calculus of variations

    • Extremal values & the Euler-Lagrange equation

    • Conservation Laws and Symmetry

    • Boundary conditions and constraints

    • Extension to vector equations and fields

  • Applications of Variational Methods

    • Classical mechanics, optics, classical field theories and quantum mechanics

  • Tensors: Motivation

    • No special observers

    • Distinction between a physical quantity and its component representation

  • Notation

    • Einstein summation notation

    • Kronecker delta

    • Levi-Civita alternator

    • Examples of vector identities

  • Examples of anisotropy, which may include the fluid stress tensor and/or the electromagnetic dielectric tensor, and their physical consequences

  • General coordinate transformations

  • Rotations and translations in two and three dimensions

  • What makes a tensor: concepts of invariance.

  • Euclidean tensors & their invariance

  • Minkowski space and Lorentzian invariance of four vectors, including the proper time and electromagnetic potential

  • Non-orthogonal coordinate systems:

    • motivation (i.e. why make things complicated?)

    • general definition of a coordinate

    • covariant and contravariant basis vectors

    • the metric tensor and its use in the evaluation of tensor quantities.

Assessment

Task Length % of module mark
Essay/coursework
Advanced Theoretical Techniques Assignment 1		 N/A 40
Essay/coursework
Advanced Theoretical Techniques Assignment 2		 N/A 60

Special assessment rules

None

Reassessment

Task Length % of module mark
Essay/coursework
Advanced Theoretical Techniques Assignment 1		 N/A 40
Essay/coursework
Advanced Theoretical Techniques Assignment 2		 N/A 60

Module feedback

Our policy on how you receive feedback for formative and summative purposes is contained in our Department Handbook.

Indicative reading

Derek F. Lawden, Introduction to Tensor Calculus, Relativity and Cosmology (Dover 2002).

William D. D'Haeseleer, Jim Callen et al., Flux Coordinates and Magnetic Field Structure: A Guide to a Fundamental Tool of Plasma Theory (Springer-Verlag 1991).

Richard Fitzpatrick: Classical Electromagnetism lecture notes:

http://farside.ph.utexas.edu/teaching/em/lectures/node106.html


The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.