Pre-requisite modules
Co-requisite modules
- None
Prohibited combinations
- None
Occurrence | Teaching cycle |
---|---|
A | Autumn Term 2022-23 |
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).
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.
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 |
None
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 |
Our policy on how you receive feedback for formative and summative purposes is contained in our Department Handbook.
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