Description
Outline:
This module aims to provide students with understanding of some advanced mathematical methods, and further experience and skills in mathematical manipulation and problem solving. Topics include: Partial Differential Equations, Series Solution of Second-Order Ordinary Differential Equations, Legendre Functions, Fourier Analysis, Lagrangian and Hamiltonian Mechanics and Special Relativity.
Aims:
This module aims to
- Provide the remaining mathematical foundations for all the second- and third-year compulsory Physics and Astronomy courses.
- Prepare students for the second-term Mathematics option MATH0043 Mathematics for Physics and Astronomy.
- Give students practice in mathematical manipulation and problem solving at second-year level.
Intended Learning Outcomes:
Students should be able to:
- Solve a variety of partial differential equations using the method of separation of variables.
- Apply series solutions to solve differential equations.听
- Use Legendre polynomials in a variety of circumstances and an understanding of their properties.
- Apply the Euler-Lagrange equation to simple problems in minimisation and solving basic mechanics problems using the Lagrangian and Hamiltonian formalism.
- Derive and apply Fourier series and transforms to simple problems.
- Apply the theory of special relativity to the numerous problems in physics where it occurs.
Teaching and Learning Methodology:
This course is delivered via weekly lectures supplemented by a series of problem solving tutorials and additional discussion.
In addition to timetabled lecture and PST hours, it is expected that students engage in self-study in order to master the material. This can take the form, for example, of practicing example questions and further reading in textbooks and online.
Indicative Topics:
- Partial Differential Equations. Superposition principle for linear homogeneous partial differential equations; separation of variables in Cartesian coordinates; boundary conditions; one-dimensional wave equation; derivation of Laplace's equation in spherical polar coordinates; separation of variables in spherical polar coordinates; the Legendre differential equation; solutions of degree zero
- Series Solution of Second-Order Ordinary Differential Equations. Series solutions: harmonic oscillator as an example; ordinary and singular points; radius of convergence; Frobenius method; Fuchs鈥 theorem; applications to second-order differential equations.0
- Legendre Functions. Application of the Frobenius method to the Legendre equation; range of convergence, quantisation of the l index; generating function for Legendre polynomials; recurrence relations; orthogonality of Legendre functions; expansion in series of Legendre polynomials; solution of Laplace's equation for a conducting sphere; associated Legendre functions; spherical harmonics.
- Lagrangian and Hamiltonian Mechanics. The Lagrangian and Lagrange's equation; variation of action; the Euler-Lagrange equations; variational principles; from the Lagrangian to the Hamiltonian; derivation of Hamilton鈥檚 equations.
- Fourier Analysis. Fourier series; periodic functions; derivation of basic formulae; simple applications; differentiation and integration of Fourier series; Parseval's identity; complex Fourier series; Fourier transforms; derivation of basic formulae and simple applications; Dirac delta function; convolution theorem.
- Special Theory of Relativity. Implications of Galilean transformation for the speed of light Michelson-Morley experiment; Einstein鈥檚 postulates; derivation of the Lorentz transformation equations and the Lorentz transformation matrix; length contraction, time dilation, addition law of velocities, 鈥減aradoxes鈥; four-vectors and invariants; transformation of momentum and energy; invariant mass; conservation of four-momentum; Doppler effect for photons; threshold energy for pair production.听
Module deliveries for 2024/25 academic year
Last updated
This module description was last updated on 19th August 2024.
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