果冻影院

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