Module MATH2741: Methods of Mathematical Physics II

Department: Mathematical Sciences

MATH2741: Methods of Mathematical Physics II

Prerequisites

• One of: Calculus I (Maths Hons) (MATH1081) OR Calculus I (MATH1061)
• AND
• one of: Linear Algebra I (Maths Hons) (MATH1091) OR Linear Algebra I (MATH1071)
• AND
• one of: Dynamics and Relativity I (MATH1XX7)* OR Foundations of Physics I (PHYS1122)

Corequisites

• Mathematical Methods II (MATH2XX1)

Excluded Combination of Modules

• Theoretical Physics 2 (PHYS2631)

Aims

• To deepen understanding of differential equations and vector calculus in the context of solving concrete problems.
• To develop principles of applied mathematics and mathematical physics which are relevant to further physical applied mathematics modules.

Content

• Systems of ODEs: phase plane analysis, linearisation.
• Action principles: Variational principles, Lagrange’s equations, Hamilton’s equations, Poisson brackets, example of a charged particle in an electromagnetic field.
• Symmetries and conservation laws: Noether’s theorem, including Hamiltonian and Poisson perspective.
• Fields and waves: Partial differential equations, action principles for continuous systems, wave equation, boundary conditions, Noether’s theorem and stress-energy tensor.
• Static electromagnetic fields: Gauss’ law, Ampere’s law and integral forms, scalar and vector potential, Poisson equation, Green’s functions, multipole expansion.
• Electrodynamics: Maxwell’s equations, Faraday’s law and integral form, electromagnetic waves, Maxwell Lagrangian and relativistic formulation.

Learning Outcomes

• Be able to analyse nonlinear systems of differential equations using linearization and phase plane analysis, as well as to understand important special classes of such systems such as Hamiltonian dynamical systems;
• Be able to understand and employ techniques from the Calculus of Variations, knowing both the mathematical ideas of deriving Euler-Lagrange equations as well as the physical interpretation of Lagrangians;
• Be able to use the formalisms of Lagrangian and Hamiltonian mechanics for particles and fields, as well as to understand their application in the development of Maxwell’s theory of electromagnetism;
• Be able to solve partial differential equations related to these field theories using a variety of methods, and to understand the physical content of their solutions.

• In addition, students will have the ability to undertake and defend the use of alternative mathematical skills in the following areas with minimal guidance: Modelling.

• Students will have basic mathematical skills in the following areas: problem solving, modelling, computation.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
• Tutorials provide active problem-solving engagement and immediate feedback to the learning process.
• Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
• Summative assignments test achievement of learning outcomes and provide feedback to students about their mastery of the topics.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Fortnightly assignments.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University