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MA40048: Analytical & geometrical theory of differential equations

Follow this link for further information on academic years Academic Year: 2014/5
Further information on owning departmentsOwning Department/School: Department of Mathematical Sciences
Further information on credits Credits: 6
Further information on unit levels Level: Masters UG & PG (FHEQ level 7)
Further information on teaching periods Period: Semester 2
Further information on unit assessment Assessment Summary: EX 100%
Further information on unit assessment Assessment Detail:
  • Examination (EX 100%)
Further information on supplementary assessment Supplementary Assessment: MA40048 Mandatory extra work (where allowed by programme regulations)
Further information on requisites Requisites: Before taking this unit you must take MA20216 and take MA20217 and take MA20218 and take MA20219 and take MA20220 and take MA30041. Students may also find it useful to take MA40062 before taking this unit.
Further information on descriptions Description: Aims:
To give a unified presention of systems of ordinary differential equations that have a Hamiltonian or Lagrangian structure. Geometrical and analytical insights will be used to prove qualitative properties of solutions. These ideas have generated many developments in modern pure mathematics, such as symplectic geometry and ergodic theory, besides being applicable to the equations of classical mechanics, and motivating much of modern physics.

Learning Outcomes:
Students should be able to state and prove general theorems for Lagrangian and Hamiltonian systems. Based on these theoretical results and key motivating examples they should be able to identify general qualitative properties of solutions of these systems.

Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (solutions to exercise sheets, problem classes)

Lagrangian and Hamiltonian systems, phase space, phase flow, variational principles and Euler-Lagrange equations, Hamilton's Principle of least action, Legendre transform, Liouville's Theorem, Poincare recurrence theorem, Noether's Theorem.
Topics from: the direct method of the Calculus of Variations, constrained variational problems, Hamilton-Jacobi equation, canonical transformations.
Further information on programme availabilityProgramme availability:

MA40048 is Optional on the following programmes:

Department of Computer Science
  • USCM-AFM14 : MComp(Hons) Computer Science and Mathematics (Year 4)
  • USCM-AAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 5)
  • USCM-AKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 5)
Department of Mathematical Sciences
* This unit catalogue is applicable for the 2014/15 academic year only. Students continuing their studies into 2015/16 and beyond should not assume that this unit will be available in future years in the format displayed here for 2014/15.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.