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MA40048: Analytical & geometrical theory of differential equations

[Page last updated: 15 October 2020]

Follow this link for further information on academic years Academic Year: 2020/1
Further information on owning departmentsOwning Department/School: Department of Mathematical Sciences
Further information on credits Credits: 6      [equivalent to 12 CATS credits]
Further information on notional study hours Notional Study Hours: 120
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:
Like-for-like reassessment (where allowed by programme regulations)
Further information on requisites Requisites: Before taking this module you must take MA20219 AND take MA20220
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.

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

Content:
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 Mathematical Sciences

Notes:

  • This unit catalogue is applicable for the 2020/21 academic year only. Students continuing their studies into 2021/22 and beyond should not assume that this unit will be available in future years in the format displayed here for 2020/21.
  • Programmes and units are subject to change 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.
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