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MA40048: Analytical & geometrical theory of differential equations

[Page last updated: 27 September 2022]

Academic Year: 2022/23
Owning Department/School: Department of Mathematical Sciences
Credits: 6 [equivalent to 12 CATS credits]
Notional Study Hours: 120
Level: Masters UG & PG (FHEQ level 7)
Semester 2
Assessment Summary: EX 100%
Assessment Detail:
  • Examination (EX 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA20219 AND take MA20220
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.

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.

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.

Programme availability:

MA40048 is Optional on the following programmes:

Department of Mathematical Sciences


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