MA40048: Analytical & geometrical theory of
differential equations
Level: Masters
Semester: 1
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.
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.