# MA5Q3 Topics in Complexity Science

**Lecturer: **Professor Robert Mackay

**Term(s): 2**

**Status for Mathematics students:** List D (also for MathSys, MASt, Maths & Interdiscipinary Maths MSc and

MASDOC)

**Commitment: **10 two hour lectures and 10 one hour classes

**Assessment: **100% by essay (15-20 pages) due beginning of term 3

**Prerequisites: **

**Content: **In 2018/9 the chosen topic is Stellarator Mathematics.

A stellarator is a magnetic confinement device for plasma (ionised gas). It has some similarities to the better known tokamak but does not require its strong toroidal current, which is problematic to drive and causes bad instabilities. But it is not close to axisymmetric so its design requires much more sophisticated mathematics to confine the plasma.

The module will address:

Charged particle motion in magnetic fields from a Hamiltonian viewpoint

Adiabatic invariance of the magnetic moment and the resulting equations for guiding centre motion

Design of magnetic field to achieve integrability of the guiding centre motion (quasi-symmetry)

Vacuum fields

Magnetohydrodynamic (MHD) equilibrium

MHD equilibrium with mean flows and electrostatic fields

Interaction of two charged particles in a magnetic field

Measures of non-integrability (conditions for non-existence of invariant tori)

Guiding-centre billiards

Other topics to be added

**Aims:**

**Objectives:**

**Books:**

**Notes: **We will use differential forms, Lie derivatives etc where it makes things tidy and easier to see but will also attempt to give parallel statements in more traditional terminology (grad, div, curl, cross product). A good book for background on this in the MHD context is

Arnold VI, Khesin BA, Topological methods in hydrodynamics (Springer, 1998) [though note that in Remark 1.4 of Chapter II, the 1-form u is not defined (it is v♭), the stationary Euler equation should be L_v u = −d(p − 1/2 |v|^2) and α = p + 1/2 i_v u.]

Another which is more expository and for Hamiltonian mechanics is

Arnold VI, Mathematical methods of classical mechanics (Springer, 1978)

You can google to find more about anything you don't understand. That's how I learn these days.