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Geometric Methods for Physics of Magnetised Plasmas

Module MA5333

This Module is offered by Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Module version of SS 2018 (current)

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
SS 2018SS 2016

Basic Information

MA5333 is a semester module in English language at Master’s level which is offered in winter semester.

This Module is included in the following catalogues within the study programs in physics.

  • Catalogue of non-physics elective courses
Total workloadContact hoursCredits (ECTS)
150 h 45 h 5 CP

Content, Learning Outcome and Preconditions


These lectures have for the purpose to show how the geometrical methods such as Hamiltonian and Lagrangian formalisms can be used for systematically reduced dynamical description of complex multi-scale physical systems on the example of magnetised plasmas.
Perturbative Lie-Transform dynamical reduction method, derivation of Ampere, Poisson and kinetic equation from the variational principle as well as derivation of conservation laws are presented.
One of the targets consists in bringing together continuous and discrete versions of variational dynamical description. As an example of numerical implementation it will be shown how that discretised form of dynamics can be implementated to the Particle-In-Cell Monte-Carlo simulations.

Learning Outcome

Students acquire
- knowledge of perturbative geometrical methods for construction of dynamical reduction procedure in multi-scaled dynamical system.
- advanced variational calculus and Noether methods for conservation laws derivation: in continuous and Monte-Carlo finite element discretisation form.


Differential Equations, Numerical Methods

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

Learning and Teaching Methods

The module consists of 2 hours of lectures each week, supplemented by a 2 hours exercise class every second week. In the lectures, the relevant models and theoretical principles for their analysis are introduced, and illustrative examples are worked out in detail. In the exercise classes, the students apply the methods to some specific examples and implement them using the python language




- V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, ISBN 978-1-4757-2063-1
- Douglas Arnold, Richard Falk, and Ragnar Winther. "Finite element exterior calculus: from Hodge theory to numerical stability." Bulletin of the American mathematical society 47.2 (2010): 281-354.
- Jasper Kreeft, Artur Palha, and Marc Gerritsma. "Mimetic framework on curvilinear quadrilaterals of arbitrary order." arXiv preprint arXiv:1111.4304 (2011).

Module Exam

Description of exams and course work

The module examination is an oral exam (20 minutes). Students have to be able to state and derive basic mathematical properties of variational methods, in particular the derivation of equations of motion, invariants an action principle. They also should know how to apply the concept of mimetic finite differences and Finite Element Exterior Calculus to different models issued from plasma physics.

Exam Repetition

The exam may be repeated at the end of the semester.

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